Integral Representation: Choquet Theory for Linear Operators on Function Spaces 9783111315478, 9783111314501

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Walter Roth Integral Representation

De Gruyter Expositions in Mathematics

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Edited by Lev Birbrair, Fortaleza, Brazil Victor P. Maslov, Moscow, Russia Walter D. Neumann, New York City, New York, USA Markus J. Pflaum, Boulder, Colorado, USA Dierk Schleicher, Marseille, France Katrin Wendland, Freiburg, Germany

Volume 74

Walter Roth

Integral Representation �

Choquet Theory for Linear Operators on Function Spaces

Mathematics Subject Classification 2020 46A03, 46A20, 46A32, 46A55, 46E40, 46G10 Author Dr. Walter Roth House 705, Purok Diamond 2 Tacip Mirafuentes, Tagum City 1800 Davao del Norte Philippines [email protected]

ISBN 978-3-11-131450-1 e-ISBN (PDF) 978-3-11-131547-8 e-ISBN (EPUB) 978-3-11-131558-4 ISSN 0938-6572 Library of Congress Control Number: 2023940890 Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available on the Internet at http://dnb.dnb.de. © 2023 Walter de Gruyter GmbH, Berlin/Boston Typesetting: VTeX UAB, Lithuania Printing and binding: CPI books GmbH, Leck www.degruyter.com

Introduction In a groundbreaking paper in 1957 [15], G. Choquet established an integral representation theorem for positive linear functionals on continuous affine functions on a compact convex metric space. Any such functional, so it states, can be represented by a regular measure that is supported by the set of extreme points. This work generated considerable interest and was later generalized to nonmetric compact spaces by E. Bishop and K. de Leeuw in [11] with substantial contributions from P. A. Meyer [17], G. Mokobozki [37] and others. The result was regarded to be profound, and the technology employed to be difficult. It broadly works as follows. Given a subspace G of the Banach space 𝒞 (X) of all continuous functions on a compact set X, an order relation is established on the dual of 𝒞 (X), that is, the space of all regular Borel measures on X. This order is defined using a suitable subcone of 𝒞 (X), and the choice of this subcone guarantees that the set of functionals that dominate a given one is weak* compact in the dual. This assures the existence of maximal elements, which deliver integral representations for positive linear functionals on the subspace G with particular properties. The support of any such maximal representation measure is localized on the Choquet boundary, a subset of X characterized by G. In the case of a compact convex set X and the space of all continuous affine functions for G, the Choquet boundary stands for the set of all extreme points of X. The details and a comprehensive presentation of the theory can, for example, be found in the excellent books by E. M. Alfsen [1] and by R. R. Phelps [41]. There have been several advances to generalize these by now classical concepts (see, e. g., [28] and [59]). The approach to integral representation in this text leans on ideas in earlier publications by the author [47] and a generalization of the techniques therein by Batty [9]. These address Choquet theory for complex- and Banach space-valued functions on a compact Hausdorff space, respectively. A more recent paper [51] contains a classification of spaces of vector-valued functions on a locally compact space and establishes an integral representation for continuous linear functionals. Our take on Choquet theory can be perceived as a generally nonconstructive optimization procedure for linear operators on spaces of vector-valued functions. Given such an operator, an optimized version is found, which preserves certain properties of the original operator and optimizes others. The details of this process depend on a highly flexible set of parameters. We deal with locally compact rather than compact domains X, spaces of vector-valued rather than real- or complex-valued functions and linear operators into a second vector space rather than linear functionals. The transfer of the classical techniques to our situation is far from trivial, and often not possible in a straightforward way. Since our operators are defined on function spaces, as in the classical version of Choquet theory, their optimized counterparts can be partially characterized by the localization of their representation measures. In order to achieve this objective, we shall introduce a suitable integration theory for vector-valued functions with respect to operatorvalued measures. Vector integration was introduced in the early twentieth century, and comprehensive presentations of the subject can, for example, be found in the works of https://doi.org/10.1515/9783111315478-201

VI � Introduction N. Dunford and J. Schwartz [26] and [25], J. Diestel and J. J. Uhl [21], N. Dinculeanu [23] or W. H. Graves [29]. In a paper in 1955, R. G. Bartle [5] formulated the basic setup for the integration of vector-valued functions with respect to vector-valued measures. It requires the utilization of a continuous bilinear form on the concerned vector spaces. Equivalently, it was observed that one might use vector-valued functions and measures whose values are continuous linear operators from the range of these functions into a second vector space. This is the formulation that best suits our intentions, but available models do not provide sufficient generality for our purpose. The first part of this text is therefore dedicated to the development of a compatible and thoroughly broad approach to integration theory. There are several substantial differences to existing concepts. We use a σ-ring of subsets, which does not necessarily contain the whole space, rather than a σ-field for the domain of our measures. This is crucial since in the vector- or operator-valued case, infinity-type elements are not available. Integrals are first defined over sets in this σ-ring, and then extended via limits to an associated σ-field. We consider functions whose values are elements or, more generally, convex subsets of a vector space E and, given a second vector space F, measures whose values are linear operators from a cone of convex subsets of E into the cone of sublinear functionals on the dual of F. These sublinear functionals can at instances be interpreted to be elements of F, its completion, its second dual or indeed convex subsets of either of these spaces. The same selections are available for the values of integrals with respect to such measures of functions that are vector-valued or convex set-valued. For our approach, we employ some elementary aspects of the theory of locally convex cones, a generalization of locally convex topological vector spaces, which was introduced by the author together with K. Keimel in [33] and [48]. Some of its basic concepts will be presented in Section 1.4. Integrals of set-valued functions have been considered before, notably in an influential paper by R. J. Aumann [3], but they arise naturally in our approach. We also offer an understanding of the modulus or semivariation of a measure as its canonical extension to the cone of convex subsets of E. Our main result in this part establishes an integral representation for continuous linear operators on spaces of vector-valued functions. The properties of the representation measures are closely related to the properties of the respective operators. This representation is then extensively used in the second part of this text, which deals with our expansion of Choquet theory. This manuscript is organized in the following way. Chapter 1 contains a variety of preparatory subjects. Most importantly, we introduce function space neighborhoods, define and provide examples for function spaces, that is, spaces of continuous vectorvalued functions endowed with particular topologies. We outline aspects of the theory of locally convex cones in preparation for the development of measure and integration theory in Chapter 2. There we define operator-valued measures, measurability and integrability of vector- and set-valued functions. The main result is an integral representation theorem for linear operators on function spaces. In Chapter 3, we develop a

Introduction

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Choquet-type representation theory for linear operators on function spaces utilizing the results of the earlier parts. Reading this text requires familiarity with the general theory of locally convex topological vector spaces, and for the second part, with classical Choquet theory. We shall recall some of the basic facts in Section 1.1 and in the sequel frequently refer to several of the more advanced results, mostly in the well-known books by H. H. Schäfer [56], N. Dunford and J. Schwartz [26], W. Rudin [54], A. Wilansky [62] and A. P. and W. Robertson [45] for the former, and by E. M. Alfsen [1] and R. R. Phelps [41] for the latter.

Contents Introduction � V 1 1.1

1.2

1.3

1.4

2 2.1

Function spaces � 1 Prerequisites and notation � 1 Locally convex topological vector spaces � 1 Linear operators � 2 The dual space � 2 Polar topologies � 3 σ-rings � 4 Vector-valued functions � 5 Extended number systems � 5 Semicontinuous functions � 6 Convex sets � 6 Spaces of vector-valued functions that vanish at infinity � 6 Neighborhood functions � 7 Function space neighborhoods � 7 Function space neighborhood systems � 8 Functions that vanish at infinity � 8 Weakly lower continuous neighborhood functions � 8 The function spaces 𝒞V (X, E) � 10 Examples � 12 Function subspaces � 19 Compactness in spaces of linear operators � 19 Bounded operators � 20 Compact and weakly compact operators � 23 Examples � 28 Vector space extensions � 29 Locally convex cones � 30 The dual cone � 31 Sub and superlinear functionals � 32 Convex sets and sublinear functionals � 33 The locally convex cone 𝒫E � 36 The locally convex cone 𝒬F � 37 Elements of 𝒬F representing convex sets � 42 Examples � 42 The cone ℱ (X, Conv(E)) � 45 Integration � 47 Operator-valued measures � 48 Definition of an operator-valued measure � 48

X � Contents

2.2

2.3

2.4

2.5

2.6

L(E, F ˚‚ )-valued measures � 48 Modulus of a measure � 50 Bounded measures � 51 Variation of a measure � 53 Measure of bounded variation � 56 Examples � 56 Compact and weakly compact measures � 60 Restriction of a measure � 60 Sets of measure zero � 60 Properties that hold almost everywhere � 61 Point evaluation measures � 62 Regularity of measures � 62 Composition measures � 67 Measurable functions � 68 Set-valued step functions � 68 Measurable vector-valued functions � 70 Measurable set-valued functions � 73 Examples � 83 Composition of measurable functions � 85 Integrals of set-valued functions � 86 Integrals of step functions � 87 Integrals of measurable functions � 88 Integrals of set-valued functions � 101 Integrals over sets in R � 102 Integrals over sets in A � 103 Integrals of vector-valued functions � 111 Examples � 115 Integrals with respect to composition measures � 117 The convergence theorems � 123 The relative topologies � 129 Sequences of set-valued functions � 130 The convergence theorem � 133 Examples � 137 Measures as linear operators � 139 Properties of operators defined by an integral � 142 Properties of measures representing an operator � 146 Examples � 148 Integral representation � 149 The representation theorem � 149 Positive operators � 156 Lattice homomorphisms � 157 Algebra homomorphisms � 158

Contents �

XI

The spectral theorem � 160 Compositions of operators with real- or complex-valued functions � 161 Point evaluation operators � 163 Examples and Remarks � 164 3 3.1

3.2

3.3

Choquet theory � 168 C(X)-convex sets and functionals � 168 𝒞 (X)-convex sets � 172 Ω-extremal point evaluations � 175 Elementary operators � 185 Examples � 187 A Choquet ordering for linear operators on function spaces � 189 Choquet cone � 189 Choquet ordering � 190 Upper and lower envelopes � 191 Minimal elements � 196 The Choquet boundary � 197 C-superharmonic sets � 204 Special cases and examples � 214 The case that X is compact and that E = F = ℝ � 214 Sample settings in the general case � 215 The case that F = ℝ or F = ℂ � 223 The case that E = ℝ or E = ℂ � 224 Examples � 227 The case that both E and F are either ℝ or ℂ � 229 Examples � 238 The case that both E and F are ordered topological vector spaces � 240 The case that both E and F are topological vector lattices � 242

List of Symbols � 243 Bibliography � 245 Further Reading � 247 Index � 249

1 Function spaces The first section of this chapter compiles some general prerequisites and frequently used terminology and notation. We do not provide proofs for the cited facts and results, but refer to standard textbooks instead. In Section 1.2, we consider vector-valued functions and introduce our concept of function space neighborhoods, which will be used throughout this text. For our purpose, we adapt a similar notion in [51] and [48] from locally convex cones to locally convex topological vector spaces. This leads to the definition of function spaces, that is, suitable spaces of continuous vector-valued functions on a locally compact space that are endowed with particular topologies. In Section 1.3, we review aspects of continuous linear operators defined on locally convex topological vector spaces, different operator space topologies and notions of compactness in such spaces. Section 1.4 briefly introduces locally convex cones in preparation for the development of operator-valued measures in Chapter 2.

1.1 Prerequisites and notation Throughout this text, we shall use the following basic facts and notation. We broadly refer to standard textbooks on topology such as [24, 81] and [63], on topological vector spaces such as [45, 56], and [62] and on functional analysis such as [26, 53, 54, 87] and [61], for details, proofs and further explanations. Locally convex topological vector spaces A locally convex topological vector space (E, 𝒱 ) is a real or complex vector space E endowed with a basis 𝒱 of balanced convex neighborhoods of the origin. We assume that the neighborhood system 𝒱 is closed for multiplication by positive scalars and downward directed with respect to set inclusion. We do not require that E is Hausdorff, that is, that ⋂V ∈𝒱 = {0}. A subset A of E is convex if αa + βb ∈ A whenever a, b ∈ A and α, β ⩾ 0 such that α + β = 1, and balanced if αa ∈ A whenever a ∈ A and |α| ⩽ 1. A subcone of E is a nonempty convex subset that is closed for multiplication by nonnegative reals. The addition and multiplication by scalars of subsets of E are defined as A + B = {a + b | a ∈ A, b ∈ B} and

αA = {αa | a ∈ A, α ⩾ 0}

for A, B ⊂ E and elements α in the scalar field of E. Sums and scalar multiples of convex or of balanced sets are again convex or balanced. For a subset of E, there are a smallest convex set, a smallest balanced set, a smallest balanced convex set and a smallest subcone of E containing it. These are called its convex hull, balanced hull, balanced convex hull and conic hull, respectively. The convex and the balanced hull of a subset A of E are given by https://doi.org/10.1515/9783111315478-001

2 � 1 Function spaces n n 󵄨󵄨 conv(A) = {∑ αi ai 󵄨󵄨󵄨 ai ∈ A, αi ⩾ 0, ∑ αi = 1} 󵄨 i=1

i=1

and bal(A) = ⋃|α|⩽1 αA. Whereas the convex hull of a balanced set is balanced, easy examples show that the balanced hull of a convex set is not necessarily convex. The balanced convex hull bconv(A) of a set A ⊂ E is therefore the convex hull of bal(A). The conic hull cone(A) of a set A ⊂ E is the convex hull of ⋃α⩾0 α(A ∪ {0}). ˝ The interior A of a subset A of E is the largest open subset of A. The closure A of A is the smallest closed subset of E containing A, and is given by A = ⋂ (A + V ). V ∈𝒱

The interior and the closure of a convex or a balanced set are again convex or balanced. The closure of a subcone is again a subcone of E. If A ⊂ E is a convex neighborhood of 0 ∈ E, that is, if V ⊂ A for some V ∈ 𝒱 , then its closure is given by A = ⋂γ>1 γA. For subsets A, B of E, we say that A absorbs B if there is α ⩾ 0 such that B ⊂ βA for all |β| ⩾ α. A set A ⊂ E is absorbing if it absorbs every finite subset of E. All neighborhoods V ∈ 𝒱 are absorbing. A set A ⊂ E is bounded if it is absorbed by all neighborhoods in 𝒱 , that is, if for every V ∈ 𝒱 there is λ ⩾ 0 such that A ⊂ λV .

Linear operators If both (E, 𝒱 ) and (F, 𝒲 ) are locally convex topological vector spaces over the same scalar field ℝ or ℂ, a linear operator from E into F is a mapping T : E → F such that T(a + b) = T(a) + T(b)

and T(αa) = αT(a)

for all a, b ∈ E and α ∈ ℝ or α ∈ ℂ. Such an operator is continuous if for every neighborhood W ∈ 𝒲 there is V ∈ 𝒱 such that T(V ) ⊂ W . The linear operators from E into F form a vector space over the same scalar field containing the continuous linear operators as a subspace.

The dual space The algebraic dual E f of E is the space of all linear operators from E into its scalar field, that is, the space of all continuous real- or complex-valued linear functionals on E. The topological dual E ˚ is the subspace of E f consisting of all continuous linear func˚ denote the corresponding spaces of tional on E. In the complex case, both Eℝf and Eℝ all real-valued real-linear, and all continuous real-valued real-linear functionals on E, that is, the real parts of the functionals in E f and in E ˚ , respectively. The polar A˝

1.1 Prerequisites and notation

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of a subset A of E is usually taken in E ˚ and consists of all linear functionals μ ∈ E ˚ such that Re μ(a) ⩽ 1 for all a ∈ A. The weak topology σ(E, E ˚ ) on E is generated by the neighborhoods V{μ1 ,...,μn } = {a ∈ E | Re μi (a) ⩽ 1 for all i = 1, . . . , n} corresponding to finitely elements μ1 , . . . , μn ∈ F ˚ . The weak topology is coarser that the given one, but creates the same dual space, the same closure of convex sets and the same notion of boundedness for subsets of E. It is Hausdorff if and only if E is Hausdorff. The weak* topology on E ˚ is the topology σ(E ˚ , E). It is generated by the neighborhoods U{a1 ,...,an } = {μ ∈ E ˚ | Re μ(ai ) ⩽ 1 for all i = 1, . . . , n} corresponding to finitely elements a1 , . . . , an ∈ E and is always Hausdorff. The dual of E ˚ in the weak topology is the quotient space E/E0 , where E0 = ⋂V ∈𝒱 V . In case that E is Hausdorff, we have E0 = {0}, hence E/E0 = E. In our settings, we will assume that the vector spaces involved are either all real or all complex. Correspondingly, all linear operators are real- or complex-linear, and all linear functionals are ℝ- or ℂ-valued.

Polar topologies A polar topology is induced by a dual pair (E, F) of vector spaces together with a realor complex-valued bilinear form ⟨ , ⟩ on E × F. The bilinear form is required to be nondegenerate, that is, ⟨a, c⟩ = 0

for a ∈ E and all c ∈ F

implies that a = 0, and likewise, ⟨a, c⟩ = 0

for c ∈ F and all a ∈ E

implies that c = 0. Every element of F defines a linear functional on E, and in this way F corresponds to a subspace of E f . The polar in F of a subset A of E is the convex set A˝ = {c ∈ F | Re⟨a, c⟩ ⩽ 1 for all a ∈ A}. The polar in E of a subset of F is similarly defined. The polar of a set and the polar of its convex hull coincide. A subset A of E is weakly bounded if sup{Re⟨a, c⟩ | a ∈ A} < +∞ for all c ∈ F. A polar topology on E or on F is a locally convex topology on one of the spaces, whose neighborhood system is rendered by the polars of a family S of weakly bounded subsets of the opposite space with the following properties: (i) αA ∈ S whenever A ∈ S and α > 0, (ii) for A, B ∈ S, there is C ∈ S such that A ∪ B ⊂ C, and (iii) the union

4 � 1 Function spaces of all sets in S spans the whole space. The coarsest polar topology on E is the weak topology σ(E, F) generated by the family S of all finite subsets of F, the finest is the strong topology β(E, F) generated by the family S of all weakly bounded subsets of F. A polar topology is consistent with the duality if it generates the opposite space as its topological dual. The weak topology is consistent, and the finest consistent polar topology on E is the Mackey topology τ(E, F) generated by the family S of all weakly compact convex subsets of F (see Section IV.3 in [56]). Every Banach space (more generally, every barreled space) E carries the strong topology β(E, E ˚ ). A Hausdorff locally convex topological vector space E is called semireflexive if the strong topology β(E ˚ , E) on E ˚ is consistent with the duality, that is, if the dual of E ˚ in the strong topology coincides with E. It is reflexive if it is semireflexive and carries the strong topology β(E, E ˚ ). Closures of convex subsets of E or F and the concept of boundedness coincide in all topologies consistent with the duality (E, F). The second polar (or bipolar) A˝˝ in E of a subset A of E is the polar of A˝ in E and coincides with the closed convex hull of A ∪ {0} (Theorem IV.1.5 in [56]). Every Hausdorff locally convex topology on a vector space E is a polar topology of the dual pair (E, E ˚ ), endowed with the nondegenerate bilinear form (a, μ) 󳨃→ μ(a). In general, since the elements of E ˚ vanish on the subspace E0 = ⋂V ∈𝒱 V and separate elements of E whose difference is not in E0 , the transfer of the bilinear form to E/E0 ×E ˚ is nondegenerate, and the preceding results apply to the dual pair (E/E0 , E ˚ ). The projection of E onto E/E0 is continuous, open and closed. Further details and clarifications can, for example, be found in Chapter IV of [56] or in Chapter 13 of [61]. Dual pairs with degenerate bilinear forms are considered in [60]. σ-rings Let X be a set. For the sake of generality, we shall formulate our take on integration theory without the imposition of a topological structure on X. Measures suitable for our purposes can generally not be defined on a σ-field containing the whole space X, since this would require such a measure to take infinite values, which are not available in the vector- or operator-valued case. We shall therefore use a σ-ring in X, that is, a nonempty family R of subsets of X, which contains finite unions, set differences and countable intersections of its elements. These properties imply that H ∈ R, and if the sets Ai ∈ R for i ∈ ℕ all are subsets of a set A ∈ R, that their union ⋃i∈ℕ Ai is also contained in R. For the latter, one easily verifies that ⋃i∈ℕ Ai = A \ ⋂i∈ℕ (A \ Ai ). Any σ-field (or σ-algebra) is of course a σ-ring in this sense, and a σ-ring is a σ-field if and only if it contains X. We may, however, associate with R in a canonical way the σ-field A = {B ⊂ X | A ∩ B ∈ R for all A ∈ R}. We shall also refer to the topological case, that is, the case that X is a locally compact Hausdorff space and that R consists of all relatively compact Borel subsets of X. Then

1.1 Prerequisites and notation

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the σ-field A contains all Borel subsets of X and exactly those if X is countably compact. By O we shall denote the subfamily of all open sets in R. For every compact subset K of X, there exists O ∈ O such that K ⊂ O (Theorem 2.7 in [55]). For a subset ˝ Y of a topological space, the sets Y and Y denote its topological closure, and interior, ˝ respectively, whereas 𝜕Y = Y \ Y stands for its topological boundary. A subset of a topological space is relatively compact if its closure in this space is compact.

Vector-valued functions Let ℱ (X, E) be the space of all E-valued functions on X, endowed with the pointwise algebraic operations. In case that E = ℝ, we write ℱ (X) for short. A function f ∈ ℱ (X, E) is supported by a subset Y of X if f (x) = 0 for all x ∈ X \ Y . For a scalarvalued function φ and f ∈ ℱ (X, E), the function φ ⋅ f ∈ ℱ (X, E) is the mapping x 󳨃→ φ(x)f (x) : X → E. If f is the constant function x 󳨃→ a for some a ∈ E, we write φ ⋅ a for the function x 󳨃→ φ(x)a. The characteristic function χY ∈ ℱ (X) of a subset Y of X takes the value 1 on Y and 0 else. Following the usual convention, a real-valued function φ ∈ ℱ (X) is said to be nonnegative if φ(x) ⩾ 0 and positive if φ(x) > 0 for all x ∈ X. In the topological case, that is, if X is a locally compact Hausdorff space, we denote by 𝒞 (X, E) the subspace of all continuous functions in ℱ (X, E). The smallest closed stet in X that supports a function f ∈ ℱ (X, E) is called the support of f and referred to by supp(f ). The subspace of all functions in 𝒞 (X, E) with compact support is denoted by 𝒞𝒦 (X, E). In case that E = ℝ, we write 𝒞 (X) and 𝒞𝒦 (X) for short. If φ ∈ 𝒞 (X) and f ∈ 𝒞 (X, E), then φ ⋅ f ∈ 𝒞 (X, E). Given a compact subset K of X and open sets O1 . . . , On ⊂ X whose union contains K, a partition of the unity (see Proposition 9.4.16 in [53] or Theorem 2.13 in [55]) yields a collection of nonnegative functions φ1 , . . . , φn ∈ 𝒞𝒦 (X) such that supp(φi ) ⊂ Oi and ∑ni=1 φi (x) = 1 for all x ∈ K. Extended number systems The set ℝ = ℝ ∪ {+∞} denotes the extended real number system with the usual order and algebraic operations. In particular, we designate α + ∞ = +∞ for all α ∈ ℝ, α ⋅ (+∞) = +∞ for all 0 < α ⩽ +∞ and 0 ⋅ (+∞) = 0. The multiplication of +∞ with negative numbers is not defined. We shall similarly use ℝ = ℝ ∪ {−∞} and ℝ = ℝ ∪ {+∞, −∞}. In ℝ, the sum of +∞ and −∞ is however not defined. ℝ+ denotes the subcone of all nonnegative numbers in ℝ, that is, ℝ+ = [0, +∞].

6 � 1 Function spaces Semicontinuous functions An ℝ-valued function φ on a topological space X is lower semicontinuous if for every α ∈ ℝ the set {x ∈ X | φ(x) > α} is open in X, or equivalently, if for every x ∈ X and α < φ(x) there is a neighborhood U of x such that φ(y) > α for all y ∈ U. Sums, positive multiples and pointwise suprema of families of lower semicontinuous functions are again lower semicontinuous. An ℝ-valued function is upper semicontinuous if its negative is lower semicontinuous. A real-valued function is continuous if and only if it is both lower and upper semicontinuous.

Convex sets Let Conv(E) be the collection of all nonempty convex subsets of E. If endowed with the addition of sets and the multiplication by nonnegative scalars, the usual associative and distributive properties hold. The singleton set {0} ∈ Conv(E) serves as the neutral element of the addition. Hence, while not being a subcone of some vector space, Conv(E) forms a cone in the sense of Section 1.4 below, where we shall also introduce suitable topologies for cones. Conv(E) is canonically ordered by set inclusion, that is, we write A⩽B

if

A⊂B

for A, B ∈ Conv(E). For a nonnegative real-valued function φ on X and A ∈ Conv(E), we denote φ ⋅ A for the function x 󳨃→ φ(x)A : X → Conv(E). If A ⩾ {0}, that is, if 0 ∈ A, we allow φ to take the value +∞, where (+∞) A is meant to be the conic hull of A, that is, (+∞) A = ⋃ρ⩾0 ρA. Thus, if A is absorbing, then (+∞) A = E. Sums, multiples by nonnegative scalars and the order relation for Conv(E)-valued functions are defined pointwise on X. There is also the multiplication by all elements of the scalar field of E in Conv(E), but it does no longer satisfy the distributive law. We have (α + β)A ⊂ αA + βA instead. In particular, (−1)A is not necessarily an additive inverse of A. We write −A for (−1)A and A − B for A + (−B). Moreover, αA ⩽ αB holds whenever A ⩽ B for A, B ∈ Conv(E) and α in ℝ or ℂ.

1.2 Spaces of vector-valued functions that vanish at infinity We shall use the previously introduced basic facts and notation. Let (E, 𝒱 ) be a locally convex topological vector space. The extended neighborhood system 𝒱c of E is the subcone of Conv(E) generated by 𝒱 ∪ {E}. That is,

1.2 Spaces of vector-valued functions that vanish at infinity

n

� 7

󵄨 󵄨

󵄨 𝒱c = {∑ Vi 󵄨󵄨󵄨 Vi ∈ 𝒱 } ⋃{E, {0}}. i=1

Neighborhood functions Let X be a set. A function n : X → 𝒱c is called a neighborhood function. Clearly, sums and positive multiples of neighborhood functions are again neighborhood functions. Function space neighborhoods A function space neighborhood v is defined by a convex family 𝒩v of neighborhood functions provided that for every A ∈ R there is V ∈ 𝒱 and n ∈ 𝒩v such that V ⊂ n(x) for all x ∈ A. The neighborhood v then determines a balanced convex set [v] of functions in ℱ (X, E) by [v] = {f ∈ ℱ (X, E) | there is n ∈ 𝒩v such that f (x) ∈ n(x) for all x ∈ X}. We shall write f ⩽ v if f ∈ [v] for functions f ∈ ℱ (X, E). The reference to an order relation in this notation will be justified later on when we shall extend the use of function space neighborhoods to Conv(E)-valued functions. Likewise, will be this less than straightforward definition altogether. Due to the absence of negatives of set-valued functions, function space neighborhoods can no longer be identified with convex subsets of functions in the more general set-valued case (see Section 1.4 below). We observe the following basic properties. 1.2.1 Lemma. Let v be a function space neighborhood. (a) If f ⩽ v, then φ ⋅ f ⩽ v for all real- or complex-valued functions φ such that |φ(x)| ⩽ 1 for all x ∈ X. In particular, χA ⋅ f ⩽ v for all A ∈ R. (b) If 𝒩v is upward directed, then φ ⋅ f + (1 − φ) ⋅ g ⩽ v whenever f , g ⩽ v and φ is a real-valued function such that 0 ⩽ φ(x) ⩽ 1 for all x ∈ X. Proof. Part (a) is obvious from the definition of a function space neighborhood, since all values of neighborhood functions are convex and balanced subsets of E. For (b), if 𝒩v is upward directed, then for f , g ⩽ v there is n ∈ 𝒩v such that both f (x), g(x) ∈ n(x) for all x ∈ X. Sums and positive multiples of function space neighborhoods are defined via the generating sets of neighborhood functions. That is, we set 𝒩v+u = {n + m | n ∈ 𝒩v , m ∈ 𝒩u }

and 𝒩αv = {αn | n ∈ 𝒩v }

for function space neighborhoods v and u and α > 0. The associative, commutative and distributive rules for these operations transfer from the corresponding ones for set-valued functions. We shall also establish an order relation among function space

8 � 1 Function spaces neighborhoods setting v ⩽ u if for every n ∈ 𝒩v there is m ∈ 𝒩u such that n ⩽ m. Recall that the latter means n(x) ⊂ m(x) for all x ∈ X. The intersection v ∩ u of v and u is generated by the neighborhood functions n ∩ m for all n ∈ 𝒩v and m ∈ 𝒩u , whereby (n ∩ m)(x) = n(x) ∩ m(x) for all x ∈ X. Function space neighborhood systems A family V of function space neighborhoods is called a function space neighborhood system if it is closed for multiplication by positive scalars and downward directed. That is, for v, u ∈ V there is w ∈ V such that both w ⩽ v and w ⩽ u. The collection V˚ of all function space neighborhoods satisfies this condition. Indeed, if u and v are function space neighborhoods, then for every A ∈ R there are n ∈ 𝒩u , m ∈ 𝒩v and VA ∈ 𝒱 such that both VA ⊂ n(x) and VA ⊂ m(x) for all x ∈ A. Then the convex hull 𝒩w of the collection of the neighborhood functions χA ⋅ VA , for all A ∈ R, generates a function space neighborhood w such that both w ⩽ u and w ⩽ v. If both V and U are function space neighborhood systems, we shall say that V is finer than U, or equivalently, that U is coarser than V if for every u ∈ U there is v ∈ V such that v ⩽ u. V and U are equivalent if V is both finer and coarser than U. Functions that vanish at infinity A function f ∈ ℱ (X, E) vanishes at infinity relative to a function space neighborhood system V if for every v ∈ V there is A ∈ R such that χ(X\A) ⋅ f ⩽ v. Clearly, sums and multiples of functions that vanish at infinity share the same property. For the remainder of this section, we shall consider the topological case, that is, the case that X is a locally compact Hausdorff space and that R consists of all relatively compact Borel subsets of X. For continuous E-valued functions on X, we observe the following as an immediate consequence of our buildup. 1.2.2 Lemma. Let v be a function space neighborhood. For every A ∈ R and f ∈

𝒞 (X, E), there is λ ⩾ 0 such that χA ⋅ f ⩽ λv.

Proof. For A ∈ R and v ∈ V, there is V ∈ 𝒱 and n ∈ 𝒩v such that V ⊂ n(x) for all x ∈ A. If f is continuous, then f (A) is relatively compact, hence bounded in E. Thus there is λ ⩾ 0 such that f (A) ⊂ λV , that is, χA ⋅ f (x) ∈ λn(x) for all x ∈ X. Weakly lower continuous neighborhood functions A neighborhood function n is said to be weakly lower continuous if for every x ∈ X, every a ∈ n(x) and every V ∈ 𝒱 there is a neighborhood U of x in X such that a ∈ n(y) + V for all y ∈ U. The suitability of this definition will become apparent in

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the later stages of this text. Clearly, sums or multiples by positive reals of weakly lower continuous neighborhood functions are again weakly lower continuous. Similarly, a function space neighborhood v is said to be weakly lower continuous if all of its defining neighborhood functions n ∈ 𝒩v are weakly lower continuous. A function space neighborhood system V is called weakly lower continuous if it consists of weakly lower continuous function space neighborhoods. 1.2.3 Lemma. If φ is a lower semicontinuous ℝ+ -valued function and if V ∈ 𝒱c , then n = φ ⋅ V is a weakly lower continuous neighborhood function. Proof. Let V ∈ 𝒱c and φ be as stated and let n = φ ⋅ V . Let x ∈ X, let a ∈ n(x) and let W ∈ 𝒱 . There is λ > 0 such that a ∈ λW . If either φ(x) = 0 or V = {0}, then a = 0 and the requirement for weak lower continuity is trivially satisfied. If V ≠ {0} and φ(x) = +∞, then φ(x)V = E, and we find a neighborhood U of x such that φ(y) ⩾ λ, hence a ∈ φ(y)V = n(y) for all y ∈ U. If 0 < φ(x) < +∞, then there is a neighborhood U of x such that (λ/(1 + λ))φ(x) ⩽ φ(y) for all y ∈ U. The latter yields (λ/(1 + λ))a ∈ n(y), and a=

λ 1 λ a+ a ∈ n(y) + W ⊂ n(y) + W . 1+λ 1+λ 1+λ

The requirement for the weak lower continuity of n is therefore met. In particular, for every open subset O of X and every V ∈ 𝒱c the neighborhood function χO ⋅ V is weakly lower continuous. So is the neighborhood function χB ⋅ V + χX\B ⋅ E for every closed subset B of X. For the following, recall that O denotes the collection of all open sets in R. 1.2.4 Lemma. Let v be a weakly lower continuous function space neighborhood. (a) If χA ⋅ f ⩽ v for A ∈ R and f ∈ 𝒞 (X, E), then for every ε > 0 there is an open set O ∈ O such that A ⊂ O and χO ⋅ f ⩽ (1 + ε)v. (b) If a ∈ n(x) for some n ∈ 𝒩v and x ∈ X, then for every ε > 0 there is a function f ∈ 𝒞𝒦 (X, E) such that f (x) = a and f ⩽ (1 + ε)v. Proof. Suppose that v is a weakly lower continuous function space neighborhood. For Part (a), suppose that χA ⋅ f ⩽ v for A ∈ R and f ∈ 𝒞 (X, E). Let U ∈ O be a relatively compact open set containing A. There is a neighborhood function n ∈ 𝒩v and V ∈ 𝒱 such that V ⊂ n(x) for all x ∈ U, and there is m ∈ 𝒩v such that (χA ⋅ f )(x) ∈ m(x) for all x ∈ X, that is, f (x) ∈ m(x) for all x ∈ A. Given ε > 0, for every point x ∈ A there is an open neighborhood Ux ⊂ U of x such that both f (y) − f (x) ∈ εV and f (x) ∈ m(y) + εV holds for all y ∈ Ux . The latter is due to the weak lower continuity of the neighborhood function m. This renders f (y) ∈ m(y) + 2εV ⊂ m(y) + 2εn(y). As a convex combination of m and n, the neighborhood function

10 � 1 Function spaces s=

2ε 1 m+ n 1 + 2ε 1 + 2ε

is contained in 𝒩v , and we have f (y) ∈ (1 + 2ε) s(y) for all y ∈ Ux . Setting O = ⋃x∈A Ux we have f (y) ∈ (1 + 2ε)s(y) for all y ∈ O, hence χO ⋅ f ∈ (1 + 2ε)v, and our claim in (a) follows. For Part (b), suppose that a ∈ n(x) for some neighborhood function n ∈ 𝒩v and x ∈ X. We shall use Part (a) with the constant function x 󳨃→ a for f and the singleton set {x} for A ∈ R. Then χA ⋅ a ⩽ v, and for every ε > O there is O ∈ O such that x ∈ O and χO ⋅ a ⩽ (1 + ε)v. Using Urysohn’s lemma (see Proposition 15.6 in [63], also Proposition III.4.15 in [53]) we find a nonnegative function φ ∈ 𝒞𝒦 (X) such that φ(x) = 1 and φ ⩽ χO . Then f = φ ⋅ a ∈ 𝒞𝒦 (X, E) and f = φ ⋅ (χO ⋅ a) ⩽ (1 + ε)v by Lemma 1.2.1(a); hence our claim in (b). Lemma 1.2.2 implies that every function in f ∈ 𝒞 (X, E), which vanishes at infinity is absorbed by the neighborhoods in V. Indeed, given v ∈ V there is A ∈ R and λ ⩾ 0 such that χ(X\A) ⋅ f ⩽ v and χA ⋅ f ⩽ λv. Thus f ⩽ (λ + 1)v. The function spaces 𝒞V (X, E) Our upcoming investigations will focus on spaces of continuous E-valued functions that vanish at infinity relative to a function space neighborhood system V. This subspace of 𝒞 (X, E) will be denoted by 𝒞V (X, E). The balanced convex and absorbing subsets ⟨v⟩ = {f ∈ 𝒞V (X, E) | f ⩽ v} of 𝒞V (X, E), for all v ∈ V, form a basis ⟨V⟩ for a locally convex topology on 𝒞V (X, E). We note that ⟨v⟩ + ⟨u⟩ ⊂ ⟨v + u⟩ and ⟨αv⟩ = α⟨v⟩ holds for all v, u ∈ V and α > 0. For the sake of keeping our notation manageable and since the meaning will be clear from the context, we shall avoid the angled brackets and use the same symbols v for function space neighborhoods and V for the neighborhood system also for the balanced convex neighborhoods and the neighborhood system, which they induce on 𝒞V (X, E). In particular, the notions f ⩽ v and f ∈ v will be synonymous for functions f ∈ 𝒞V (X, E). In this way, (𝒞V (X, E), V) is a locally convex topological vector space. We shall abbreviate 𝒞V (X) if E = ℝ. Note that for any choice of the neighborhood system V all functions with compact support vanish at infinity, hence 𝒞𝒦 (X, E) is a subspace of 𝒞V (X, E). Since for every x ∈ X, there is φ ∈ 𝒞𝒦 (X) such that φ(x) = 1 and because φ ⋅ a ∈ 𝒞𝒦 (X, E) for all a ∈ E, the latter implies in particular that {f (x) | f ∈ 𝒞V (X, E)} = E. Moreover, if f ∈ 𝒞V (X, E) and if φ is a bounded continuous real-or complex valued (depending on the scalar field of E) function on X, then φ ⋅ f ∈ 𝒞V (X, E) according to Lemma 1.2.1(a). We shall make use of the following observations.

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1.2.5 Proposition. Every function f ∈ 𝒞V (X, E) can be approximated with respect to the neighborhoods in V by: (i) elementary functions e = φ1 ⋅ a1 + ⋅ ⋅ ⋅ + φn ⋅ an ∈ 𝒞𝒦 (X, E) for φi ∈ 𝒞𝒦 (X) and ai ∈ f (X). (ii) step functions h = χA1 ⋅ a1 + ⋅ ⋅ ⋅ + χAn ⋅ an ∈ ℱ (X, E) for Ai ∈ R and ai ∈ f (Ai ). Proof. Let f ∈ 𝒞V (X, E) and v ∈ V. There is a compact subset K of X such that χX\K ⋅ f ⩽ v. We choose an open set O ∈ O containing K. According to the definition of a function space neighborhood, there is V ∈ 𝒱 and n ∈ 𝒩v such that V ⊂ n(x) for all x ∈ O. There are open sets O1 , . . . , On ⊂ O, which cover K and such that f (x)−f (y) ∈ V whenever x, y ∈ Oi . We choose ai ∈ f (Oi ). Now for (i) we select a corresponding partition of the unity (see Section 1.1), that is, a collection of nonnegative functions φ1 , . . . , φn ∈ n 𝒞𝒦 (X) such that supp(φi ) ⊂ Oi and φ(x) = 1 for all x ∈ K, where φ = ∑i=1 φi . We may also assume that 0 ⩽ φ(x) ⩽ 1 holds for all x ∈ X, because otherwise we can divide each of the functions φi by the continuous function x 󳨃→ max{φ(x), 1}. Let e = ∑ni=1 φi ⋅ ai . We have φi (x)(f (x) − ai ) ⩽ φi (x)V for all x ∈ X and i = 1, . . . , n, hence φ(x)f (x) − e(x) ⩽ φ(x)V . This renders f (x) − e(x) ∈ V for all x ∈ K, hence χK ⋅ (f − e) ⩽ v by our choice of V . Moreover, since O contains the support of e and since X \ O ⊂ X \ K, we have χX\O ⋅ (f − e) = χX\O ⋅ f ⩽ v. For all x ∈ O \ K, on the other hand we have φ(x)f (x) − e(x) ⩽ φ(x)V ⊂ V by the above, hence χO\K ⋅ (φ ⋅ f − e) ⩽ v by our choice of the neighborhood V ∈ 𝒱 . Furthermore, since 0 ⩽ 1 − φ(x) ⩽ 1 for all x ∈ X and since χO\K f ⩽ v, Lemma 1.2.1(a) yields that χO\K ⋅ (1 − φ) ⋅ f ⩽ v. Combining the last two observations, we obtain χO\K ⋅ (f − e) = χO\K ⋅ (φ ⋅ f − e) + χO\K ⋅ (1 − φ) ⋅ f ⩽ v + v = 2v.

Finally, summarizing all of the above, f − e = χX\O ⋅ (f − e) + χO\K ⋅ (f − e) + χK ⋅ (f − e) ⩽ v + 2v + v = 4v.

This proves our claim for the approximation by elementary functions. The argument for step functions is more straightforward. With the same choice for K, V , O, Oi from above, we set A1 = O1 and Ai = Oi \ ⋃i−1 k=1 Ai for i = 2, . . . , n. The sets Ai are pairwise disjoint, their union A is a subset of O and contains X. We choose ai ∈ f (Ai ) and

12 � 1 Function spaces set h = ∑ni=1 χAi ⋅ ai . Then f (x) − h(x) ∈ V for all x ∈ A, hence χA ⋅ (f − h) ⩽ v. Since X \ A ⊂ X \ K, we also have χX\A ⋅ (f − h) = χX\A ⋅ f ⩽ v. Thus f − h ∈ 2v. Our claim follows. 1.2.6 Proposition. (a) If 𝒞V (X, E) is Hausdorff, then E is Hausdorff and for every nonempty set O ∈ O and 0 ≠ a ∈ E there is v ∈ V such that χO ⋅ a ⩽̸ v. (b) If for every x ∈ X and 0 ≠ a ∈ E, there is v ∈ V such that χ{x} ⋅ a ⩽̸ v, then 𝒞V (X, E) is Hausdorff. Proof. For Part (a), suppose that 𝒞V (X, E) is Hausdorff. Let a ∈ ⋂V ∈𝒱 V and 0 ≠ φ ∈ 𝒞𝒦 (X). For every v ∈ V, there is V ∈ 𝒱 and n ∈ 𝒩v such that V ⊂ n(x) for all x ∈ supp(φ). Hence φ ⋅ a ⩽ v. The latter holds true for all v ∈ V and implies that φ ⋅ a = 0. Thus a = 0, and we infer that E is Hausdorff. For the second part of our claim in (a), suppose to the contrary that there are H ≠ O ∈ O and 0 ≠ a ∈ E such that χO ⋅ a ⩽ v for all v ∈ V. We choose 0 ≠ φ ∈ 𝒞𝒦 (X) such that |φ| ⩽ χO . Then 0 ≠ φ ⋅ a ∈ 𝒞V (X, E) and φ ⋅ a = φ ⋅ (χO ⋅ a) ⩽ v for all v ∈ V by Lemma 1.2.1(a). Hence 𝒞V (X, E) is not Hausdorff, contradicting the assumption of Part (a). For Part (b), let 0 ≠ f ∈ 𝒞V (X, E). There is x ∈ X such that a = f (x) ≠ 0. Under the assumptions of Part (b), there is v ∈ V such that χ{x} ⋅ a ⩽̸ v. Then f ⩽̸ v, because otherwise we had χ{x} ⋅ f = χ{x} ⋅ a ⩽ v by Lemma 1.2.1(a). We infer that 𝒞V (X, E) is Hausdorff. We shall list a few examples of function spaces. Our very general approach allows for a wide variety of settings. More details for some of these can be found in [51], Examples 2.2. 1.2.7 Examples. Examples (a) to (d) below deal with cases where the function space neighborhoods are generated by singleton sets 𝒩v = {nv }, that is, by a single weakly lower continuous neighborhood function. (a) The topology of uniform convergence on X is generated by the function space neighborhoods vV = {f ∈ ℱ (X, E) | f (x) ∈ V for all x ∈ X}, rendered by the constant neighborhood function x 󳨃→ V , where V ∈ 𝒱 . They give rise to the neighborhood system V = {vV | V ∈ 𝒱 }. The function space 𝒞V (X, E) consists of all functions f ∈ 𝒞 (X, E) such that for every V ∈ 𝒱 there is a compact subset K of X and f (x) ∈ V holds for all x ∈ X \ K. (b) The topology of compact convergence is generated by the function space neighborhoods

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v(K,V ) = {f ∈ ℱ (X, E) | f (x) ∈ V for all x ∈ K}, where V ∈ 𝒱 and K is a compact subset of X, rendered by the weakly lower continuous neighborhood function x 󳨃→ V for x ∈ K and x 󳨃→ E else. They give rise to the neighborhood system V = {v(K,V ) | V ∈ 𝒱 , K ⊂ X compact}. We have 𝒞V (X, E) = 𝒞 (X, E) in this case. (c) The topology of pointwise convergence is generated by the function space neighborhoods v(Y ,V ) = {f ∈ ℱ (X, E) | f (x) ∈ V for all x ∈ Y }, where V ∈ 𝒱 and Y is a finite subset of X, rendered by the weakly lower continuous neighborhood function x 󳨃→ V for x ∈ Y and x 󳨃→ E else. They give rise to the neighborhood system V = {v(Y ,V ) | V ∈ 𝒱 , Y ⊂ X finite}. We have 𝒞V (X, E) = 𝒞 (X, E). (d) Examples (a), (b) and (c) render weakly lower continuous function space neighborhood systems and are special cases for weighted space topologies, which were introduced for real-valued functions by Nachbin and Prolla (see [39] and [42]). A family Ω of nonnegative real-valued upper semicontinuous functions on X is called a family of weights if for all ω1 , ω2 ∈ Ω there are ω3 ∈ Ω and ρ > 0 such that ω1 ⩽ ρ ω3 and ω2 ⩽ ρ ω3 . The functions ρω = 1/ω then are (0, +∞]-valued and lower semicontinuous, and for V ∈ 𝒱 and ω ∈ Ω the function space neighborhood v(ρω ,V ) = {f ∈ ℱ (X, E) | f (x) ∈ ρΩ V for all x ∈ X}, where V ∈ 𝒱 , is rendered by the neighborhood function ρω ⋅V . (Following our convention from Section 1.1, we set (+∞) V = E.) According to Lemma 1.2.3, this neighborhood function is weakly lower continuous. The neighborhoods of this type give rise to the function space neighborhood system V = {v(ρω ,V ) | V ∈ 𝒱 , ω ∈ Ω} and a corresponding function space 𝒞V (X, E). (e) We can modify the preceding examples replacing the singleton sets 𝒩v = {nv } by the upward directed sets 𝒩v consisting of all convex combinations of the weakly lower continuous neighborhood functions χO ⋅ nv , where O ∈ O. Since all involved neighborhood functions have compact support, any function in 𝒞 (X, E) that vanishes at infinity must itself have a compact support. Thus 𝒞V (X, E) = 𝒞𝒦 (X, E) in this setting.

14 � 1 Function spaces (f) The finest function space topology for ℱ (X, E) is created by the neighborhood system V˚ consisting of all function space neighborhoods. It is straightforward to construct an easily accessible basis for this topology: We consider collections of neighborhood functions χA ⋅V , where A ∈ R and V ∈ 𝒱 , with the following property: For every A ∈ R, there is B ∈ R and V ∈ 𝒱 such that A ⊂ B and χB ⋅V is contained in the collection. The convex hull 𝒩 of such a collection defines a function space neighborhood, and the system V‚ of all function space neighborhoods of this type is closed for multiplication by positive scalars, downward directed and easily seen to be equivalent to V˚ . If in the topological case one considers only collections of neighborhood functions χO ⋅ V for open sets O ∈ O, these functions are weakly lower continuous and define weakly lower continuous function space neighborhoods. The system Vτ of these neighborhoods is therefore weakly lower continuous, clearly coarser than V‚ , but also finer since every set in R is contained in some set in O. Consequently, both systems V‚ and Vτ are equivalent to V˚ . In summary, in the topological case the finest function space topology admits a weakly lower continuous basis. Furthermore, all neighborhood functions involved in the systems V‚ and Vτ are supported by sets in R. Thus, in the topological case, any function in 𝒞 (X, E) that vanishes at infinity relative to V˚ has a compact support. We infer that 𝒞V‹ (X, E) = 𝒞𝒦 (X, E). The finest function space topology is indeed the usual strict inductive limit topology on 𝒞𝒦 (X, E), generated by the family of subspaces, each consisting of those functions with support in a fixed compact subset of X and endowed with the topology of uniform convergence on this subset (see II.6 in [56]). (g) Let μ be a regular real-valued Borel measure on X, let 1 ⩽ p ⩽ +∞ and consider the function space neighborhood vp generated by the collection of all weakly lower continuous neighborhood functions nρ , that is, functions x 󳨃→ ρ(x)𝔹, where 𝔹 = [−1, +1] is the unit ball in E = ℝ and ρ is a nonnegative real-valued lower semicontinuous function such that ‖ρ‖p ⩽ 1. Then V = {λvp | λ > 0} forms a function space neighborhood system, and 𝒞V (X) consists of all continuous functions in Lp (μ) that vanish at infinity with respect to vp . For p < +∞, the latter is guaranteed if either X is σ-compact or if μ(X) is finite. The neighborhood vp in 𝒞V (X) is the unit ball with respect to the Lp -norm. (h) The case X = ℕ with the discrete topology covers a variety of sequence spaces. All function space systems in the preceding Examples 1.2.7 are weakly lower continuous. In Parts (a) to (e), the families 𝒩v of neighborhood functions generating the respective function space neighborhoods are upward directed. Thus Lemma 1.2.1(b) applies. If E is Hausdorff, then according to Proposition 1.2.6(b), examples (a) to (f) render Hausdorff function space topologies for 𝒞V (X, E), provided that in (d) for every x ∈ X there is ω ∈ Ω such that ω(x) > 0. Several special cases appear worth mentioning. An ordered topological vector space E over ℝ or ℂ is a locally convex topological vector space that carries an order relation ⩽ defined by the subcone E+ = {a ∈ E | a ⩾ 0}

1.2 Spaces of vector-valued functions that vanish at infinity

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of its positive elements. That is, a ⩽ b for a, b ∈ E if and only if b − a ∈ E+ . Note that 0 ∈ E+ , hence the element 0 ∈ E is called positive, other than in the conventional notation for real numbers. The positive cone E+ is supposed to be closed in E (see [56] V.4). We do however not require that E+ ∩ (−E+ ) = {0}. The latter holds true if and only if the order is antisymmetric and implies that the set {0} is closed in E and, therefore, that the topology of E is Hausdorff. The positive cone in the dual E ˚ of E is E+˚ = {μ ∈ E | Re μ(a) ⩾ 0 for all a ∈ E+ }, that is, the polar of −E+ in E ˚ . We have a ⩽ b for a, b ∈ E if and only if Re μ(a) ⩽ Re μ(b) for all μ ∈ E+˚ . The order of E transfers pointwise to the functions in 𝒞V (X, E). If convergence in 𝒞V (X, E) implies pointwise convergence, that is, if V is finer than the function space topology of pointwise convergence from Example 1.2.7(c), then the subcone of all E+ -valued functions is closed. Otherwise, we use its closure for the positive cone 𝒞V (X, E)+ of 𝒞V (X, E). A topological vector lattice E is an orderd topological vector space over ℝ with an antisymmetric order and lattice operations ∨ and ∧ satisfying the usual rules (see [99] or [95]). The operator a 󳨃→ |a| : E → E is supposed to be continuous. This implies that the lattice operations a 󳨃→ a+ = a ∨ 0, a 󳨃→ a− = (−a) ∨ 0 are continuous from E to E and both (a, b) 󳨃→ a ∨ b and (a, b) 󳨃→ a ∧ b are continuous from E × E to E. Antisymmetry of the order implies that E is Hausdorff. The lattice operations in E transfer pointwise to the functions in ℱ (X, E). A topological algebra E over ℝ or ℂ is a locally convex topological vector space with a multiplication satisfying the rules of an algebra, and such that for every fixed element a the operators b 󳨃→ ab and b 󳨃→ ba are continuous from E to E. The multiplication transfers pointwise to the functions in ℱ (X, E). An involution is a continuous real-linear operator a 󳨃→ a˚ on E such that (αa)˚ = αa˚ , (ab)˚ = b˚ a˚ and (a˚ )˚ = a for a, b ∈ E and α in ℝ or ℂ. Again, this operation transfers pointwise to the functions in ℱ (X, E). 1.2.8 Proposition. Let (𝒞V (X, E), V) be a function space. (a) If E is an ordered topological vector space, then 𝒞V (X, E) is also an ordered topological vector space. Its positive cone 𝒞V (X, E)+ is the closure of the cone of all E+ -valued functions. (b) If E is a topological vector lattice over ℝ, if the cone of all E+ -valued functions is closed in 𝒞V (X, E) and if for every v ∈ V there is u ∈ V such that f ⩽ v whenever |f | ⩽ |g| for some g ⩽ u, for f , g ∈ 𝒞V (X, E), then 𝒞V (X, E) is also a topological vector lattice. (c) If E is a topological algebra over ℝ or ℂ and if for every v ∈ V, there is u ∈ V such that fg ⩽ v whenever f , g ⩽ u, for f , g ∈ 𝒞V (X, E), then 𝒞V (X, E) is also a topological algebra. If E contains an involution and if for every v ∈ V there is u ∈ V such that f ˚ ⩽ v whenever f ∈ u, then f 󳨃→ f ˚ is an involution on 𝒞V (X, E).

16 � 1 Function spaces Proof. Part (a) was argued before. For Part (b), that is, the case of a topological vector lattice E, we first observe that, due to our assumption that the cone of all E+ -valued functions is closed in 𝒞V (X, E), it coincides with the positive 𝒞V (X, E)+ . Hence the order of 𝒞V (X, E) is antisymmetric. Indeed, if both f ⩽ 0 and f ⩾ 0 holds for a function f ∈ 𝒞V (X, E), then f (x) ⩽ 0 and f (x) ⩾ 0 for all x ∈ X. Since the order of E is antisymmetric, we infer that f = 0. Because of the continuity of the lattice operations in E, the functions f ∨ g, f ∧ g and |f | are continuous whenever f and g are continuous. Moreover, if f ∈ ℱ (X, E) vanishes at infinity relative to V, then for every v ∈ V there is u ∈ V, as in the condition of (b). Hence there is A ∈ R such that χX\A ⋅ |f | ⩽ v, as χX\A ⋅ |f | ⩽ u. The function |f | therefore also vanishes at infinity. If both f , g ∈ ℱ (X, E) vanish at infinity, since |f ∨ g| ⩽ |f | + |g| and |f ∧ g| ⩽ |f | + |g|, the same argument shows that both functions f ∨ g and f ∧ g also vanish at infinity. Thus 𝒞V (X, E) is indeed a lattice, and the condition of (b) implies that the operation f 󳨃→ |f | is continuous. Likewise, for Part (c) the continuity of the multiplication in E guarantees that the product of two continuous functions is again continuous, and the condition in (c) implies that the product of functions that vanish at infinity also vanishes at infinity. Hence 𝒞V (X, E) is an algebra, and the continuity of its multiplication follows in a similar way. If E contains an involution satisfying the condition in (c), then the continuity of the involution operator guarantees that the function f ˚ is continuous, whenever f is continuous. Given v ∈ V and A ∈ R, there is u ∈ V such that χX\A ⋅ f ˚ ⩽ v whenever χX\A ⋅ f ∈ u. Thus if f ∈ ℱ (X, E) vanishes at infinity relative to V, so does f ˚ . The involution is therefore a real-linear operator on 𝒞V (X, E) with the required properties. Its continuity follows directly from the condition in (c). Suppose that E is indeed a function space 𝒞U (Y , F), where Y is another locally compact Hausdorff space, (F, 𝒲 ) is a locally convex topological vector space and U is a function space neighborhood system for F-valued functions on Y . With a function f ∈ ℱ (X, ℱ (Y , F)), we associate in a canonical way a function ̃f ∈ ℱ (X × Y , F) setting ̃f (x, y) = f (x)(y). This correspondence is one-to one and onto. If we assume that U is the function space neighborhood system of compact convergence on Y (see Example 1.2.7(b)), then 𝒞U (Y , F) = 𝒞 (Y , F), and we claim that ̃f ∈ 𝒞 (X × Y , F) if and only if f ∈ 𝒞 (X, 𝒞 (Y , F)). Indeed, suppose that the function f ∈ ℱ (X, 𝒞 (Y , F)) is continuous. Let (x, y) ∈ X × Y and let W ∈ 𝒲 be a neighborhood in F. We find a compact neighborhood L of y in Y such that f (x)(y′ ) − f (x)(y) = ̃f (x, y′ ) − ̃f (x, y) ∈ (1/2)W for all y′ ∈ L. Since convergence in 𝒞U (Y , F) means compact convergence, there is u ∈ U such that g ⩽ u for g ∈ 𝒞U (Y , F) implies that g(y′ ) ∈ (1/2)W holds for all y′ ∈ L. In turn, there is a neighborhood U of x in X such that f (x ′ ) − f (x) ⩽ u for all x ′ ∈ U. Hence ̃f (x ′ , y′ ) − ̃f (x, y′ ) ∈ (1/2)W

1.2 Spaces of vector-valued functions that vanish at infinity

� 17

holds for all x ′ ∈ U and y′ ∈ L. Combining the above, we obtain that ̃f (x ′ , y′ ) − ̃f (x, y) = (̃f (x ′ , y′ ) − ̃f (x, y′ )) + (̃f (x, y′ ) − ̃f (x, y)) ∈ W holds for all (x ′ , y′ ) ∈ U × L. Hence the function ̃f is indeed continuous. For the converse, suppose that the function ̃f ∈ ℱ (X × Y , F) is continuous and for every x ∈ X define the function f (x) ∈ ℱ (Y , F) by f (x)(y) = ̃f (x, y). The continuity of ̃f implies that f (x) ∈ 𝒞 (Y , F). We proceed to establish that the function f ∈ ℱ (X, 𝒞 (Y , F)), that is, the mapping x 󳨃→ f (x) is continuous. Let x ∈ X and u ∈ U. There is a compact subset K of Y and W ∈ 𝒲 such that g(y) ∈ W for all y ∈ K implies that g ⩽ u for a function g ∈ ℱ (Y , F). By the continuity of the function ̃f , for every y ∈ K, there are open neighborhoods Uy of x in X and Ly of y ∈ Y such that f (x ′ , y′ ) − f (x, y) ∈ (1/2)W for all (x ′ , y′ ) ∈ Uy × Ly . Hence there are y1 , . . . , yn ∈ K such that the open sets Lyi cover K. Then U = ⋂ni=1 Uyi is a neighborhood of x. Every element y ∈ K is contained in one of the sets y ∈ Lyi and, therefore, (x ′ , y) ∈ U × Lyi ⊂ Uyi × Lyi holds for all x ′ ∈ U. Thus ̃f (x ′ , y) − ̃f (x, y) = (̃f (x ′ , y) − ̃f (x, y )) + (̃f (x, y ) − ̃f (x, y)) ∈ W i i holds for all (x ′ , y) ∈ U × K. This shows that f (x ′ )(y) − f (x)(y) ∈ W , and consequently, that f (x ′ ) − f (x) ⩽ u for all x ′ ∈ U. Our claim follows. Going forward with this special case, we further narrow our assumptions on the function space 𝒞U (Y , F) and assume that F is a normed space, that Y is compact and that U renders the topology of uniform convergence on Y (see Example 1.2.7(a)). Let 𝔹 denote the unit ball of F and u the unit ball of 𝒞U (Y , F), that is, the function space neighborhood defined by the constant neighborhood function y 󳨃→ 𝔹. Then U consists of all positive multiples of u and induces the neighborhood system ⟨U⟩ on 𝒞U (Y , F). We have ⟨U⟩c = {λ⟨u⟩ | λ ∈ ℝ+ }, where 󵄩 󵄩 ⟨u⟩ = {g ∈ 𝒞U (Y , F) | 󵄩󵄩󵄩g(y)󵄩󵄩󵄩 ⩽ 1 for all y ∈ Y } and (+∞)⟨u⟩ = 𝒞U (Y , F) (see Section 1.1). We continue to outline a natural correspoñ dence between 𝒞V (X, 𝒞U (Y , F)) and a function space 𝒞V ̃ (X × Y , F) where V is a suitable function space neighborhood system for F-valued functions on X ×Y . We note that the neighborhood functions on X that define the function space neighborhood system

18 � 1 Function spaces V and are ⟨U⟩c -valued, while neighborhood functions on X ×Y are 𝒲c -valued, where 𝒲c = {λ𝔹 | λ ∈ ℝ+ }. Every neighborhood function n on X is defined by an ℝ+ -valued function φn on X such that n(x) = φn (x)⟨u⟩ for all x ∈ X. In a canonical way, we associate with n a neighborhood function ñ on X × Y defined as ñ (x, y) = φn (x)𝔹 for all (x, y) ∈ X × Y . Using this correspondence, we can now define the function space ̃ for X × Y as follows. With every v ∈ V, we associate a neighborhood system V ̃ defined by the convex family of neighborhood functions neighborhood ṽ ∈ V ̃ | n ∈ 𝒩v }. 𝒩ṽ = {n The requirements for a function space neighborhood are satisfied for ṽ. Indeed, let K be a compact subset of X. There is n ∈ 𝒩v and λ > 0 such that λ⟨u⟩ ⊂ n(x) for all x ∈ K. Hence λ𝔹 ⊂ ñ(x, y) for all (x, y) ∈ K × Y . Since every relatively compact Borel subset of X × Y is contained in some set of this type, our claim follows. The family ̃ = {ṽ | v ∈ V} then forms a function space neighborhood system for ℱ (X × Y , F). It V is now obvious that f ⩽ v holds if and only if ̃f ∈ ṽ for all f ∈ ℱ (X, ℱ (Y , F)). Combining all of the above, we realize that the mapping f 󳨃→ ̃f is indeed a linear bijection between 𝒞V (X, 𝒞U (Y , F))

and

𝒞V ̃ (X × Y , F),

which preserves the respective function space neighborhoods. Moreover, if the func̃ Indeed, let tion space neighborhood system V is weakly lower continuous, so is V. ̃ and let ñ ∈ 𝒩ṽ . For weak lower continuity of the neighborhood function ñ, let ṽ ∈ V (x, y) ∈ X × Y , let c ∈ ñ(x, y) and let W = ε𝔹 ∈ 𝒲 for ε > 0. We set λ = φn (x), that is, n(x) = λ⟨u⟩. Hence ‖c‖ ⩽ λ. There is a function g ∈ 𝒞U (Y , F) such that g ⩽ λu and g(y) = c. We have g ∈ n(x), and by the weak lower continuity of the neighborhood function n there is a neighborhood U of x such that g ∈ n(x ′ ) + ε⟨u⟩ for all x ′ ∈ U. The latter means that ‖g(y′ )‖ ⩽ φn (x ′ ) + ε holds for all (x ′ y′ ) ∈ U × Y . Hence c = g(y) ∈ ñ(x ′ , y′ ) + ε𝔹. Since U × Y is a neighborhood of (x, y), our claim follows. We summarize. 1.2.9 Proposition. If 𝒞U (Y , F) is a function space, where F is a normed space, Y is compact and U renders the topology of uniform convergence on Y , then 𝒞V (X, 𝒞U (Y , F)) is isomorphic to a function space 𝒞V ̃ (X × Y , F). If the function space neighborhood system ̃ V is weakly lower continuous, so is V.

1.3 Compactness in spaces of linear operators

� 19

Function subspaces We conclude this section with the following observation. If G is a subspace of E, then the sets VG = V ∩ G for V ∈ 𝒱 form a neighborhood basis for G. Thus if n is a neighborhood function for ℱ (X, E), then the intersections of its values with G define a neighborhood function for ℱ (X, G). In this way, a functions-space neighborhood system V induces a functions-space neighborhood system for ℱ (X, G) in a canonical way. Accordingly, (𝒞V (X, G), V) is again a function space and indeed a subspace of (𝒞V (X, E), V).

1.3 Compactness in spaces of linear operators Let (E, 𝒱 ) and (F, 𝒲 ) be real or complex locally convex topological vector spaces such that F is Hausdorff. Let F ˚ denote the topological dual of F. The second dual (or bidual) F ˚˚ of F is the dual of F ˚ if the latter is endowed with its strong topology, that is, the topology generated by the polars of balanced bounded subsets of F. We consider the following standard topologies on the space L(E, F) of all continuous realor complex- (depending on the common scalar field) linear operators T : E → F: (i) the uniform operator topology generated by the neighborhoods W

𝒰A = {T ∈ L(E, F) | T(A) ⊂ W }

for bounded subsets A of E and W ∈ 𝒲 ; (ii) the strong operator topology generated by the neighborhoods W

𝒰A = {T ∈ L(E, F) | T(A) ⊂ W }

for finite subsets A of E and W ∈ 𝒲 ; (iii) the weak operator topology generated by the neighborhoods ϒ

󵄨

󵄨

𝒰A = {T ∈ L(E, F) | 󵄨󵄨󵄨ν(T(a))󵄨󵄨󵄨 ⩽ 1 for all a ∈ A and μ ∈ ϒ}

for finite subsets A of E and ϒ of F ˚ . If F = G˚ is the dual of a locally convex ordered topological vector space G, then we also refer to (iv) the weak* operator topology generated by the neighborhoods ϒ

󵄨

󵄨

𝒰A = {T ∈ L(E, F) | 󵄨󵄨󵄨ν(T(a))󵄨󵄨󵄨 ⩽ 1 for all a ∈ A and μ ∈ ϒ}

for finite subsets A of E and ϒ of G. Using any of the above topologies L(E, F) is again a locally convex topological vector space. It is Hausdorff since F is supposed to be Hausdorff. For later use, we recall the well-known fact that the operators in L(E, F) are also weakly continuous, that is, continuous if we consider the spaces E and F in their respective weak topologies σ(E, E ˚ ) and σ(F, F ˚ ) (see IV.7.4 in [56] or II.6.13 in [45]).

20 � 1 Function spaces This follows with a simple argument. Let T ∈ L(E, F). A neighborhood W in the σ(F, F ˚ )-topology of F is defined by finitely many elements μ1 , . . . , μn ∈ F ˚ , that is, W = {c ∈ F | Re μi (c) ⩽ 1, for i = 1, . . . , n}. The linear functionals μi ˝ T on E, that is, a 󳨃→ μi (T(a)), are continuous, hence elements of E ˚ . Thus V = {a ∈ F | Re(μi ˝ T)(a) ⩽ 1, for i = 1, . . . , n} defines a neighborhood in the σ(E, E ˚ )-topology of E. Our claim follows, because T(V ) ⊂ W .

Bounded operators A linear operator T : E → F is called bounded if it maps bounded subsets of E into bounded subsets of F. Clearly, all continuous operators are bounded. The converse holds true if E is a normed space, but not in general. The families of bounded subsets of E are identical for all topologies consistent with the dual pair (E, E ˚ ). The same holds true for F and (F, F ˚ ) (see IV.3.2, Corollary 2 in [56] and the remark below). We shall use the same set of topologies (i) to (iv) from above for the space Lb (E, F) of all bounded linear operators from E to F. While we always assume that the locally convex space F is Hausdorff, we do not impose the same requirement on E. The set E0 = ⋂V ∈𝒱 V forms a closed subspace of E, and we have E0 = {a ∈ E | μ(a) = 0 for all μ ∈ E ˚ } by the Hahn–Banach theorem. Thus A = A + E0 for every subset A of E that is either open or closed. We abbreviate Ẽ for the quotient space E/E0 = {a + E0 | a ∈ E}, endowed with the quotient space topology (see I.2 in [56]). The space Ẽ is Hausdorff, and the canonical projection from E to Ẽ is linear, onto, continuous and both open and closed. The space E itself is Hausdorff if and only if E0 = {0}, that is, if and only if ̃ For an element a ∈ E, we write ã for a + E0 ∈ E, ̃ that is the image of a under E = E. ̃ the canonical projection of E onto E, and ̃ = {ã | a ∈ A} ⊂ Ẽ A ̃ = A? for a subset A of E. Clearly, A + E0 . If A is open, closed, convex or balanced in ̃ ̃ Conversely, if A ̃ is open, closed, E, then A is open, closed, convex or balanced in E. ̃ then A+E0 is open, closed, convex or balanced in E. A subset convex or balanced in E,

1.3 Compactness in spaces of linear operators

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̃ is bounded, compact A is bounded, compact or relatively compact in E if and only if A ̃ or relatively compact in E. Moreover, the subspace E0 is contained in the kernel of every operator T ∈ Lb (E, F). Indeed, E0 is a bounded subset of E, hence T(E0 ) is bounded in F and, therefore, T(E0 ) = {0}, since F is supposed to be Hausdorff. We infer that T takes the same value at all elements of an equivalence class ã = a + E0 in Ẽ and corresponds to ̃ F), where an operator T̃ ∈ Lb (E, ̃ ã) = T(a) T( ̃ Conversely, for every operator T̃ ∈ Lb (E, ̃ F) the formula T(a) = T( ̃ ã), for all ã ∈ E. for all a ∈ E, defines an operator in Lb (E, F). Thus there is a bijective linear cor̃ F) and Lb (E, F), which also preserves the respective oprespondence between Lb (E, erator space topologies. The same observations apply to the spaces of continuous liñ F) and L(E, F.) In particular, E and Ẽ share the same dual, that ear operators L(E, ˚ ˚ is, Ẽ = E . Every operator 𝒯 ∈ Lb (E, F) renders a bilinear form on Ẽ × F ˚ , that is, (ã, μ) 󳨃→ ̃ Thus the elements of μ(𝒯 (a)), where a is any element of the equivalence class ã ∈ E. ˚ ̃ the tensor product E ⊗ F are linear functionals on Lb (E, F), and (Lb (E, F), Ẽ ⊗ F ˚ ) constitutes a dual pair with the bilinear form n

n

i=1

i=1

⟨𝒯 , ∑ ãi ⊗ μi ⟩ = ∑ μi (𝒯 (ai )), where ai ∈ ãi . According to the entry about polar topologies in Section 1.1 (see also IV.1 in [56]), we need to establish that the above bilinear form is nondegenerate, that is, (i) ⟨𝒯 , ω⟩ = 0 for all ω ∈ Ẽ ⊗ F ˚ implies hat 𝒯 = 0, and (ii) ⟨𝒯 , ω⟩ = 0 for all 𝒯 ∈ Lb (E, F) implies that ω = 0. While (i) is obvious, (ii) requires a brief argument. Let n

ω = ∑ ãi ⊗ μi ∈ E ⊗ F ˚ . i=1

We may assume that both sets {ãi } and {μi } are linearly independent in Ẽ and F ˚ , respectively. If ω ≠ 0, then n ⩾ 1, and there is ν ∈ Ẽ ˚ = E ˚ such that ν(ã1 ) = 1 and ν(ãi ) = 0 for all i > 1 (see IV.1.1 in [56]). Since F is Hausdorff, there is c ∈ F such that μ1 (c) = 1. We define 𝒯 (a) = ν(a)c ∈ F for all a ∈ E and observe that 𝒯 ∈ L(E, F). Since ⟨𝒯 , ω⟩ = 1, our claim follows. For the following, recall our brief review of polar topologies in Section 1.1. 1.3.1 Proposition. The weak, strong and uniform operator topologies of Lb (E, F) are polar topologies of the dual pair (Lb (E, F), Ẽ ⊗ F ˚ ). The weak and the strong operator

22 � 1 Function spaces topologies are both consistent with this duality, and in either of these L(E, F) is a dense subspace of Lb (E, F), hence features the same dual Ẽ ⊗ F ˚ . Proof. For our first statement, we observe that the weak, strong and uniform operator topologies of Lb (E, F) are all generated by families of polars of subsets ̃ μ ∈ ϒ} ΩϒA = {ã ⊗ μ | ã ∈ A, of Ẽ ⊗ F ˚ . For the weak operator topology, we allow all finite subsets of E for A and all finite subsets of F ˚ for ϒ. For the strong operator topology, we choose all finite subsets for A and the polars of neighborhoods W ∈ 𝒲 for ϒ. We obtain the uniform operator topology if we choose all bounded subsets of E for A and again all polars of neighborhoods in 𝒲 for ϒ. In all of these instances, ϒ is compact in the weak* topology of F ˚ . We shall first verify that all the sets ΩϒA involved are weakly bounded in Ẽ ⊗ F ˚ , that is, families of their polars in Lb (E, F) create polar topologies of the dual pair (Lb (E, F), Ẽ ⊗ F ˚ ). Indeed, for any of the above choices for A and ϒ, the set A is bounded in E and ϒ ⊂ W ˝ for some W ∈ 𝒲 . Thus for every 𝒯 ∈ Lb (E, F) there is λ ⩾ 0 such that 𝒯 (A) ⊂ λW , hence ⟨𝒯 , ã ⊗ μ⟩ = μ(𝒯 (a)) ⩽ λ for all ã ⊗ μ ∈ ΩW A . This yields our claim. For our second statement, we shall prove that the sets ΩϒA that generate the strong or the weak operator topology of Lb (E, F), respectively, are weakly compact in Ẽ ⊗ F ˚ . That is, the family of their polars in Lb (E, F) creates a topology consistent with the dual pair (Lb (E, F), Ẽ ⊗ F ˚ ). For this, let (ãi ⊗ μi )i∈ℐ be a net in ΩϒA . Since A is finite and ϒ is weak* compact, there is a subnet (ã ⊗ μj )j∈𝒥 such that a ∈ A and μj converges to μ ∈ ϒ. Thus ã ⊗ μ ∈ ΩϒA and for every 𝒯 ∈ Lb (E, F) we have lim⟨𝒯 , ã ⊗ μj ⟩ = lim μj (𝒯 (a)) = μ(𝒯 (a)) = ⟨𝒯 , ã ⊗ μ⟩. j∈𝒥

j∈𝒥

We conclude that the subnet (ã ⊗μj )j∈𝒥 converges to ã ⊗μ ∈ ΩϒA in the weak topology of this duality, thus completing our argument. Our final assertion that L(E, F) is a dense subspace of Lb (E, F) in the strong topology and, therefore, also in the weak operator topology follows directly from the observation (ii) from above: Only the zero element in Ẽ ⊗ F ˚ , that is, the dual of Lb (E, F) in the strong operator topology vanishes on all operators in L(E, F). The Hahn–Banach theorem then renders our claim. Proposition 1.3.1 shows in particular that the weak operator topology on Lb (E, F) is indeed the weak topology of the duality (Lb (E, F), Ẽ ⊗ F ˚ ). 1.3.2 Corollary. If Lb (E, F) is endowed with the weak or the strong operator topology, then for every linear functional ω ∈ Lb (E, F)˚ there are a1 , . . . , an ∈ E and μ1 , . . . , μn ∈ F ˚ such that n

ω(𝒯 ) = ∑ μi (𝒯 (ai )) i=1

1.3 Compactness in spaces of linear operators

� 23

for all 𝒯 ∈ Lb (E, F). Proof. We have Lb (E, F)˚ = Ẽ ⊗F ˚ by Proposition 1.3.1. Thus ω = ∑ni=1 ãi ⊗μi for ãi ∈ Ẽ and μi ∈ F ˚ . If we choose ai ∈ ãi , then n

n

i=1

i=1

ω(𝒯 ) = ⟨𝒯 , ∑ ãi ⊗ μi ⟩ = ∑ μi (𝒯 (ai )) holds for all 𝒯 ∈ Lb (E, F), our claim. Compact and weakly compact operators An operator in Lb (E, F) is called compact (or weakly compact) if it maps bounded subsets of E into relatively compact (or relatively weakly compact) subsets of F, that is, the closure in the topology of F (or in the weak topology of F) of the image of a bounded set is compact. Because the balanced convex hull of a bounded set in E is again bounded, it suffices to verify these conditions for balanced convex subsets of E. Consequently, since the closures of a convex set in the given and in the weak topologies of F coincide and since compactness implies weak compactness, this reflection shows in particular that every compact operator is also weakly compact. The compact and the weakly compact operators form respective subspaces of Lb (E, F). We observe the following. 1.3.3 Proposition. Let τ be a polar topology of the dual pair (F, F ˚ ) on F. (a) Every weakly compact operator in Lb (E, F) is also bounded if F is endowed with τ. (b) If every subset of F that is bounded with respect to τ is relatively weakly compact in the given topology, then every operator 𝒯 ∈ Lb (E, F) that is also bounded if F is endowed with τ, is weakly compact. Proof. (a) Suppose that 𝒯 ∈ Lb (E, F) is weakly compact and let A be a bounded subset of E. Then 𝒯 (A) is relatively compact in F with respect to its weak topology. Thus the polar of T(A) in F ˚ is a neighborhood consistent with the duality (F, F ˚ ). Let U be a closed convex neighborhood in F for the polar topology τ. Then its polar U ˝ is bounded in F ˚ . Thus there is λ > 0 such that U ˝ ⊂ λT(A)˝ . This yields T(A) ⊂ λU, and our claim follows. (b) Suppose that τ satisfies our assumption and that 𝒯 ∈ Lb (E, F) is also bounded if F is endowed with τ. Let A be a bounded subset of E. Then 𝒯 (A) is bounded in F with respect to τ, hence relatively weakly compact in the given topology of F. If F is semireflexive, that is, if the canonical embedding of F into F ˚˚ is onto (see IV.5 in [56]), then the condition on τ in 1.3.3(b) holds with τ for the given topology of F (see IV.5.5 in [56]). Consequently, all operators in Lb (E, F) are weakly compact in this case.

24 � 1 Function spaces In general, no straightforward description appears to be available for the dual of Lb (E, F) under the uniform operator topology. However, we list the following useful observation concerning its second dual, that is, the dual of Lb (E, F)˚ endowed with the strong topology from its duality with Lb (E, F). 1.3.4 Proposition. If Lb (E, F) is endowed with the uniform operator topology, then for every functional Ω ∈ Lb (E, F)˚˚ there is an operator 𝒯Ω ∈ Lb (E, F ˚˚ ), where F ˚˚ is endowed with its weak* topology, such that ̃ ⊗ μ) 𝒯Ω (a)(μ) = Ω(a for all a ∈ E and μ ∈ F ˚ , that is, ã ⊗ μ ∈ Ẽ ⊗ F ˚ ⊂ Lb (E, F)˚ . Proof. Following Proposition 1.3.1, the dual Lb (E, F)˚ of Lb (E, F) with respect to the uniform operator topology contains Ẽ ⊗ F ˚ as a subspace. Let Ω ∈ Lb (E, F)˚˚ . There is a neighborhood B˝ in Lb (E, F)˚ , the polar of a balanced bounded subset B in Lb (E, F) such that |Ω(Θ)| ⩽ 1 for all Θ ∈ B˝ , that is, for all Θ ∈ Lb (E, F)˚ such that |Θ(T)| ⩽ 1 for all T ∈ B. If A is a bounded subset of E, we note that BA = {T(a) | a ∈ A and T ∈ B} is a balanced bounded subset of F, since for every W ∈ 𝒲 the uniform neighborhood 𝒰AW in Lb (E, F) absorbs B. Its polar B˝A is therefore a neighborhood in the strong topology of F ˚ , and for every μ ∈ B˝A and a ∈ A we have |(ã ⊗ μ)(T)| = |μ(T(a))| ⩽ 1 for all T ∈ B. Thus ã ⊗ μ ∈ B˝ and |Ω(ã ⊗ μ)| ⩽ 1 by the above. Next, for every fixed element a ∈ E we associate with Ω the linear functional Ωa on F ˚ setting Ωa (μ) = Ω(ã ⊗ μ) for all μ ∈ F ˚ . Then Ωa is an element of F ˚˚ . Surely, if μ is an element of the neighborhood B˝{a} , then |Ωa (μ)| = |Ω(ã ⊗ μ)| ⩽ 1 by the above. The mapping 𝒯Ω such that 𝒯Ω (a) = Ωa is therefore a linear operator from E into F ˚˚ . Moreover, this operator is bounded and, therefore, an element of Lb (E, F ˚˚ ), provided that F ˚˚ is endowed with its weak* topology. Indeed, let A be a bounded subset of E and let μ ∈ F ˚ , that is, the dual of F ˚˚ . The polar B˝A of BA is a neighborhood in F ˚ and, therefore, absorbs μ, that is, μ ∈ ρB˝A for some ρ ⩾ 0. Thus |μ(Ωa )| = |Ω(ã ⊗ μ)| ⩽ ρ for all a ∈ A by the above. Consequently, the set {𝒯Ω (a) | a ∈ A} is bounded in F ˚˚ as claimed. Proposition 1.3.4 is of particular interest in the case that F is reflexive, that is, if the canonical embedding of F into F ˚˚ is both onto and an isomorphism (see IV.5 in [56]). If combined with the inverse of the embedding of F into F ˚˚ , the operator 𝒯Ω becomes F-valued and bounded with respect to the weak, hence also the given topology of F. That is, we have 𝒯Ω ∈ Lb (E, F) in this case. We go forward to investigate spaces of continuous linear operators and for our purposes are notably interested in identifying compact subsets of Lb (E, F). We consider both the strong and weak operator topologies. We note that any subset of L(E, F) that

1.3 Compactness in spaces of linear operators

� 25

is relatively compact in L(E, F) is also relatively compact in Lb (E, F). The reverse conclusion does however not hold true in general. 1.3.5 Proposition. If H ⊂ Lb (E, F) is relatively compact in the weak operator topology, then it is bounded in the uniform operator topology. Proof. Suppose that H ⊂ Lb (E, F) is relatively compact in the weak operator topology and let UAW be a neighborhood in the uniform operator topology. The polar H˝ of H in Ẽ ⊗F ˚ is a neighborhood consistent with the duality (Lb (E, F), Ẽ ⊗F ˚ ), whereas (UAW )˝ ̃ ˚ since U W is a neighborhood in a polar topology of this duality (see is bounded in E⊗F A Proposition 1.3.1). Thus (UAW )˝ ⊂ λH˝ for some λ > 0 and, therefore, H ⊂ λUAW . For compactness in the weak operator topology, a complete characterization is available in case that F is reflexive using Proposition 1.3.4. 1.3.6 Proposition. Suppose that F is reflexive. A subset of Lb (E, F) is relatively compact in the weak operator topology if and only if it is bounded with respect to the uniform operator topology. Proof. In order to verify that a uniformly bounded subset B of Lb (E, F) is relatively compact in the weak operator topology, it suffices to verify that its polar in Ẽ ⊗ F ˚ is consistent with the duality (Lb (E, F), Ẽ ⊗ F ˚ ). That is, we shall verify that every linear functional on Ẽ ⊗ F ˚ that is bounded on B˝ is indeed an element of Lb (E, F). Let Ω be such a functional. Since B is bounded with respect to the uniform operator topology, its polar is a neighborhood in the strong topology of Lb (E, F)˚ , the dual of Lb (E, F) with respect to the uniform topology, and we have Ω ∈ Lb (E, F)˚˚ . According to Proposition 1.3.4, there is 𝒯Ω ∈ Lb (E, F ˚˚ ), where F ˚˚ is endowed with its weak* topology, such that 𝒯Ω (a)(μ) = Ω(ã ⊗ μ) for all a ∈ E and μ ∈ F ˚ . Since F is reflexive, we have 𝒯Ω ∈ Lb (E, F) by the observation following Proposition 1.3.4, and our claim follows. The general case is however less accessible. We shall in the sequel restrict ourselves to the space L(E, F) of continuous linear operators rather than bounded ones. The following proposition establishes a useful sufficient criterion for compactness in the strong and weak operator topologies. A subset H of L(E, F) is equicontinuous if for every W ∈ 𝒲 there is V ∈ 𝒱 such that T(V ) ⊂ W for all T ∈ H. 1.3.7 Proposition. If H ⊂ L(E, F) is equicontinuous, closed in the strong (or weak) operator topology of L(E, F) and if for every a ∈ E the set {T(a) | T ∈ H} is relatively compact (or relatively weakly compact) in F, then H is compact in the strong (or weak) operator topology of L(E, F). Proof. Let (Ti )i∈ℐ be an ultranet in H. For every a ∈ E, the ultranet (Ti (a))i∈ℐ is contained in the relatively compact (or relatively weakly compact) subset {T(a) | T ∈ H} of F and, therefore, converges toward an element T(a) ∈ F. The mapping a 󳨃→ T(a) : E → F is obviously linear. Because H is equicontinuous, given W ∈ 𝒲 there is V ∈ 𝒱 such that Ti (V ) ⊂ W for all i ∈ ℐ . Thus T(a) ∈ W whenever a ∈ V , hence

26 � 1 Function spaces T(V ) ⊂ W and, therefore, of T ∈ L(E, F). Furthermore, our construction implies that (Ti )i∈ℐ converges to T in the strong (or weak) operator topology. Thus T ∈ H, which was supposed to be closed. Hence every ultranet in H is convergent, and our claim follows. A locally convex topological vector space E is barreled (see II.7 in [56]) if every barrel, that is, every closed absorbing balanced convex subset of E, is a neighborhood of the origin. This property holds if and only if E carries the strong topology β(E, E ˚ ) of the dual pair (E, E ˚ ). Every Banach space is barreled. 1.3.8 Corollary. If E is barreled, then H ⊂ L(E, F) is compact in the strong (or weak) operator topology of L(E, F) if and only if it is closed in the strong (or weak) operator topology of L(E, F) and for every a ∈ E the set {T(a) | T ∈ H} is relatively compact (or relatively weakly compact) in F. Proof. The necessity of the conditions for the compactness of H is obvious. On the other hand, if E is barreled, H is closed and for all a ∈ E the sets {T(a) | T ∈ H} are relatively compact, then for every W ∈ 𝒲 the set U = ⋂T∈H T −1 (W ) is closed, convex and absorbing, hence a barrel in E. The latter follows since for every a ∈ E there is λ ⩾ 0 such that {T(a) | T ∈ H} ⊂ λW , hence a ∈ λU. Since E is barreled, there is V ∈ 𝒱 such V ⊂ U. Thus T((1/2)V ) ⊂ (1/2)W ⊂ W for all T ∈ H, and H is understood to be equicontinuous. Our claim follows with Proposition 1.3.7. For a subset U of E and a convex subset S of F, we denote by LSU (E, F) the subset of operators T ∈ L(E, F) such that T(U) ⊂ S. Clearly, LSU (E, F) is convex, and closed in the weak operator topology of L(E, F) provided that S is closed in F. The balanced convex hull of a subset A of a locally convex topological vector space was defined in Section 1.1 and is given by n n 󵄨󵄨 bconv(A) = {∑ αi ai 󵄨󵄨󵄨 ai ∈ A, ∑ |αi | ⩽ 1}. 󵄨 i=1

i=1

It is the smallest balanced convex set containing A and coincides with the convex hull of its balanced hull, that is, the convex hull of the set bal(A) = ⋃|γ|⩽1 γA. If A is convex and relatively compact, so is bconv(A). For the latter, we observe that bconv(A) is a subset of (A − A) in the real, and of (A − A) + i(A − A) in the complex case, hence is also relatively compact, since multiples and sums of relatively compact sets are known to be relatively compact (Lemma III.7 in [45]). There is also a largest balanced subset of A, called its balanced core and given by

1.3 Compactness in spaces of linear operators

⋂|γ|⩾1 γA,

bcore(A) = {

H,

� 27

if 0 ∈ A, if 0 ∉ A.

If A is convex, compact or relatively compact, so is bcore(A). 1.3.9 Corollary. Let U be a subset of E and let S be a closed convex subset of F. If either U is a neighborhood of 0 ∈ E and bcore(S) is compact (or weakly compact), or bconv(U) is a neighborhood of 0 ∈ E and S is compact (or weakly compact), then the set LSU (E, F) = {T ∈ L(E, F) | T(U) ⊂ S} is convex and compact in the strong (or weak) operator topology of L(E, F). The operators in LSU (E, F) are themselves compact (or weakly compact). Proof. Since S is supposed to be closed in F, the set LSU (E, F) is closed in L(E, F) in both the strong and weak operator topology. If U is a neighborhood of 0 ∈ E, then it contains a balanced neighborhood V ∈ 𝒱 . Thus T(V ) ∈ bcore(S) for every operator T ∈ LSU (E, F) and, therefore, LSU (E, F) ⊂ Lbcore(S) (E, F). V We shall verify the criterion of Proposition 1.3.7 for H = Lbcore(S) (E, F) provided that V bcore(S) is compact (or weakly compact). Indeed, H is closed in L(E, F) and given W ∈ 𝒲 there is α > 0 such that bcore(S) ⊂ αW . Hence T((1/α)V ) ⊂ W for all T ∈ H, and H is equicontinuous. Furthermore, for every a ∈ E we have a ∈ αV for some α > 0, hence T(a) ∈ α bcore(S) for all T ∈ H. Consequently, the set {T(a) | T ∈ H} is relatively compact (or relatively weakly compact) in F as required. As a closed subset of H, the set LSU (E, F) is therefore also compact. Finally, for every bounded subset A of E there is α > 0 such that A ⊂ αV . Thus T(A) ⊂ α bcore(S) for all T ∈ LSU (E, F). Since α bcore(S) is relatively compact (or relatively weakly compact), all operators in LSU (E, F) are for that reason compact (or weakly compact). For the second case, that is, if bconv(U) is a neighborhood of 0 ∈ E and S is compact, we observe that LSU (E, F) ⊂ Lbconv(S) (E, F). bconv(U) Since bconv(S) is relatively compact, we may use the first case with bconv(U) in place of U and the closure of bconv(S) in place of S. If S ⊂ F is the polar in F of a subset Π of F ˚ (this will be the case in many of our applications below), then an easy argument using IV.1.3 in [56] shows that ˝

bcore(S) = ⋂ γΠ˝ = ⋂ (γΠ)˝ = ( ⋃ γΠ) = bconv(Π)˝ . |γ|⩾1

|γ|⩽1

|γ|⩽1

28 � 1 Function spaces The last of the above equalities uses the aforementioned fact that the balanced convex hull of Π coincides with the convex hull of bal(Π) = ⋃|γ|⩽1 γΠ, and then the fact that the polar of a set equals the polar of its convex hull. The requirement from Corollary 1.3.9 that bcore(S) is weakly compact therefore implies that its polar in F ˚ , that is, the weak* closure of bconv(Π) (see Theorem IV.1.5 in [56]), is absorbing in F ˚ . 1.3.10 Proposition. If H ⊂ L(E, F) is compact in the strong (or weak) operator topology, ∞ then for any choice of Ti ∈ H and λi ⩾ 0 such that ∑∞ i=1 λi = 1 the series ∑i=1 λi Ti converges in the strong (or weak) operator topology. Its sum is an element of H. Proof. First, assume that 0 ∈ H. For n ∈ ℕ, let σn = ∑ni=1 λi and n

n

i=1

i=1

Sn = ∑ λi Ti = ∑ λi Ti + (1 − σn ) 0 ∈ H, that is, the partial sums of the given operator series. Given a neighborhood 𝒰 in the strong (or weak) operator topology of L(E, F), there is ρ > 0 such that H ⊂ ρ 𝒰 and n0 ∈ ℕ such that 1 − σn ⩽ 1/ρ. Then for all n0 ⩽ n < m we have m

Sm − Sm = ∑ λi Ti ∈ (1 − σn )H ⊂ 𝒰 . i=n+1

Hence (Sn )n∈ℕ is a Cauchy sequence and, therefore, convergent in H. Now for the general case choose T0 ∈ H and set H0 = H−T0 , and Si = Ti −T0 . The series ∑∞ i=1 λi Si then ∞ converges by the preceding to ∑∞ λ T − T ∈ H . Hence λ T ∈ H as claimed. ∑ 0 0 i=1 i i i=1 i i We shall illustrate these results in a few examples. 1.3.11 Examples. (a) If F = ℝ or ℂ, that is, the scalar field of E, then L(E, F) = E ˚ with both the strong and weak operator topologies coinciding with the weak* topology of E ˚ , that is, the topology σ(E ˚ , E). We have Ẽ ⊗ F = Ẽ in this case, and Propõ Recall that sition 1.3.1 confirms that the dual of E ˚ under its weak* topology is E. ̃ E = E, provided that E is Hausdorff. The uniform operator topology on L(E, F), on the other hand, is the strong topology β(E ˚ , E) on E ˚ . The statements of Propositions 1.3.5 and 1.3.6 are of some interest in this case. While every continuous linear functional in E ˚ is bounded, that is, it maps bounded subsets of E into bounded subsets of ℝ or ℂ, the reverse conclusion does not hold in general. It does hold true however, if E is a normed space or, more generally, if E is bornological, that is, if every balanced convex subset of E that absorbs every bounded subset of E is a neighborhood (see II.8 in [56]). In general, we have E ˚ ⊂ Eb˚ , whereby the latter denotes the space of all bounded linear functionals on E. Proposition 1.3.6 says that a subset of Eb˚ is σ(Eb˚ , E)-relatively compact if and only if it β(Eb˚ , E)-bounded, a statement that does not hold true in general, if one replaces Eb˚ by E ˚ .

1.4 Vector space extensions

� 29

If U is a convex neighborhood of 0 ∈ E and S = {z ∈ F | Re (z) ⩽ 1}, then LSU (E, F) = U ˝ . Since bcore(S) is the compact unit ball of ℝ or ℂ, Corollary 1.3.9 recovers the familiar result that U ˝ is weak* compact. (b) If F is semireflexive, that is, if the canonical embedding of F into F ˚˚ is onto, then every closed convex bounded subset S of F is weakly compact. Thus for any neighborhood U of 0 ∈ E the set LSU (E, F) is compact in the weak operator topology. In case that both E and F are normed spaces with the respective unit balls 𝔹E and 𝔹 𝔹F , then the unit ball L𝔹FE (E, F) is compact in the weak operator topology. In case that F is indeed reflexive, the last observation is transcended by the result of Proposition 1.3.6 since Lb (E, F) and L(E, F) coincide in this case. (c) Suppose that F is the dual of a normed space G and endowed with the weak topology of the dual pair (F, G). Then the strong topology β(F, G) on F, that is, the topology of the dual norm, satisfies the assumption of Proposition 1.3.3 by the Banach– Alaoglu theorem. Hence any operator 𝒯 ∈ Lb (E, F) that is bounded with respect to the strong topology of F is weakly compact. If E is also a normed space, then all of these operators are continuous with respect to the respective norm topologies of E and F. For a concrete example, consider an integration space L1 (μ) for G and L∞ (μ) for F. (d) Suppose that both E and F are ordered topological vector spaces with positive cones E+ and F+ , respectively. If U is a neighborhood of 0 ∈ E, which contains E+ , and if S is a closed convex subset S of F such that bcore(S) is weakly compact, then according to the first alternative in Corollary 1.3.9 the set LSU (E, F) is compact in the weak operator topology. Its elements are weakly compact operators, and positive, provided that ⋂λ>0 λS ⊂ F+ . Alternatively, if U is a convex subset of E+ such that bconv(U) is a neighborhood of 0 ∈ E and if S is a weakly compact convex subset of F+ , then according to the second alternative in Corollary 1.3.9 the set LSU (E, F) is compact in the weak operator topology. Its elements are weakly compact operators, and positive, provided that U absorbs all elements of E+ . In an equivalent setup, we may replace the positive cones E+ and F+ by their negative counterparts E− = −E+ and F− = −F+ in all of the above settings.

1.4 Vector space extensions Our upcoming integration theory involving operator-valued measures will require to consider extensions of locally convex topological vector spaces into locally convex cones, a concept developed in [33] and [48]. We shall provide the outlines of this theory as far as it is essential for our progress and globally refer to the above mentioned texts for further details. Locally convex cones are generalizations of locally convex topological vector spaces.

30 � 1 Function spaces Locally convex cones In this vein, a cone is a set 𝒫 endowed with an addition (a, b) 󳨃→ a + b and a scalar multiplication (α, a) 󳨃→ αa for a, b ∈ 𝒫 and real numbers α ⩾ 0. The addition is supposed to be associative and commutative, and there is a neutral element 0 ∈ 𝒫 . For the scalar multiplication, the usual associative and distributive properties hold, that is, α(βa) = (αβ)a, (α + β)a = αa + βa, α(a + b) = αa + αb, 1a = a and 0a = 0 for all a, b ∈ 𝒫 and α, β ⩾ 0. The cancellation law, stating that a + c = b + c implies a = b, however, is not required in general. It holds if and only if the cone 𝒫 can be embedded into a real vector space. An ordered cone 𝒫 carries a reflexive transitive relation ⩽ such that a ⩽ b implies a + c ⩽ b + c and αa ⩽ αb for all a, b, c ∈ 𝒫 and α ⩾ 0. Antisymmetry is not required. Equality on 𝒫 is obviously such an order. The extended real number system ℝ and the collection Conv(E) of nonempty subsets of a vector space E are examples of ordered cones. Both do not satisfy the cancellation law. A full locally convex cone (𝒫 , 𝒱 ) is an ordered cone 𝒫 that contains an abstract neighborhood system 𝒱 , that is, a subset of positive (that is ⩾ 0) elements that is directed downward and closed for multiplication by positive scalars. The elements V of 𝒱 define upper, lower and symmetric neighborhoods for the elements a ∈ 𝒫 by V (a) = {b ∈ 𝒫 | b ⩽ a + V },

(a)V = {b ∈ 𝒫 | a ⩽ b + V }

and V s (a) = V (a) ∩ (a)V , which create the upper, lower and symmetric topologies on 𝒫 , respectively. All elements of 𝒫 are required to be bounded below, that is, for every a ∈ 𝒫 and V ∈ 𝒱 we have 0 ⩽ a + λV for some λ ⩾ 0. They need however not be bounded above. An element a ∈ 𝒫 is called bounded (above) if for every V ∈ 𝒱 there is λ ⩾ 0 such that a ⩽ λV . The presence of unbounded elements constitutes the main difference between locally convex cones and subcones of locally convex topological vector spaces and accounts for much of the richness and subtlety of this setting. The upper, lower and symmetric topologies are each compatible with the algebraic operations, that is, the mappings (a, b) 󳨃→ a + b : 𝒫 2 → 𝒫 and a 󳨃→ αa : 𝒫 → 𝒫 for a fixed α ⩾ 0 are continuous (see Proposition I.1.1 in [48]). Hence the usual limit rules apply for nets in 𝒫 with sums and multiples by positive scalars. The mapping α 󳨃→ αa : ℝ+ → 𝒫 is however assured to be continuous only if the element a ∈ 𝒫 is bounded. Finally, a locally convex cone (𝒫 , 𝒱 ) is a subcone of a full locally convex cone not necessarily containing the abstract neighborhood system 𝒱 . Every locally convex ordered topological vector space over ℝ is a locally convex cone in this sense, as it can be canonically embedded into a full locally convex cone (Example I.2.7 in [33]). We shall further elaborate on the embedding of vector spaces into locally convex cones below. First examples also include the cone 𝒫 = ℝ with the neighborhood system 𝒱 = {ε > 0},

1.4 Vector space extensions

� 31

cones of convex subsets of a locally convex topological vector space and cones of ℝ-valued functions with suitable neighborhood systems. These are indeed the types of locally convex cones that we shall employ for our purposes. We note that the symmetric topology on ℝ is finer than the usual one-point compactification topology on this set. Instead, it is the usual topology of ℝ with +∞ adjoined as an isolated point. Thus convergence toward +∞ requires that this value is taken cofinally. For a wide range of examples of locally convex cones, we refer to Examples I.1.4 in [50]. Subcones of a locally convex cone, if endowed with the inherited neighborhood system are again locally convex cones. A subset A of 𝒫 is called bounded below (or above) if for every V ∈ 𝒱 there is λ ⩾ 0 such that 0 ⩽ a + λV (or a ⩽ λV ) for all a ∈ A, and bounded if it satisfies both. We remark that other than in the case of topological vector spaces the mapping (α, a) 󳨃→ αa : [0, +∞) × 𝒫 → 𝒫 is not guaranteed to be continuous in any of the cone topologies at points (α, a) where the element a ∈ 𝒫 is unbounded (see Proposition I.1.1 in [48]). This shortcoming is at least partially remedied by the use of the slightly coarser relative topologies, which we shall briefly introduce and employ in Section 2.4 below. If (𝒫 , 𝒱 ) and (𝒬, 𝒲 ) are locally convex cones, an operator T : 𝒫 → 𝒬 is called linear if T(a + b) = T(a) + T(b) and T(αa) = αT(a) holds for all a, b ∈ 𝒫 and α ⩾ 0. T is continuous if for every W ∈ 𝒲 there is V ∈ 𝒱 such that T(a) ⩽ T(b) + W whenever a ⩽ b + V for a, b ∈ 𝒫 . By L(𝒫 , 𝒬), we denote the cone of all continuous linear operators from 𝒫 to 𝒬. The operators in L(𝒫 , 𝒬) map bounded subsets of 𝒫 into bounded subsets of 𝒬 and are monotone in the sense that a ≤ b + V for all V ∈ 𝒱 implies that T(a) ≤ T(b) + W for all W ∈ 𝒲 . The dual cone The dual cone 𝒫 ˚ of a locally convex cone 𝒫 is the cone L(𝒫 , ℝ) of all continuous ℝ-valued linear functionals on 𝒫 . The polar V ˝ of a neighborhood V ∈ 𝒱 consists of all μ ∈ 𝒫 ˚ such that μ(a) ⩽ μ(b) + 1 whenever a ⩽ b + V for a, b ∈ 𝒫 . The polars of neighborhoods are known to be compact in the weak* topology of 𝒫 ˚ (Proposition II.2.4 in [33]), that is, the topology of pointwise convergence of the elements of 𝒫 ˚ considered as ℝ-valued functions on 𝒫 . In this context, ℝ is meant to carry its usual one-point compactification topology. We have 𝒫 = ⋃V . ˚

˝

v∈𝒱

If 𝒫 is indeed a locally convex topological vector space E, then 𝒫 ˚ coincides with its ˚ real dual Eℝ . If E is ordered and if this order is implemented in its embedding into a full cone (see I.2.7 in [33]), then 𝒫 ˚ consists of the real parts of the functionals in E+˚ , that is, 𝒫 ˚ is the cone of all real-valued monotone real-linear functionals on E.

32 � 1 Function spaces Locally convex cones yield a rich duality theory, including powerful Hahn–Banachtype extension and separation theorems (see [33, 48] and [50]), some of which we shall use. We recall some facts about sublinear and superlinear functionals on locally convex cones. Sub and superlinear functionals A sublinear functional on a locally convex cone 𝒫 is a mapping p : 𝒫 → ℝ such that p(λa) = λp(a)

and p(a + b) ⩽ p(a) + p(b)

for all a, b ∈ 𝒫 and λ ⩾ 0. A superlinear functional on 𝒫 is a mapping q : 𝒫 → ℝ such that −q is sublinear. An extended superlinear functional is allowed to take both values +∞ and −∞. The inequality q(a + b) ⩾ q(a) + q(b) is expected to hold if the sum on the right-hand side is defined in ℝ. The Sandwich theorem for locally convex cones (Theorem 3.1 in [50]) states the following. 1.4.1 Theorem. For a sublinear functional p : 𝒫 → ℝ and an extended superlinear functional q : 𝒫 → ℝ, there exists a linear functional μ ∈ 𝒫 ˚ such that q ⩽ μ ⩽ p if and only if there is V ∈ 𝒱 such that q(a) ⩽ p(a) + 1 whenever

a ⩽b+V

for a, b ∈ 𝒫 . The relation q ⩽ μ ⩽ p is meant pointwise on 𝒫 . With suitable insertions for p and q, this result guarantees in particular that every continuous linear functional on a subcone of 𝒫 can be extended to a continuous linear functional on 𝒫 . We shall also make use of the powerful Range theorem (Theorem 5.1 in [50]). 1.4.2 Theorem. Let p and q be sublinear and extended superlinear functionals on 𝒫 such that there is at least one linear functional μ ∈ 𝒫 ˚ satisfying q ⩽ μ ⩽ p. Then for all a ∈ 𝒫 , sup μ(a) = sup inf{p(b) − q(c) | b, c ∈ 𝒫 , q(c) ∈ ℝ, a + c ⩽ b + V },

μ∈𝒫 ˚ q⩽μ⩽p

V ∈𝒱

and for all a ∈ 𝒫 such that μ(a) is finite for at least one μ ∈ 𝒫 ˚ satisfying q ⩽ μ ⩽ p, inf μ(a) = inf sup{q(c) − p(b) | b, c ∈ 𝒫 , p(b) ∈ ℝ, c ⩽ a + b + V }.

μ∈𝒫 ˚ q⩽μ⩽p

V ∈𝒱

These results apply of course also to a locally convex topological vector space (E, 𝒱 ), where a ⩽ b + V for a, b ∈ E and V ∈ 𝒱 reads as a ∈ b + V , that is, a − b ∈ V . The continuous linear functionals in consideration then are real-valued and real-linear ˚ on E, that is, elements of Eℝ . The classical Hahn–Banach theorem, although purely

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algebraic, is an immediate consequence of Theorem 1.4.1. Indeed, if p is a real-, that is, finite-valued sublinear functional on a real or complex vector space E, then the set V = {a ∈ E | p(γa) ⩽ 1 for all |γ| = 1} is balanced, convex and absorbing in E. The first two properties are obvious. For the latter, given a ∈ E, let ρ = max{p(a), p(−a)} in the real and ρ = max{p(a), p(−a), p(ia), p(−ia)} in the complex case. Then for any γ in the scalar field such that |γ| = 1, one easily calculates that p(γa) ⩽ 2ρ. Thus a ∈ 2ρV , our claim. Hence the neighborhood system 𝒱 = {ρV | ρ > 0} defines a locally convex topology on E. We use Theorem 1.4.1 with the given sublinear functional p, any ℝ-valued superlinear functional q such that q ⩽ p, and the neighborhood V ∈ 𝒱 . Surely, for a − b ∈ V we have q(a) ⩽ p(a) ⩽ p(b) + p(a − b) ⩽ p(b) + 1. Thus by 1.4.1 there is a linear functional μ ∈ Eℝf such that q ⩽ μ ⩽ p. In particular, given a subspace F of E and a linear functional ν ∈ Fℝf that is dominated by p on F, we set q(a) = ν(a) for a ∈ F, and q(a) = −∞ else. Then q ⩽ μ means that μ coincides with ν on F, hence is an extension of ν. This is the version of the Hahn–Banach theorem most popular in standard texts (see, e. g., Theorem II.3.10 in [26] or Theorem II.3.2 in [56]). We note that these particular assertions hold no longer true in general if p is allowed to take the value +∞ (see Example 1.4.8 below). In fact, Theorem 1.4.1 implies that an ℝ-valued sublinear functional p on a vector space E dominates a linear functional in Eℝf if and only if it is bounded below on some balanced, convex and absorbing subset of E (see Theorem 1.8 in [2]). Indeed, if p dominates a linear functional, the latter is bounded below on some balanced, convex and absorbing subset of E, and so is p. Conversely, if p(a) ⩾ −1 for all elements a of such a subset V of E, this subset generates a locally convex topology on E, and we use 1.4.1 with p and the superlinear functional q defined as q(0) = 0 and q(a) = −∞ for all a ≠ 0. Then for a − b ∈ V we have either q(a) = −∞ or a = 0 and p(b) ⩾ −1, hence q(0) = 0 ⩽ p(b) + 1, and Theorem 1.4.1 yields our claim. In case that p is finite-valued, since p(a) ⩾ −p(−a) holds for all a ∈ E, this condition is satisfied with the set V = {a ∈ E | p(γa) ⩽ 1 for all |γ| = 1} from above. Convex sets and sublinear functionals Let (E, F) be a dual pair of real or complex vector spaces endowed with a nondegenerate real- or complex-valued bilinear form ⟨ , ⟩, and let 𝒱 and 𝒲 be neighborhood systems for polar topologies on E and F, respectively, both consistent with this duality. There is a canonical correspondence between nonempty convex subsets of E and lower semicontinuous sublinear functionals on F, and vice versa. For A ∈ Conv(E), we set pA (c) = sup{Re⟨a, c⟩ | a ∈ A}

34 � 1 Function spaces for all c ∈ F. Clearly, pA is sublinear and lower semicontinuous on F, and we have pαA = αpA and pA+B = pA + pB for A, B ∈ Conv(E) and α ⩾ 0. For A ∈ Conv(E) and scalars α in ℝ or in ℂ in general, we have pαA (c) = sup{Re⟨αa, c⟩ | a ∈ A} = sup{Re⟨a, αc⟩ | a ∈ A} = pA (αc) for all c ∈ F. Conversely, given a lower semicontinuous sublinear functional p on F, we set Ap = {a ∈ E | Re⟨a, c⟩ ⩽ p(c) for all c ∈ F}. It follows from Theorem 1.4.1 that Ap is not empty, hence an element of Conv(E). Indeed, we use F in place of 𝒫 , the given sublinear functional p on F, and set q(0) = 0 and q(c) = −∞ for 0 ≠ c ∈ F. Due to the lower semicontinuity of p at 0 ∈ F, there is W ∈ 𝒲 such that p(c) ⩾ −1 for all c ∈ W . Then c − d ∈ W for c, d ∈ F = E ˚ implies that q(c) ⩽ p(d) + 1. This is obvious for c ≠ 0, and for c = 0 it follows from the above. Hence there is a linear functional a ∈ E = F ˚ such that Re⟨a, c⟩ ⩽ p(c) for all c ∈ F. That is a ∈ Ap . We shall confirm that pAp = p. Surely, the inequality pAp ⩽ p, pointwise on F, is evident. The reverse inequality is however less trivial and uses the first statement in Theorem 1.4.2 with the same insertions for p and q from above. We have a ∈ Ap if and only if q(d) ⩽ Re⟨a, d⟩ ⩽ p(d) for all d ∈ F. That is to say, pAp (c) = sup Re⟨a, c⟩ = sup inf{p(d) | d ∈ F, c − d ∈ W } a∈Ap

W ∈𝒲

holds by 1.4.2 for every c ∈ F. As p is supposed to be lower semicontinuous on F, given c ∈ F and any α < p(c) there is a neighborhood W ∈ 𝒲 such that p(d) > α for all d ∈ c + W , that is, c − d ∈ −W = W . Hence inf{p(d) | d ∈ F, c − d ∈ W } ⩾ α. This proves pAp (c) ⩾ p(c), and our claim follows. Furthermore, according to the Hahn–Banach separation theorem (see Theorem II.9.2 in [56]), for every A ∈ Conv(E) and b ∉ A, the topological closure of A in E, there is c ∈ F such that Re⟨b, c⟩ > sup{Re⟨a, c⟩ | a ∈ A}. Therefore, we have b ∈ A if and only if Re⟨b, c⟩ ⩽ sup{Re⟨a, c⟩ | a ∈ A} = pA (c) holds for all c ∈ F. We infer that pA ⩽ pB holds for A, B in Conv(E) if and only if A ⊂ B, and that pA = pB if and only if A = B. 1.4.3 Proposition. Let (E, F) be a dual pair, and let 𝒱 and 𝒲 be neighborhood systems for polar topologies on E and F, respectively, both consistent with the duality.

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(a) The embedding A 󳨃→ pA from Conv(E) into the cone of all lower semicontinuous sublinear functionals on F is linear, onto and pA ⩽ pB holds for A, B in Conv(E) if and only if A ⊂ B. (b) For A ∈ Conv(E), the sublinear functional pA is real-valued on F if and only if A is bounded. (c) A sublinear functional on F is continuous if and only if it is upper semicontinuous at 0 ∈ F, that is, if and only if it is bounded above on some neighborhood W ∈ 𝒲 . (d) For A ∈ Conv(E), the sublinear functional pA is continuous in the Mackey topology τ(F, E) of F if and only if A is relatively compact in the weak topology σ(E, F) of E. Proof. Part (a) was argued before. Part (b) is obvious since the topology of E is supposed to be consistent with the duality (E, F). Therefore, boundedness of sets in Conv(E) in the given and in the weak topology σ(E, F) of E coincides. For (c), let p be a sublinear functional on F. Clearly, continuity of p implies upper semicontinuity at 0 ∈ F. Hence there is W ∈ 𝒲 such that p is bounded above on W . Conversely, suppose that there is W ∈ 𝒲 such that p(c) ⩽ 1 for all c ∈ W . Let c ∈ F and ε > 0. For every d ∈ c + εW , that is, d − c ∈ εW , we have both p(d) ⩽ p(c) + p(d − c) ⩽ p(c) + ε and p(c) ⩽ p(d) + p(c − d) ⩽ p(d) + ε since W is supposed to be balanced. Thus p is continuous at c. For Part (d), suppose that A ∈ Conv(E) is relatively compact in the weak topology σ(E, F) of E. Then the polar A˝ of A is a neighborhood in the Mackey topology τ(F, E) of F and sup{pA (c) | c ∈ A˝ } = sup{Re⟨a, c⟩ | a ∈ A, c ∈ A˝ } ⩽ 1. Thus pA is bounded above on A˝ and, therefore, continuous in the Mackey topology τ(F, E) of F by Part (c). Conversely, if pA is τ(F, E)-continuous on F, then there is a balanced σ(E, F)-compact set B ∈ Conv(E) such that pA (c) ⩽ 1 for all c ∈ B˝ . Thus B˝ ⊂ A˝ and A ⊂ B˝˝ = B. Hence A is relatively σ(E, F)-compact. Proposition 1.4.3(c) implies in particular that a sublinear functional, if dominated by a continuous one, is itself continuous. 1.4.4 Corollary. Let E be a Hausdorff locally convex topological vector space. A sublinear functional on E is lower semicontinuous in the given topology of E if and only if it is lower semicontinuous in the weak topology σ(E, E ˚ ). Proof. Clearly, lower semicontinuity in the weak topology implies lower semicontinuity in the given topology. For the converse, we apply Proposition 1.4.3(a) with the dual pair

36 � 1 Function spaces (E ˚ , E), the given topology on E and the weak* topology, that is, σ(E ˚ , E), on E ˚ . If a sublinear functional p on E is lower semicontinuous in the given topology, then according to Proposition 1.4.3(a) there is a convex subset Ω of E ˚ such that p(a) = sup{Re μ(a) | μ ∈ Ω} for all a ∈ E. As the pointwise supremum of σ(E, E ˚ )-continuous functions on E, the functional p itself is therefore σ(E, E ˚ )-lower semicontinuous. We remark that the ordered cone Conv(E) contains the supremum of any family {Ai }i∈ℐ of its elements. Indeed we observe that ⋁ Ai = conv(⋃ Ai ),

i∈ℐ

i∈ℐ

that is, the convex hull of the union of the sets Ai . We have p ⋁i∈ℐ Ai = ⋁ pAi , i∈ℐ

that is, the pointwise supremum on F of the sublinear functionals pAi . The distributive law is easily verified, that is to say, ⋁ (Ai + B) = ⋁ Ai + B

i∈ℐ

i∈ℐ

holds for all B ∈ Conv(E). Similar statements are generally not available for infima in Conv(E), even if the family {Ai }i∈ℐ is directed downward and its intersection is not empty.

The locally convex cone 𝒫E We shall utilize these concepts to develop our approach to integration theory. The following extensions of a vector space are of our particular interest. Given a locally convex topological vector space (E, 𝒱 ), the family Conv(E) of all nonempty convex subsets of E endowed with its usual addition and multiplication by nonnegative scalars forms a cone for which the cancellation does not hold in general. It is canonically ordered by set inclusion. A subcone 𝒫E of Conv(E) that contains all singleton sets {a} for a ∈ E is called a cone extension of E. It is a full cone extension provided that 𝒱 ⊂ 𝒫E . With the set inclusion as its canonical order, we use the vector space neighborhoods in 𝒱 to generate a corresponding neighborhood system for 𝒫E . That is, we set A ∈ V (B) if

A⩽B+V

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for A, B ∈ 𝒫E and V ∈ 𝒱 . Clearly, for every A ∈ Conv(E) and V ∈ 𝒱 there is λ ⩾ 0 such that {0} ⩽ A+λV . In this way, (𝒫E , 𝒱 ) becomes a locally convex cone as previously introduced. It is a full locally convex cone provided that 𝒱 ⊂ 𝒫E . The embedding a 󳨃→ {a} : E → 𝒫E is one-to-one and preserves the algebraic operations of E and 𝒫E . The upper, lower and symmetric topologies of 𝒫E all coincide with the given topology of E on the image of this embedding. In this manner, (E, 𝒱 ) becomes a real-linear subspace of 𝒫E . We shall identify the elements a of E with the singleton sets {a} ∈ 𝒫E . There is also a multiplication by negative numbers in Conv(E), which does however not result in additive inverses (see Section 1.1). For every linear functional ν ∈ E ˚ , the mapping C 󳨃→ sup{Re μ(a) | a ∈ C} : 𝒫E → ℝ defines an element of the dual cone 𝒫E˚ of 𝒫E . The locally convex cone 𝒬F We shall also make use of a second cone extension of a vector space, called its lattice cone extension. For this, let (F, 𝒲 ) be a Hausdorff locally convex topological vector space and set 󵄨󵄨 φ is sublinear and bounded below 󵄨 }. 󵄨󵄨 on the polars of all neighborhoods in 𝒲

𝒬F = { φ : F → ℝ 󵄨󵄨󵄨 ˚

Endowed with the pointwise addition and multiplication by nonnegative scalars for functions on F ˚ , together with the pointwise order on F ˚ , 𝒬F forms an ordered cone. With every neighborhood W ∈ 𝒲 , we associate an ℝ+ -valued sublinear functional σW ∈ 𝒬F defined as σW (μ) = sup{Re μ(c) | c ∈ W } = inf{λ ⩾ 0 | μ ∈ λW ˝ },

for μ ∈ F ˚ . (As usual, we set inf H = +∞.) The second equality in this expression is easily verified. For this, we abbreviate α = sup{Re μ(c) | c ∈ W } and β = inf{λ ⩾ 0 | μ ∈ λW ˝ }. We have Re μ(c) ⩽ λ whenever c ∈ 𝒲 and μ ∈ λW ˝ . Thus α ⩽ β. The inverse inequality is obvious if α = +∞. Otherwise, we observe that μ ∈ ρW ˝ holds for every ρ > α. Hence β ⩽ ρ and, therefore, β ⩽ α, our claim. We also notice that σW (μ) > 0 at all nonzero elements μ ∈ F ˚ , which follows from the fact that W is absorbing in F. The elements of W ∈ 𝒲 then define a corresponding neighborhood system on 𝒬F , that is to say, we set φ ∈ W (ψ)

if

φ ⩽ ψ + σW

38 � 1 Function spaces for φ, ψ ∈ 𝒬F and W ∈ 𝒲 . In this way, (𝒬F , 𝒲 ) becomes a full locally convex cone. Indeed, every φ ∈ 𝒬F is bounded below on the polar of every neighborhood W ∈ 𝒲 . Hence there is λ ⩾ 0 such that φ(μ) ⩾ −λ for all μ ∈ W ˝ . We claim that 0 ⩽ φ(μ) + λσW (μ) holds for all ν ∈ F ˚ . There is nothing to prove if σW (μ) = +∞. Otherwise, we have μ ∈ ρW ˝ for all ρ > σW (μ). Thus φ(μ) ⩾ −λρ, that is, 0 ⩽ φ(μ) + λρ. Our claim follows, and we infer that 0 ⩽ φ + λσW . Hence φ ∈ 𝒬F is bounded below as is required for the elements of a locally convex cone. We further note that φ ⩽ ψ holds for φ, ψ ∈ 𝒬F if and only if φ ⩽ ψ + σW for all W ∈ 𝒲 , and that φ ⩽ ψ + σW holds if and only if φ(μ) ⩽ ψ(μ) + 1 for all μ ∈ W ˝ . Indeed, the former obviously implies the latter, and if the latter holds and if μ ∈ λW ˝ for λ > 0, then (1/λ)μ ∈ W ˝ , hence φ((1/λ)μ) ⩽ ψ((1/λ)μ) + 1, that is, φ(μ) ⩽ ψ(μ) + λ. The symmetric topology of 𝒬F is the topology of uniform convergence on the polars in F ˚ of all neighborhoods W ∈ 𝒲 . In other words, convergence of a net (φi )i∈ℐ in 𝒬F toward an element φ ∈ 𝒬F in the symmetric topology requires that for every W ∈ 𝒲 there is i0 ∈ ℐ such that for every μ ∈ W ˝ we have either |φi (μ) − φ(μ)| ⩽ 1 or φi (μ) = φ(μ) = +∞ for all i ⩾ i0 . Any Cauchy net (φi )i∈ℐ in 𝒬F with respect to the symmetric topology is therefore a pointwise Cauchy net on F ˚ with values in ℝ endowed with its symmetric topology. Accordingly, it is pointwise convergent to an ℝ-valued function φ on F ˚ , which is easily seen to be sublinear and bounded below on the polars of all neighborhoods in 𝒲 , hence an element of 𝒬F . The convergence of (φi )i∈ℐ to φ is indeed uniform on the polars of all neighborhoods in 𝒲 and, therefore, refers to the symmetric topology of 𝒬F . We summarize. 1.4.5 Proposition. The locally convex cone 𝒬F is complete in its symmetric topology. If F is a normed space with unit ball 𝔹, then the dual unit ball 𝔹˝ in F ˚ , which determines the topology of 𝒬F also defines the strong topology β(F ˚ , F), that is, the dual norm topology, of F ˚ . A sublinear functional φ on F ˚ is an element of 𝒬F if and only if it is bounded below on 𝔹˝ , that is, if and only if it is β(F ˚ , F)-lower semicontinuous at 0 ∈ F ˚ . Because the topology β(F ˚ , F) is consistent with the duality (F ˚ , F ˚˚ ), according to Corollary 1.4.4 a sublinear functional φ on F ˚ is β(F ˚ , F)-lower semicontinuous on all of F ˚ if and only if it is lower semicontinuous in the weak topology σ(F ˚ , F ˚˚ ). According to Proposition 1.4.3(c), a sublinear functional φ on F ˚ is β(F ˚ , F)-continuous on F ˚ if and only if it is bounded above on 𝔹˝ . 1.4.6 Proposition. Let F be a normed space. (a) The subcone of all β(F ˚ , F)-continuous functionals on F ˚ is closed in the lower topology of 𝒬F . (b) The subcone of all β(F ˚ , F)-lower semicontinuous functionals on F ˚ is closed in the symmetric topology of 𝒬F . Proof. Suppose that F is a normed space with unit ball 𝔹. Both sets 𝒬c of β(F ˚ , F)continuous and 𝒬l of β(F ˚ , F)-lower semicontinuous sublinear functionals on F ˚

1.4 Vector space extensions

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form subcones of 𝒬F . First, let φ ∈ 𝒬F be an element of the lower closure of 𝒬c . There is ψ ∈ 𝒬c such that ψ ∈ (φ)𝔹, that is, φ ⩽ ψ + σ𝔹 , and since ψ is β(F ˚ , F)continuous, there is λ ⩾ 0 such that ψ(μ) ⩽ λ for all μ ∈ 𝔹˝ . Thus φ(μ) ⩽ ψ(μ) + σ𝔹 (μ) ⩽ λ + 1 for all μ ∈ 𝔹˝ . Hence φ is also β(F ˚ , F)-continuous by 1.4.3(c), our claim. Now let φ ∈ 𝒬F be an element of the symmetric closure of 𝒬l . We shall verify that φ is also lower semicontinuous on F ˚ with respect to β(F ˚ , F). For this, let μ ∈ F ˚ and α < φ(μ). Let λ > 0 such that μ ∈ λ𝔹˝ and choose ε > 0 such that β = α + 3ελ < φ(μ). There is ψ ∈ 𝒬l such that ψ ∈ (ε𝔹)s (φ), that is, both φ ⩽ ψ + εσ𝔹

and

ψ ⩽ φ + εσ𝔹 .

We infer that β < φ(μ) ⩽ ψ(μ) + εσ𝔹 (μ) ⩽ ψ(μ) + ελ, hence β − ελ < ψ(μ). Because ψ is β(F ˚ , F)-lower semicontinuous at μ, we find 0 < δ ⩽ λ such that β − ελ < ψ(ν) ⩽ φ(ν) + εσ𝔹 (ν) for all ν ∈ μ + δ𝔹˝ . Since μ + δ𝔹˝ ⊂ 2λ𝔹˝ , we have σ𝔹 (ν) ⩽ 2λ and, therefore, α = β − 3ελ ⩽ φ(ν) for all ν ∈ μ + δ𝔹˝ . This demonstrates that φ is lower semicontinuous at μ, hence on all of F ˚ , in the strong topology β(F ˚ , F). Our claim follows. We shall however mostly use the coarser topology of pointwise convergence for the ℝ-valued sublinear functionals in 𝒬F . That is, convergence of a net (φi )i∈ℐ in 𝒬F toward an element φ ∈ 𝒬F in this topology requires that (φi (μ))i∈ℐ converges to φ(μ) for every μ ∈ F ˚ in the usual one-point compactification topology of ℝ. The topology of pointwise convergence for 𝒬F is of course still Hausdorff. The following observations are apparent from the definition of 𝒬F . We recall that a subset of Φ of 𝒬F is bounded below if for every W ∈ 𝒲 there is λ ⩾ 0 such that φ ∈ (0)(λW ), that is, 0 ⩽ φ + λσW , for all φ ∈ Φ. The following observation is evident. 1.4.7 Proposition. (a) Every nonempty subset Φ ⊂ 𝒬F has a supremum sup Φ ∈ 𝒬F . We have sup(Φ + φ) = sup Φ + φ for all φ ∈ 𝒬F . (b) The pointwise limit of a bounded below net in 𝒬F is contained in 𝒬F . The intersection of 𝒬F and −𝒬F consists of all real-valued linear functionals on F ˚ that are bounded on the polars of all neighborhoods in 𝒲 . For this reason, we shall also consider the space F ˚‚ of all real- or complex-valued linear functionals on F ˚ that are bounded on the polars of all neighborhoods in 𝒲 , endowed with the

40 � 1 Function spaces topology of uniform convergence on these polars. That is, the topology generated by the second polars W ˝˝ in F ˚‚ of the neighborhoods W ∈ 𝒲 . We have F ⊂ F ˚˚ ⊂ F ˚‚ ⊂ F ˚ , f

where F ˚˚ is the second dual of F, that is, the dual of F ˚ under its strong topology, and F ˚f is the algebraic dual of F ˚ . Indeed, for every element ω ∈ F ˚˚ there is a bounded subset B of F such that ω is bounded on the polar B˝ of B in F ˚ . For every W ∈ 𝒲 , we have B ⊂ λW for some λ ⩾ 0, hence W ˝ ⊂ λB˝ , and ω is seen to be bounded on W ˝ , hence an element of F ˚‚ . If F is indeed a normed space, that is, if F admits a bounded neighborhood, then F ˚‚ = F ˚˚ . The topology of F ˚‚ induces the given topology on F and the natural topology on F ˚˚ (see IV.5.3 in [56]). We note that the natural topology is generally weaker than the strong topology of F ˚˚ , which is generated by the polars of all (strongly) bounded subsets of F ˚ . Via the embedding ω 󳨃→ φω : F ˚‚ → 𝒬F , where φω (μ) = Re ω(μ) for all μ ∈ F ˚ , the spaces F, F ˚˚ and F ˚‚ may be considered to be real-linear subspaces of 𝒬F . The elements φω are real-linear functionals on F ˚ , bounded on the polars of the neighborhoods in 𝒲 . The functional φω is continuous with respect to the strong topology of F ˚ provided that ω ∈ F ˚˚ , and continuous with respect to the weak* topology provided that ω ∈ F. The element ω ∈ F ˚‚ can be reconstructed from φω by ω(μ) = φω (μ) in the real and ω(μ) = φω (μ) − iφω (iμ) in the complex case. The embedding ω 󳨃→ φω of F, F ˚˚ and F ˚‚ into 𝒬F is reallinear and one-to-one. The image of F ˚‚ is the intersection of 𝒬F and −𝒬F . The upper, lower and symmetric topologies of 𝒬F all coincide with the given topology of F and the natural topology of F ˚˚ and F ˚‚ on this embedding. The completion F̂ of F is generally not a subspace of F ˚˚ , but of F ˚‚ since it carries the same neighborhood system and the same dual F ˚ as F. Thus we may use the same insertion c 󳨃→ φc : F̂ → 𝒬F as for F ˚‚ . The topology of pointwise convergence on 𝒬F induces the weak topology on F and F̂ and the weak* topology on F ˚˚ . Similarly, using the embedding of F, F ˚˚ and F ˚‚ into 𝒬F , the spaces L(E, F), L(E, F ˚˚ ) and L(E, F ˚‚ ) may be considered to be real-linear subspaces of L(E, 𝒬F ). Indeed, with an operator T ∈ L(E, F ˚‚ ) we associate the real-linear operator a 󳨃→ φT(a) : E → 𝒬F . Again, F ˚‚ is supposed to carry the topology of uniform convergence on the polars of the neighborhoods in 𝒲 . The linearity of T ∈ L(E, F ˚‚ ) returns that φT(αa) (μ) = φαT(a) (μ) = φT(a) (αμ) holds for all a ∈ E, all α in the common scalar field of E and F and all μ ∈ F ˚ . If F = ℂ, for example, then F ˚ = ℂ and 𝒬F consists of all ℝ-valued functions on ℂ that are sublinear and bounded below on the unit ball of ℂ. The elements z ∈ ℂ are embedded into 𝒬F via z 󳨃→ φz ∈ 𝒬F , where φz (w) = Re zw for all w ∈ ℂ. The case that F = ℝ is similar.

1.4 Vector space extensions

� 41

As was established before, the embedding c 󳨃→ φc : F → 𝒬F can be extended canonically to an embedding C 󳨃→ φC of Conv(F) into 𝒬F setting φC = sup{φc | c ∈ C} ∈ 𝒬F , that is, φC (μ) = sup{Re μ(c) | c ∈ C} for all C ∈ Conv(F) and μ ∈ F ˚ . The sublinear functional φC ∈ 𝒬F is lower semicontinuous on F ˚ with respect to its weak* topology. Proposition 1.4.3(a) if applied to the dual pair (F, F ˚ ) with the given topology of F and the weak* topology for F ˚ yields that this embedding is linear, and that its range consists of all sublinear functionals on F ˚ that are lower semicontinuous in a topology consistent with the duality (F ˚ , F) (see Corollary 1.4.4). The operator C 󳨃→ φC : Conv(F) → 𝒬F is continuous if both Conv(F) and 𝒬F are endowed with their respective locally convex cone topologies. Indeed, given W ∈ 𝒲 , we notice that φW = σW . Then C ∈ W (D), that is, C ⩽ D + W for C, D ∈ Conv(F) implies by 1.4.3(a) that φC ⩽ φD+W = φD + φW = φD + σW , that is, φC ∈ W (φD ). A similar embedding can be established for Conv(F ˚˚ ) if we use Proposition 1.4.3 for the dual pair (F ˚˚ , F ˚ ) with the weak* topology of F ˚˚ and the strong topology β(F ˚ , F) for F ˚ . The elements of Conv(F ˚˚ ) correspond to the sublinear functionals in 𝒬F that are lower semicontinuous on F ˚ in a topology consistent with the duality (F ˚ , F ˚˚ ), and the operator C 󳨃→ φC : Conv(F ˚˚ ) → 𝒬F is continuous if Conv(F ˚˚ ) is endowed with the locally convex cone topology generated by the second polars W ˝˝ in F ˚˚ of the neighborhoods in 𝒲 , in other words, by the natural topology of F ˚˚ . Indeed, given W ∈ 𝒲 , again we notice that φW ˝˝ = σW since φW ˝˝ (μ) = sup{Re ω(μ) | ω ∈ W ˝˝ } ⩽ 1 holds for all μ ∈ W ˝ . Then C ∈ W ˝˝ (D), that is, C ⩽ D + W ˝˝ for C, D ∈ Conv(F ˚˚ ) implies that φC ⩽ φD+W ˝˝ = φD + φW ˝˝ = φD + σW . For C ∈ Conv(F) and an element α in the scalar field of F, we have φαC (μ) = φC (αμ) for all μ ∈ F ˚ .

42 � 1 Function spaces Elements of 𝒬F representing convex sets Let F ˚0 be a subspace of F ˚‚ containing F. In the light of the preceding considerations, we shall say that an element φ of 𝒬F represents a set in Conv(F ˚0 ) if φ = φC for some C ∈ Conv(F ˚0 ), that is, if the sublinear functional φ is lower semicontinuous on F ˚ in a topology that is consistent with the duality (F ˚ , F ˚0 ). We are most interested in the cases that F ˚0 = F,

F ˚0 = F ˚˚ ,

F ˚0 = F̂

or

F ˚0 = F ˚‚ .

Obviously, if F ˚1 is a subspace of F ˚‚ containing F ˚0 , then lower semicontinuity in σ(F ˚ , F ˚0 ) implies lower semicontinuity in σ(F ˚ , F ˚1 ). Thus, if an element φ ∈ 𝒬F represents a set C0 in Conv(F ˚0 ), then it also represents a set C1 in Conv(F ˚1 ). According to Proposition 1.4.3, however, the closures of C0 and C1 as subsets of F ˚1 coincide in any topology that is consistent with the duality (F ˚1 , F ˚ ). The following example will demonstrate that in general not every element of 𝒬F represents a convex subset of some subspace of F ˚‚ . 1.4.8 Example. Let (F, 𝒲 ) be the space of all sequences (αi )i∈ℕ in ℝ endowed with the topology of pointwise convergence (see Example 1.2.7(c)). The neighborhoods in 𝒲 are defined by finite subsets Y of ℕ and ε > 0 as W(Y ,ε) = {(αi )i∈ℕ ∈ F | |αi | ⩽ ε for all i ∈ Y }. If αi = 0 for all i ∈ Y for a sequence (αi )i∈ℕ ∈ F, then every linear functional μ in ˝ the polar W(Y ,ε) of W(Y ,ε) takes the value 0 on this sequence. Hence μ concurs on any two sequences in F whose terms with indexes in Y coincide. If en denotes the n-th unit sequence in F, then μ(ei ) = 0 for all i ∈ ̸ Y and μ((αi )i∈ℕ ) = μ(∑ αi ei ) = ∑ αi μ(ei ) i∈Y

i∈Y

holds for all (αi )i∈ℕ ∈ F. Thus the dual F ˚ of F corresponds to the space of all se˝ quences (γi )i∈ℕ in ℝ with only finitely many nonzero elements, and the polar W(Y ,ε) of W(Y ,ε) consists of all sequences (γi )i∈ℕ in F ˚ such that γi = 0 for all i ∉ Y and ∑i∈Y |γi | ⩽ 1/ε. For every c = (γi )i∈ℕ ∈ F ˚ , we set +∞,

φ(c) = {

n(c) ∑i∈ℕ γi ,

if at least one γi > 0, if all γi ⩽ 0.

whereby n(c) = max{i ∈ ℕ | γi ≠ 0} if c ≠ 0 and n(0) = 0. It is easily verified that φ defines a sublinear functional on F ˚ , and indeed an element of 𝒬F as it is bounded

1.4 Vector space extensions

� 43

below on the polar of every neighborhood in F. Surely, for every W(Y ,ε) ∈ 𝒲 we have ˝ n(c) ⩽ max{i ∈ Y } for all c ∈ W(Y ,ε) , hence φ(c) ⩾ −(1/ε) max{i ∈ Y }. There is however no linear functional ω ∈ F ˚f on F ˚ such that ω ⩽ φ, because assuming otherwise, we set εn = ω(cn ), where cn is the n-th unit sequence in F ˚ and obtain a contradiction as follows. We choose n ∈ ℕ and λ ⩾ 0 such that n ⩾ ε1 + 1 and λ > εn − n and set c = −(λc1 + cn ) ∈ F ˚ . Then ω(c) = −(λε1 + εn ), and as ω(c) ⩽ φ(c) = −n(λ+1), we conclude that λε1 +εn ⩾ n(λ+1), that is, εn −n ⩾ λ(n−ε1 ) ⩾ λ, contradicting the above. As a consequence of Propositions 1.4.5 and 1.4.6, we obtain the following. 1.4.9 Proposition. If E is a reflexive normed space, then the locally convex cone Conv(E) is complete in its symmetric topology. Proof. We shall use Proposition 1.4.6 with E in place of F. Let E be a reflexive normed space with unit ball 𝔹 and let (Ai )i∈ℐ be a symmetric Cauchy net in the locally convex cone 𝒫E = Conv(E). For every ε > 0, there is i0 ∈ ℐ such that Ai ∈ (ε𝔹)s (Ak ), that is, Ai ⩽ Ak + ε𝔹

and

Ak ⩽ Ai + ε𝔹

for i, k ⩾ i0 . The corresponding sublinear functionals φAi ∈ 𝒬E are lower semicontinuous in σ(E ˚ , E), hence also in the finer topology β(E ˚ , E). The net (φAi )i∈ℐ is Cauchy in the symmetric topology of 𝒬E and, therefore, convergent by Proposition 1.4.5. Its limit φ is also β(E ˚ , E)-lower semicontinuous by Proposition 1.4.6(b). Because E is supposed to be reflexive, the strong topology β(E ˚ , E) is consistent with the duality (E ˚ , E). Thus following Proposition 1.4.3(a) there is a convex set A ∈ Conv(E) such that φ = φA . It is now straightforward to verify that limi∈ℐ Ai = A holds in the symmetric topology of Conv(E). We proceed with a few observations about continuous linear operators from the locally convex cone 𝒫E into the locally convex cone 𝒬F . 1.4.10 Lemma. Every operator T ∈ L(𝒫E , 𝒬F ) is monotone in the sense that A ⊂ B for A, B ∈ 𝒫E implies that T(A) ⩽ T(B). Proof. Let T ∈ L(𝒫E , 𝒬F ) and A ⊂ B for A, B ∈ 𝒫E . For μ ∈ F ˚ and ε > 0, there is W ∈ 𝒲 such that μ ∈ εW ˝ . Because T is continuous, there is V ∈ 𝒱 such that T(C) ∈ W (T(D)), that is, T(C) ⩽ T(D) + σW , whenever C ∈ V (D) for C, D ∈ 𝒫E . But A ⊂ B implies that A ⊂ B + V , and we infer that T(A)(μ) ⩽ T(B)(μ) + σW (μ) ⩽ T(B)(μ) + ε holds for all ε > 0. Our claim follows.

44 � 1 Function spaces We note that every operator T ∈ L(𝒫E , 𝒬F ) is real-linear on the subspace E of 𝒫E . Indeed, for a ∈ E we have {a} + {−a} = {0}. Hence T(−a) = −T(a) and T(αa) = (−α)T(−a) = αT(a) for α < 0. Thus T(αa) = αT(a) holds for all α ∈ ℝ. The image T(a) of an element a ∈ E is therefore contained in the embedding of F ˚‚ as a subspace of 𝒬F . 1.4.11 Theorem. For an operator T ∈ L(𝒫E , 𝒬F ) and A ∈ Conv(E), the formula T(A) = sup{T(a) | a ∈ A} ∈ 𝒬F defines an operator in L(Conv(E), 𝒬F ) such that T(A) ⩽ T(A) for all A ∈ 𝒫E . We have T(A) ⩽ S(A) for all A ∈ Conv(E) for every other operator S ∈ L(Conv(E), 𝒬F ) with this property. Proof. Let T ∈ L(𝒫E , 𝒬F ). Let us first verify that the mapping A 󳨃→ T(A) : Conv(E) → 𝒬F as defined in our statement is linear: For A, B ∈ Conv(E), for λ ⩾ 0 and every μ ∈ F ˚ , we have T(A + B)(μ) = sup{T(a + b)(μ) | a ∈ A, b ∈ B} = sup{T(a)(μ) | a ∈ A} + sup{T(b)(μ) | b ∈ B} = T(A)(μ) + T(B)(μ) and T(λA)(μ) = sup{T(a)(μ) | a ∈ λA} = λ sup{T(a)(μ) | a ∈ A} = λT(A)(μ). For continuity, let W ∈ 𝒲 and choose V ∈ 𝒱 such that T(C) ∈ W (T(D)) whenever C ∈ V (D) for C, D ∈ 𝒫E . Let A, B ∈ Conv(E) such that A ∈ V (B), that is, A ⩽ B + V . We shall verify that T(A) ∈ W (T(B)). Indeed, for every a ∈ A we have a ∈ b + V , that is, a ∈ V (b), for some b ∈ B. Since a, b ∈ 𝒫E , this yields T(a) ∈ W (T(b)), that is, T(a) ⩽ T(b) + σW . Now using Proposition 1.4.7(a), we infer that T(a) ⩽ sup{T(b) + σW | b ∈ B} = sup{T(b) | b ∈ B} + σW = T(B) + σW holds for all a ∈ A and, therefore, T(A) ⩽ T(B) + σW , our claim. We conclude that T ∈ L(Conv(E), 𝒬F ). Moreover, for every A ∈ 𝒫E we have T(A) = sup{T(a) | a ∈ A} ⩽ T(A) by Lemma 1.4.10. Every other operator S ∈ L(Conv(E), 𝒬F ) with this property coincides with T on the elements of E, hence

1.4 Vector space extensions

� 45

T(A) = sup{T(a) | a ∈ A} = sup{S(a) | a ∈ A} ⩽ S(A) holds for all A ∈ Conv(E). The restriction of an operator T ∈ L(𝒫E , 𝒬F ) to the subspace E of 𝒫E is reallinear, that is, 0 = T(a) + T(−a) implies that T(a) ∈ F ˚‚ and T(−a) = −T(a) holds for all a ∈ E. In other words, we have T(−a)(μ) = T(a)(−μ) for all μ ∈ F ˚ . If, in the complex case, the restriction of T to E is indeed complex-linear, that is, an element of L(E, F ˚‚ ), then we have T(αa)(μ) = T(a)(αμ) for all a ∈ E, all α ∈ ℂ and all μ ∈ F ˚ .

The cone ℱ (X, Conv(E)) Let X be a set. By ℱ (X, Conv(E)), we denote the cone of all Conv(E)-valued functions on X, endowed with the pointwise operations and order, that is, f ⩽ g for f , g ∈ ℱ (X, Conv(E)) if f (x) ⩽ g(x) in Conv(E) for all x ∈ X. On the subspace ℱ (X, E) of ℱ (X, Conv(E)), this order is of course the equality. A function f ∈ ℱ (X, Conv(E)) is said to be positive if f (x) ⩾ 0 for all x ∈ X. Unfortunately, this notion conflicts with the conventional notion of positivity for real-valued functions, where the value 0 is not permitted. We continue to use the latter for functions in ℱ (X). The definition of function space neighborhoods in Section 1.2 extends in a canonical way to the setvalued functions in ℱ (X, Conv(E)). In this vein, for a function space neighborhood v defined by a convex family 𝒩v of neighborhood functions and for functions f , g ∈ ℱ (X, Conv(E)) we write f ⩽ g + v if f ⩽ g + n, that is, f (x) ⩽ g(x) + n(x) for all x ∈ X, for some neighborhood function n ∈ 𝒩v . On the subspace ℱ (X, E) of ℱ (X, Conv(E)), this notion is consistent with our definition in Section 1.2, that is, f ⩽ g + v for f , g ∈ ℱ (X, E) means that both f − g ⩽ v and g − f ⩽ v. We note that other than for E-valued functions the action of a function space neighborhood v on ℱ (X, Conv(E)) is not fully determined by the convex subset {f ∈ ℱ (X, Conv(E)) | f ⩽ v} of ℱ (X, Conv(E)). One would have to employ the subset {(f , g) | f ⩽ g + v} of ℱ (X, Conv(E))2 instead. We also write 𝒜 ⩽ ℬ + v for nonempty subsets 𝒜, ℬ of ℱ (X, Conv(E)) if for every f ∈ 𝒜 there is g ∈ ℬ such that f ⩽ g + v. Using this notation for Conv(E)-valued functions, the condition on function space neighborhoods from Section 1.2 now reads that for every A ∈ R there is V ∈ 𝒱 such that χA ⋅ V ⩽ v. Statements (a) and (b) of Lemma 1.2.1 transfer as follows to Conv(E)valued functions. (a): if f ⩽ g + v, then φ ⋅ f ⩽ φ ⋅ g + v for all φ such that |φ| ⩽ 1, and (b): if 𝒩v is upward directed, then φ ⋅ f + (1 − φ) ⋅ g ⩽ φ ⋅ f ′ + (1 − φ) ⋅ g ′ + v whenever f ⩽ f ′ +v, g ⩽ g ′ +v and 0 ⩽ φ ⩽ 1. We shall take advantage of this simplified notation in the sequel.

46 � 1 Function spaces A function f ∈ ℱ (X, Conv(E)) is said to be bounded above (or bounded below) if for every function space neighborhood v there is λ ⩾ 0 such that f ⩽ λv (or 0 ⩽ f + λv). A function is bounded if it is both bounded above and below. These concepts of boundedness are also used for subsets of ℱ (X, Conv(E)) provided that the above conditions hold with the same λ ⩾ 0 for all functions in this subset. Since all function space neighborhoods are balanced, boundedness above and below coincide for E-valued functions. Given a subset B of X, we shall also use the symbols =B , ⩽B or ⩾B in order to signify that the relation =, ⩽ or ⩾ holds for Conv(E)-valued functions pointwise on the set B. For a function f ∈ ℱ (X, Conv(E)) and an element α of the scalar field of E, we shall denote the function x 󳨃→ αf (x) by αf ∈ ℱ (X, Conv(E)). For an E-valued function, −f is of course the additive inverse of f , but in general we only have 0 ⩽ f + (−f ) and −(−f ) = f . Moreover, αf ⩽ αg holds whenever f ⩽ g, and αf ⩽ αg + |α|v holds whenever f ⩽ g + v for f , g ∈ ℱ (X, Conv(E)), a scalar α in ℝ or ℂ and a function space neighborhood v.

2 Integration Using the notation, facts and methodology from the preceding chapter, we shall proceed to investigate continuous linear operators from a function space (𝒞V (X, E), V) into a locally convex Hausdorff space (F, 𝒲 ). The space of these operators will be denoted by L(𝒞V (X, E), F). Section 2.6 below will be concerned with integral representations for this type of operators, based on the classical result by Riesz [44], which states that every positive linear functional on the space of all continuous real-valued functions with compact support on a locally compact Hausdorff space X can be represented by the integral with respect to some regular positive Borel measure on X (see Theorem 13.23 in [53]). There are a number of generalizations of this result, most of them are concerned with operators on spaces of continuous real-valued functions on a compact space and representations by vector-valued measures. One of the better-known versions is the representation theorem by Bartle–Dunford–Schwartz [7] (see Theorem 1 in Chapter VI.2 in [21]). It states that a weakly compact continuous linear operator from a space of continuous real- or complex-valued functions on a compact space, endowed with the supremum norm, into some Banach space F can be represented by the integral with respect to a regular F-valued Borel measure. In case that the operator T is not weakly compact, the representing measure has to be allowed to take values in the second dual of F. These results will be recovered as special cases in the later sections of this chapter. Our approach to integral representations follows the method used for locally convex cones in [48]. In the special case of topological vector spaces, results can be transferred using vector space terminology, but at some expense of refinement and simplicity. Our more specialized setting, however, allows at times for more direct arguments and at instances more advanced results. Ideally, one would aim for a representation of an operator in L(𝒞V (X, E), F) by an integral using an L(E, F)-valued measure on X. This is, however, not possible in general as has been established by earlier approaches, even in the special case when X is compact, E = ℝ and 𝒞 (X) is endowed with the topology of uniform convergence. We define and investigate operator-valued measures in Section 2.1. In Section 2.2 we establish a rather narrow measure independent notion of measurability of functions, which is used in Section 2.3 for the definition of integrals of vector- and set-valued functions. The general convergence theorems are treated separately in Section 2.4. In Section 2.5, we investigate the linear operators on function spaces defined by the integral. Section 2.6 then contains an integral representation theorem for linear operators on function spaces together with some of its consequences, one of our main results. We broadly refer to the sources in [6, 10, 19] and [31] for real-valued, and to [5, 20, 21, 29, 43] and [58] for vector-valued integration theory. Set-valued functions and their integrals are considered in [3, 4, 27] and [57]. Results about integral representation can be found in [7, 8, 12, 21, 35] and [44]. https://doi.org/10.1515/9783111315478-002

48 � 2 Integration

2.1 Operator-valued measures For the sake of generality, we shall formulate our take on integration theory in this and the following sections for a set X without the imposition of a topological structure. Measures utilized for integral representation of linear operators on function spaces can generally not be defined on a σ-field containing the whole space X. We shall therefore employ a σ-ring R in X instead. This notion was introduced and developed in Sections 1.1 and 1.2. In the topological case, that is, if X is a locally compact Hausdorff space, we assume that R consists of all relatively compact Borel subsets of X. As remarked earlier, the σ-field A associated with R contains all Borel subsets of X in this case, and exactly those if X is countably compact. Definition of an operator-valued measure The measures θ on X in our approach will be defined on the sets in R and will be L(𝒫E , 𝒬F )-valued, where 𝒫E is a cone extensions of E and 𝒬F is the lattice cone extension of F. Both were introduced in Section 1.4. More precisely, an L(𝒫E , 𝒬F )-valued measure θ on R is a set function A 󳨃→ θA : R → L(𝒫E , 𝒬F ) such that θH = 0 and that θ is countably additive, that is, ∞

θ⋃i∈ℕ Ai = ∑ θAi i=1

holds whenever the sets Ai ∈ R are disjoint and ⋃i∈ℕ Ai ∈ R. Convergence for the series on the right-hand side is meant pointwise for the elements of 𝒫E and F ˚ , that is, for every element C ∈ 𝒫E the series ∑∞ i=1 θAi (C) converges in the topology of pointwise convergence of 𝒬F to θ⋃i∈ℕ Ai (C). L(E, F ˚‚ )-valued measures If 𝒫E = E, then for every A ∈ R and a ∈ E we have 0 = θA (a) + θ(−a), hence θA (a) = −θA (a) ∈ 𝒬F and, therefore, θA (a) is an element of the embedding of F ˚‚ into 𝒬F . Thus θA (a) is a real-linear functional on F ˚ , and θA defines a real-linear operator from E into F ˚‚ . We have θA (−a)(μ) = −θA (a)(μ) = θA (a)(−μ) for all a ∈ E and μ ∈ F ˚ . However, if both E and F are vector spaces over ℂ, not every L(E, 𝒬F )-valued measure θ defines a family of complex-linear operators θA from E into F ˚‚ . For θ to be L(E, F ˚‚ )-valued, we require in addition that

2.1 Operator-valued measures

� 49

θA (αa)(μ) = θA (a)(αμ) holds for all A ∈ R, all a ∈ E, α ∈ ℂ and μ ∈ F ˚ . This condition then guarantees that θA (αa) = αθA (a) if interpreted as elements of F ˚‚ . Considering our prior observations, it needs to be verified only in the complex case and then only for α = i. The same considerations apply to L(E, F)- and L(E, F ˚˚ )-valued measures. If the measure θ is L(E, F ˚˚ )-valued, pointwise convergence on F ˚ of its evaluations on sets in R means convergence of the operators θA in the weak* operator topology of L(E, F ˚˚ ). In case that θ is L(E, F)-valued, this means convergence in the weak operator topology of L(E, F). The last statement may however be strengthened using the classical Orlicz–Pettis theorem about conditional convergence of series in a topological vector space. 2.1.1 Proposition. If 𝒫E = E and θ is an L(E, F)-valued measure, then for disjoint sets Ai ∈ R such that ⋃i∈ℕ Ai ∈ R the series θ⋃i∈ℕ Ai = ∑∞ i=1 θAi converges in the strong operator topology of L(E, F). Proof. We recall from Proposition 1.3.1 that the dual of L(E, F) under both its weak and strong operator topologies is the tensor product Ẽ ⊗ F ˚ , where Ẽ denotes the quotient space E/E0 and E0 = ⋂V ∈𝒱 V . The weak and strong operator topologies on L(E, F) are both Hausdorff. Suppose that θ is an L(E, F)-valued measure. Let Ai be disjoint sets in R such that ⋃i∈ℕ Ai ∈ R. According to the above definition of a measure, the series ˚ ̃ in ∑∞ i=1 θAi in L(E, F) converges in the weak topology of the duality (L(E, F), E ⊗ F ) to the operator θ⋃i∈ℕ Ai ∈ L(E, F). The strong operator topology of L(E, F) is consistent with this duality. Thus we may employ the Orlicz–Pettis theorem from [40] (also see [36] or [46]), which states that a series in a Hausdorff locally convex topological vector space that is weakly subseries convergent is convergent in the given topology. In this context, a series is weakly subseries convergent if the series formed by any subsequence of its terms is weakly convergent. This is clearly the case for the series ∑∞ i=1 θAi in L(E, F) from above which is therefore seen to converge in the strong operator topology. The strong additivity of an L(E, F)-valued measure θ as established in Proposition 2.1.1 does not extend to L(𝒫E , 𝒬F )-valued measures in general. In this context, we shall say that an L(𝒫E , 𝒬F )-valued measure θ is strongly additive at an element C ∈ 𝒫E if for all Ai ∈ R such that ⋃i∈ℕ Ai ∈ R the series ∑∞ i=1 θAi (C) in 𝒬F converges in the symmetric topology of 𝒬F (see Section 1.4) to θ⋃i∈ℕ Ai (C). That is, this series of ℝ-valued functionals on F ˚ converges uniformly on the polars of all neighborhoods in 𝒲 , whereby convergence toward +∞ ∈ ℝ at a point in F ˚ requires that this value is taken cofinally. Unless mentioned otherwise, the following results refer to pointwise convergence in 𝒬F for the evaluation of a measure at an element of 𝒫E . 2.1.2 Lemma. Let A ∈ R. (a) If Ai ∈ R such that Ai ⊂ Ai+1 for all i ∈ ℕ and A = ⋃∞ i=1 Ai , then θE = limi→∞ θAi .

50 � 2 Integration (b) If Ai ∈ R such that A ⊃ Ai ⊃ Ai+1 for all i ∈ ℕ, and ⋂∞ i=1 Ai = H, then limi→∞ θAi (C)(μ) = 0 for all C ∈ 𝒫E and μ ∈ F ˚ such that θA (C)(μ) < +∞. Proof. For Part (a), let B1 = A1 and Bi = Ai \ Bi−1 for i > 1. The sets Bi are disjoint, An = ⋃ni=1 Bi and A = ⋃∞ i=1 Bi . From the countable additivity of the measure θ, we infer that θAn = ∑ni=1 θBi and θA = ∑∞ i=1 θBi , hence our claim. For Part (b), let Bi = A \ Ai for i ∈ ℕ. Thus Bi ⊂ Bi+1 and ⋃∞ i=1 Bi = A. This shows θA = limi→∞ θBi by Part (a) and θA = θBi + θAi . If θA (C)(μ) < +∞ for C ∈ 𝒫E and μ ∈ F ˚ , our claim follows. If 𝒫E = E and θ is L(E, F)-valued, then according to Proposition 2.1.1 the convergence statements of Lemma 2.1.2 hold indeed with respect to the strong operator topology of L(E, F). More generally, if θ is L(𝒫E , 𝒬F )-valued and strongly additive at an element C ∈ 𝒫E , then the convergence statements of Lemma 2.1.2, with the terms evaluated at C, hold with respect to the symmetric topology of 𝒬F . Modulus of a measure Other than in the classical approach, we proceed to define the semivariation of a measure as its extension from the elements of E to the cone of all nonempty convex subsets of E. Given an L(𝒫E , 𝒬F )-valued measure θ, we define its modulus or semivariation |θ| as follows: For A ∈ R and a set C ∈ Conv(E), we set (see also I.3.2 in [48]) n n 󵄨󵄨 |θ|A (C) = sup{∑ θAi (ci ) 󵄨󵄨󵄨 ci ∈ C, Ai ∈ R disjoint, ⋃ Ai = A}, 󵄨 i=1

i=1

where the elements ci ∈ E are identified with the singleton sets {ci } ∈ 𝒫E . The supremum on the right-hand side is taken in 𝒬F , that is, pointwise on F ˚ . Hence |θ|A (C) is an element of 𝒬F . Clearly, |θ|A (a) = θA (a) holds for all a ∈ E, and we shall argue in 2.1.3 below that |θ|A (C) ⩽ θA (C) holds for all C ∈ 𝒫E . We have |θ|A (C) ⩽ |θ|A (D) whenever C ⊂ D for C, D ∈ Conv(E). If 0 ∈ C, then 0 ⩽ |θ|A (C) ⩽ |θ|B (C) provided that A ⊂ B for A, B ∈ R, and in the definition of |θ|A (C) we only require that the disjoint sets Ai ∈ R are contained in A, not that their union is all of A, since the set An+1 = A \ ⋃ni=1 Ai with cn+1 = 0 may be added. We recognize that the modulus of an L(𝒫E , 𝒬F )-valued measure is determined by its values on E alone. If 𝒫E = E and θ is an L(E, F ˚‚ )-valued measure, our definitions of the modulus and the multiplication of the elements of Conv(E) with general scalars yield immediately that |θ|A (αC)(μ) = |θ|A (C)(αμ) holds for all A ∈ R, all C ∈ Conv(E), α in ℝ or ℂ and μ ∈ F ˚ . If θ is L(E, F ˚˚ )valued, then for every A ∈ R and C ∈ Conv(E) the element θA (C) ∈ 𝒬F is a σ(F ˚ , F ˚˚ )-lower semicontinuous sublinear functional on F ˚ , hence in the light of Proposition 1.4.3(a) represents a convex subset of F ˚˚ . If θ is L(E, F)-valued, then a similar argument shows that θA (C) represents a convex subset of F.

2.1 Operator-valued measures

� 51

Bounded measures An L(𝒫E , 𝒬F )-valued measure θ is bounded or of bounded semivarion if for every A ∈ R and W ∈ 𝒲 there is a neighborhood V ∈ 𝒱 such that |θ|A (V ) ⩽ σW , that is, |θ|A (V )(μ) ⩽ 1 for all μ ∈ W ˝ . Since every element μ ∈ F ˚ is contained in the polar of some neighborhood in 𝒲 , boundedness of a measure θ implies in particular that for every A ∈ R and μ ∈ F ˚ there is a neighborhood V ∈ 𝒱 such that |θ|A (V )(μ) < +∞. 2.1.3 Theorem. The modulus |θ| of a bounded L(𝒫E , 𝒬F )-valued measure θ is itself a bounded L(Conv(E), 𝒬F )-valued measure. The modulus of |θ| is |θ|. We have |θ|A (C) ⩽ θA (C) for all A ∈ R and C ∈ 𝒫E , and for any other L(Conv(E), 𝒬F )-valued measure ϑ with this property we have |θ|A (C) ⩽ ϑA (C) for all A ∈ R and C ∈ Conv(E). Proof. First, we shall argue that for a fixed set A ∈ R the mapping C 󳨃→ |θ|A (C) : Conv(E) → 𝒬F is linear. Indeed, let C, D ∈ Conv(E), let Ai , Bk ∈ R such that ⋃ni=1 Ai = ⋃m k=1 Bk = A. For any choice of elements ci ∈ C and dk ∈ D, we calcun late that θAi (ci ) = ∑m k=1 θAi ∩Bk (ci ) for all i = 1, . . . , n and θBk (dk ) = ∑i=1 θAi ∩Bk (dk ) for all k = 1, . . . , m. Thus n

m

i=1

k=1

n

m

∑ θAi (ci ) + ∑ θBk (dk ) = ∑ ∑ θAi ∩Bk (ci + dk ) i=1 k=1

The sets Ai ∩ Bk are disjoint and we have ⋃ni=1 ⋃m k=1 Ai ∩ Bk = A. The above therefore yields that |θ|A (C) + |θ|A (D) ⩽ |θ|A (C + D). For the reverse inequality, let Ai ∈ R such that ⋃ni=1 Ai = A and let ci +di ∈ C+D. Then n

n

n

i=1

i=1

i=1

∑ θAi (ci + di ) = ∑ θAi (ci ) + ∑ θAi (di ). We infer that |θ|A (C + D) ⩽ |θ|A (C) + |θ|A (D). The equality |θ|A (λC) = λ|θ|A (C) for λ ⩾ 0 is evident from the definition of the modulus. Next, we shall verify that |θ|A ∈ L(Conv(E), 𝒬F ), that is, that the linear operator |θ|A is continuous. Clearly, |θ|A is monotone, that is, |θ|A (C) ⩽ |θ|A (D) holds whenever C ⩽ D for C, D ∈ Conv(E). Now let W ∈ 𝒲 . Since θ is supposed to be bounded, there is V ∈ 𝒱 such that |θ|A (V ) ⩽ σW . Accordingly, C ⩽ D + V for C, D ∈ Conv(E) implies that |θ|A (C) ⩽ |θ|A (D + V ) = |θ|A (V ) + |θ|A (D) ⩽ |θ|A (D) + σW as required. Next, we shall demonstrate that the set function A 󳨃→ |θ|A : R → L(Conv(E), 𝒬F )

52 � 2 Integration is indeed a measure. Clearly, |θ|H = 0. For countable additivity, suppose that ⋃∞ i=1 Ai = A ∈ R for disjoint sets Ai ∈ R and let C ∈ Conv(E). In a first step, let us assume that ˚ 0 ∈ C. The series ∑∞ i=1 |θ|Ai (C) then has positive terms and converges (pointwise on F ) in 𝒬F . Let B1 , . . . , Bn ∈ R be disjoint sets such that ⋃nk=1 Bk = A and let ck ∈ C. Then ∞

θBk (ck ) = ∑ θBk ∩Ai (ck ) i=1

for each k = 1, . . . , n by the countable additivity of θ, hence n

n

∞

k=1

k=1

i=1

∞

n

∞

i=1

k=1

i=1

∑ θBk (ck ) = ∑ (∑ θBk ∩Ai (ck )) = ∑( ∑ θBk ∩Ai (ck )) ⩽ ∑ |θ|Ai (C).

This renders |θ|A (C) ⩽ ∑∞ i=1 |θ|Ai (C). For the reverse inequality, let n ∈ ℕ and for ni i each i = 1, . . . , n, let B1 , . . . , Bni i ∈ R be disjoint sets such that ⋃k=1 Bki = Ai and let

c1i , . . . , cni i ∈ C. Then

n

ni

i=1

k=1

∑( ∑ θBi (cki )) ⩽ |θ|⋃ni=1 Ai (C) ⩽ |θ|A (C), k

as the sets Bki and are pairwise disjoint and their union is ⋃ni=1 Ai . Now taking the supremum over all such choices of sets Bik and elements cki ∈ C yields n

∑ |θ|Ai (C) ⩽ |θ|A (C). i=1

This holds for all n ∈ ℕ and yields that ∑∞ i=1 |θ|Ai (C) ⩽ |θ|A (C). Our claim follows. Now in a second step, for any C ∈ Conv(E) we choose c ∈ C and set D = C − c. Then 0 ∈ D and our first step applies. That is to say, we have ∑∞ i=1 |θ|Ai (D) = |θ|A (D). Moreover, for the element c ∈ E we have |θ|B (c) = θB (c) for all B ∈ R, hence also ∑∞ i=1 |θ|Ai (c) = |θ|A (c) by the countable additivity of the measure θ. Summarizing, and using the previously established linearity of the operators |θ|A , |θAi |, this renders ∞

∞

i=1

i=1 ∞

∞

i=1

i=1

∑ |θ|Ai (C) = ∑ |θ|Ai (D + c) = ∑ |θ|Ai (D) + ∑ |θ|Ai (c) = |θ|A (D) + |θ|A (c) = |θ|A (C). Thus |θ| is indeed an L(Conv(E), 𝒬F )-valued measure. The remaining statements of the theorem are easily verified. First, for every C ∈ 𝒫E and every element c ∈ C we have θA (c) ⩽ θA (C) for all A ∈ R by Lemma 1.4.10. Thus

2.1 Operator-valued measures n

n

i=1

i=1

� 53

∑ θAi (ci ) ⩽ ∑ θAi (C) = θA (C) whenever ci ∈ C and Ai ∈ R are disjoint sets such that ⋃ni=1 Ai = A. This yields |θ|A (C) ⩽ θA (C), our claim. Next, let ϑ be any L(Conv(E), 𝒬F )-valued measure with this property, let A ∈ R and C ∈ Conv(E). Again, we argue that for every c ∈ C we have θA (c) = ϑA (c) ⩽ ϑA (C) and ∑ni=1 θAi (ci ) ⩽ ϑA (C) whenever ci ∈ C and Ai ∈ R are disjoint sets such that ⋃ni=1 Ai = A. Thus indeed |θ|A (C) ⩽ ϑA (C). Now consider the modulus ‖θ‖ of |θ| for ϑ in this last argument. We have ‖θ‖A (C) ⩽ |θ|A (C) for all A ∈ R and C ∈ Conv(E) by the above, hence |θ|A (C) ⩽ ‖θ‖A (C), also by the above. This shows ‖θ‖ = |θ|. Finally, for the boundedness of |θ| let A ∈ R and W ∈ 𝒲 . Because θ itself is bounded, we find V ∈ 𝒱 such that ‖θ‖A (V ) = |θ|A (V ) ⩽ σW . This completes our arguments. We shall subsequently require that all L(𝒫E , 𝒬F )-valued measures are bounded without explicitly listing this condition. It is guaranteed to hold if 𝒫E is a full cone extension of E. Indeed, let A ∈ R and W ∈ 𝒲 . Because θA ∈ L(𝒫E , 𝒬F ), there is V ∈ 𝒱 such that θA (C) ⩽ θA (D) + σW whenever C ⩽ D + V for C, D ∈ 𝒫E . Since V ∈ 𝒫E , this yields |θ|A (V ) ⩽ θA (V ) ⩽ σW . In particular, the modulus |θ| of a bounded L(𝒫E , 𝒬F )valued measure is a bounded L(Conv(E), 𝒬F )-valued measure. The strong additivity of an L(E, F)-valued measure as established in Proposition 2.1.1 does not necessarily extend to is modulus. But we observe the following. 2.1.4 Lemma. If |θ| is strongly additive at every neighborhood V ∈ 𝒱 , then it is strongly additive at every bounded element of Conv(E). Proof. Suppose that |θ| is strongly additive at every V ∈ 𝒱 , let C ∈ Conv(E) be bounded and let Ai ∈ R be disjoint sets such that A = ⋃i∈ℕ Ai ∈ R. Given W ∈ 𝒲 , there is V ∈ 𝒱 such that |θ|A (V ) ⩽ σW and λ ⩾ 0 such that both 0 ⩽ C + λV and C ⩽ λV . We set Bn = ⋃∞ i=n Ai ∈ R. Given ε > 0, by the strong additivity of |θ| at V there is n0 ∈ ℕ such that |θ|Bn (V ) ⩽ εσW for all n ⩾ n0 . The latter implies that both 0 ⩽ |θ|Bn (C) + εσW and |θ|Bn (C) ⩽ εσW . Hence 󵄨󵄨 󵄨 󵄨󵄨|θ|A (C)(μ) − |θ|⋃ni=1 Ai (C)(μ)󵄨󵄨󵄨 ⩽ ε holds for all μ ∈ W ˝ and all n ⩾ n0 .

Variation of a measure For a bounded L(𝒫E , 𝒬F )-valued measure θ on R, we define its variation relative to a function space neighborhood v and a nonempty subset Π of F ˚ on a set A ∈ R as (see III.1.4 in [26] or I.1 in [21]) by

54 � 2 Integration n 󵄨󵄨 A ∈ R disjoint, A ⊂ A, 󵄨 i }. ‖θ‖(v,Π) = sup {∑ θAi (ai )(μi ) 󵄨󵄨󵄨 i A 󵄨󵄨 ai ∈ E, χA ⋅ ai ⩽ v and μi ∈ Π i i=1

Since 0 ⩽ v, the variation of a measure is ℝ+ -valued and related to its earlier defined modulus or semivariation in the following way. 2.1.5 Lemma. Let θ be an L(𝒫E , 𝒬F )-valued measure, let v be a function space neighborhood and let Π ⊂ F ˚ . (a) If χA ⋅ C ⩽ v for C ∈ Conv(E) and A ∈ R, then |θ|A (C)(μ) ⩽ ‖θ‖(v,Π) holds for all A μ ∈ Π. (b) If v is defined by the single constant neighborhood function χX ⋅ V for V ∈ 𝒱 and (v,{μ}) if Π = {μ} ⊂ F ˚ is a singleton set, then |θ|A (V )(μ) = ‖θ‖A holds for all A ∈ R. Proof. Our claim in Part (a) is immediate from the respective definitions of the modulus (v,{μ}) and the variation of a measure; so is the inequality |θ|A (V )(μ) ⩽ ‖θ‖A in Part (b). For the reverse relation, let Ai ∈ R be disjoint subsets of A and let ai ∈ E such that χAi ⋅ ai ⩽ v, that is, ai ∈ V . Then n

∑ θAi (ai )(μ) ⩽ |θ|A (V )(μ) i=1

(v,{μ})

and, therefore, ‖θ‖A

⩽ |θ|A (V )(μ).

2.1.6 Proposition. Let θ be an L(𝒫E , 𝒬F )-valued measure, let v be a function space neighborhood and let Π ⊂ F ˚ . The variation ‖θ‖(v,Π) of θ defines a countably additive ℝ+ -valued measure on R. Proof. We observed before that the values of ‖θ‖(v,Π) are nonnegative, and ‖θ‖(v,Π) =0 H is apparent. For disjoint sets A, B ∈ R, our definition returns immediately that ‖θ‖(v,Π) + ‖θ‖(v,Π) ⩽ ‖θ‖(v,Π) A B A∪B . In particular, we have (v,Π) ‖θ‖(v,Π) ⩽ ‖θ‖(v,Π) + ‖θ‖(v,Π) A A B\A ⩽ ‖θ‖B

whenever A ⊂ B. Now let ⋃∞ i=1 Ai = A ∈ R for disjoint sets Ai ∈ R. Then n

⩽ ‖θ‖(v,Π) ⩽ ‖θ‖(v,Π) ∑ ‖θ‖(v,Π) A A (⋃n A ) i=1

i

i=1

i

(v,Π) holds for all n ∈ ℕ by the above. This yields ∑∞ ⩽ ‖θ‖(v,Π) . For the reverse i=1 ‖θ‖Ai A inequality, let Bk ∈ R be disjoint subsets of A, let χBk ⋅ bk ⩽ v and μk ∈ Π for k = 1, . . . , n. For every k, we have Bk = ⋃∞ i=1 (Bk ∩ Ai ) and, therefore, θBk (bk )(μk ) = θ (b )(μ ). Since the sets B ∩ A ⊂ A are disjoint and since χ(Bk ∩Ai ) ⋅ bk ⩽ v ∑∞ (B ∩A ) k k k i i=1 k i for all k and i, we conclude that

2.1 Operator-valued measures n

n ∞

k=1

k=1 i=1 ∞ n

� 55

∑ θBk (bk )(μk ) = ∑ ∑ θ(Bk ∩Ai ) (bk )(μk ) = ∑ ∑ θ(Bk ∩Ai ) (bk )(μk ) i=1 k=1 ∞

⩽ ∑ ‖θ‖(v,Π) . A i=1

i

(v,Π) (v,Π) Hence ‖θ‖(v,Π) ⩽ ∑∞ is understood to be countably additive as k=1 ‖θ‖Ai , and ‖θ‖ A claimed.

2.1.7 Proposition. Let θ be an L(𝒫E , 𝒬F )-valued measure, which coincides with its modulus on 𝒫E . If for every A ∈ R and W ∈ 𝒲 , there is a function space neighborhood v ˝ ) < +∞, then θ is strongly additive at every bounded element of 𝒫E . such that ‖θ‖(v,W A Proof. Let A = ⋃i∈ℕ Ai ∈ R for disjoint sets Ai ∈ R and let W ∈ 𝒲 . There is a func˝ ) < +∞. Let C be a bounded element tion space neighborhood v such that ‖θ‖(v,W A of 𝒫E . There is V ∈ 𝒱 such that χA ⋅ V ⩽ v. Hence |θ|B (V )(μ) ⩽ ‖θ‖(v,W B

˝

)

< +∞

for all μ ∈ W ˝ and all subsets B ∈ R of A by Lemma 2.1.5(a). There is λ > 0 such that both 0 ⩽ C + λV and C ⩽ λV . Because θ and |θ| are supposed to coincide at C, we calculate that θB (C)(μ) = |θ|B (C)(μ) ⩽ λ|θ|B (V ) ⩽ λ‖θ‖(v,𝕎 B

˝

)

and 0 ⩽ |θ|B (C + λV )(μ) = |θ|B (C)(μ) + λ|θ|B (V )(μ) ⩽ θB (C)(μ) + λ‖θ‖(v,𝕎 B

˝

)

) holds for all μ ∈ W ˝ and all subsets B ∈ R of A. Hence |θB (C)|(μ) ⩽ λ‖θ‖(v,𝕎 . Set B ∞ Bn = ⋃i=n+1 Ai , that is, θBn = θA − θ(⋃ni=1 Ai ) . According to Proposition 2.1.6 for every ˝

ε > 0 there is n0 ∈ ℕ such that ‖θ‖(v,W B n

˝

)

⩽ ε/λ for all n ⩾ n0 . Thus

sup{θBn (C)(μ) | μ ∈ W ˝ } ⩽ ε, that is, θBn (C) ⩽ σW and θBn (C) ∈ W s (0). Our claim follows. In a straightforward way ‖θ‖(v,Π) can be extended to an ℝ+ -valued measure on the σ-field A that is associated with R if we set ‖θ‖(v,Π) = sup ‖θ‖(v,Π) B A∩B A∈R

56 � 2 Integration for all B ∈ A. Countable additivity on A is immediate since for disjoint sets Bi ∈ A we calculate ‖θ‖(v,Π) = sup ‖θ‖(v,Π) ⋃ B (A∩⋃ i∈ℕ

i

A∈R

i∈ℕ

Bi )

= sup ∑ ‖θ‖(v,Π) A∩B i

A∈R i∈ℕ

(v,Π) = sup sup ‖θ‖(v,Π) A∩B = sup sup ‖θ‖A∩B

= =

A∈R i∈ℕ sup ‖θ‖(v,Π) Bi i∈ℕ . ∑ ‖θ‖(v,Π) Bi i∈ℕ

i

i∈ℕ A∈R

i

Measure of bounded variation An L(𝒫E , 𝒬F )-valued measure θ is of bounded variation relative to a function space neighborhood v and a nonempty subset Π of F ˚ if sup ‖θ‖(v,Π) < +∞. A

A∈R

2.1.8 Examples and Remarks. (a) Boundedness of a measure θ implies that for every A ∈ R and every bounded set C ∈ Conv(E) the functional |θ|A (C) is a bounded element of the locally convex cone 𝒬F . Indeed, for 𝒲 ∈ 𝒲 there is V ∈ 𝒱 such that |θ|A (V ) ⩽ σW . Since C ⊂ λV for some λ ⩾ 0, this yields |θ|A (C) ⩽ λ|θ|A (V ) < λσW . Because for every μ ∈ F ˚ there is W ∈ 𝒲 such that μ ∈ W ˝ , this implies in particular that |θ|A (C) is real-valued on F ˚ . Consequently, if θ is L(𝒫E , F)-valued, then the set n

n

i=1

i=1

󵄨󵄨 {∑ θAi (ci ) 󵄨󵄨󵄨 ci ∈ C, Ai ∈ R disjoint, ⋃ Ai = A} 󵄨 is bounded in the given topology of F. Similarly, if θ is L(𝒫E , F ˚˚ )-valued, then this set is bounded in the natural topology of F ˚˚ . (b) If X is compact, if both E and F are normed spaces with unit balls 𝔹E and 𝔹F , respectively, and if 𝒞V (X, E) carries the topology of uniform convergence, then the variation of an L(E, F)-valued measure θ relative to the unit ball 𝔹𝒞 of 𝒞V (X, E) and the polar of 𝔹F for Π ⊂ F ˚ is given by (𝔹𝒞 ,𝔹˝F )

‖θ‖A

n 󵄨󵄨 = sup{∑ ‖θAi ‖ 󵄨󵄨󵄨 Ai ∈ R disjoint, Ai ⊂ A} 󵄨 i=1

for A ∈ A, where ‖θA ‖ is the operator norm of the operator θA ∈ L(E, F). This is of course the usual definition of the variation of a vector-valued measure (see III.1.4 in [26]).

2.1 Operator-valued measures

� 57

An example of a bounded measure, that is, a measure with bounded semivariation, but unbounded variation can be found in [21] I.1.6. (c) If an L(𝒫E , 𝒬F )-valued measure θ is indeed L(𝒫E , F)-valued (or L(𝒫E , F ˚˚ )valued), then its values on sets A ∈ R are linear operators from 𝒫E into the subspace of 𝒬F , which consists of all real-valued linear functionals on F ˚ that are continuous with respect to the weak* topology (or, equivalently, to the strong topology β(F ˚ , F)) of F ˚ . The values of its modulus |θ| then are linear operators from Conv(E) into the subcone of 𝒬F consisting of all ℝ-valued sublinear functionals on F ˚ that are lower semicontinuous with respect to the weak* topology (or the strong topology) of F ˚ . This follows immediately from the above definition of |θ|. In the light of the embedding of Conv(F) (or of Conv(F ˚˚ )) into 𝒬F , which was elaborated on in Section 1.4, the values of |θ| may therefore be considered to be closed, convex subsets of F (or of F ˚˚ ). Moreover, as for all A ∈ R the elements θA (C) for all C ∈ 𝒫E are real-valued linear functionals on F ˚ in this case, C ⊂ D for C, D ∈ 𝒫E implies that |θ|A (D) = θA (D) = θA (C) = |θ|A (C). Indeed, we have |θ|A (C) ⩽ θA (C) by Theorem 2.1.3. Since |θ|A (C) is sublinear and θA (C) is linear, hence finite on F ˚ , we deduce that |θ|A (C) = θA (C) and likewise, |θ|A (D) = θA (D). The relation θA (C) ⩽ θA (D) follows from the monotonicity of the operator θA ∈ L(𝒫E , 𝒬F ) (Lemma 1.4.10). Hence θA (C) = θA (D) since both are linear functionals on F ˚ . For elements of D, that is for singleton subsets of D, we conclude in particular that θA (d) = θA (D) = |θ|A (D) holds for all d ∈ D in this case. (d) If F = ℂ, then F ˚ = ℂ and 𝒬F consists of all ℝ-valued sublinear functionals on ℂ that are bounded below on the unit ball of ℂ. If an L(𝒫E , 𝒬F )-valued measure θ is indeed L(𝒫E , F)-valued, then its values on sets A ∈ R and C ∈ 𝒫E are complex numbers, operating as real-valued linear functionals on F ˚ = ℂ as z 󳨃→ Re(θA (C) z) : ℂ → ℝ. The values of its modulus |θ| are computed for every A ∈ R and C ∈ Conv(E) as an ℝ-valued sublinear functional on ℂ by n

n

i=1

i=1

󵄨󵄨 |θ|A (C)(z) = sup{∑ Re(zθAi (ci )) 󵄨󵄨󵄨 ci ∈ C, Ai ∈ R disjoint, ⋃ Ai = A}, 󵄨 for z ∈ ℂ. For balanced sets C ∈ Conv(E), this expression simplifies to n 󵄨 󵄨 󵄨󵄨 |θ|A (C)(z) = sup{∑󵄨󵄨󵄨θAi (ci )󵄨󵄨󵄨 󵄨󵄨󵄨 ci ∈ C, Ai ∈ R disjoint, Ai ⊂ A}|z|. 󵄨 i=1

The latter applies in particular to the neighborhoods in 𝒱 and reveals that our requirement of boundedness for the measure θ corresponds to Dieudonné’s notion of

58 � 2 Integration p-domination in [22] and to Prolla’s of finite p-semivariation in Chapter 5.5 of [42] for measures with values in the dual of a locally convex topological vector space. The case that F = ℝ is similar. (e) If E = ℂ and F is a locally convex topological vector space over ℂ, then every

F ‚ -valued measure θ that is countably additive pointwise on the elements of F ˚ ˚

may be considered to be an operator-valued measure in our sense since every element ω ∈ F ˚‚ constitutes a continuous linear operator from ℂ into F ˚‚ via z 󳨃→ zω. Hence θ is L(ℂ, 𝒬F )-valued, and we have θA (z)(μ) = Re μ(zθA ) for A ∈ R, z ∈ ℂ and μ ∈ F ˚ . For C ∈ Conv(ℂ), the modulus of θ computes as n

n

i=1

i=1

󵄨󵄨 |θ|A (C)(μ) = sup{∑ Re μ(zi θAi ) 󵄨󵄨󵄨 zi ∈ C, Ai ∈ R disjoint, ⋃ Ai = A} 󵄨 for all μ ∈ F ˚ . For the unit ball 𝔹 of ℂ, in particular, we have n 󵄨 󵄨 󵄨󵄨 |θ|A (𝔹)(μ) = sup{∑󵄨󵄨󵄨μ(θAi )󵄨󵄨󵄨 󵄨󵄨󵄨 Ai ∈ R disjoint, Ai ⊂ A}, 󵄨 i=1

and in case that θ is F-valued and that F is a normed space, 󵄩󵄩 n 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄨󵄨 sup |θ|A (𝔹)(μ) = sup{󵄩󵄩󵄩∑ zi θAi 󵄩󵄩󵄩 󵄨󵄨󵄨 |zi | ⩽ 1, Ai ∈ R disjoint, Ai ⊂ A}. 󵄩󵄩 󵄩󵄩 󵄨 ‖μ‖⩽1 󵄩i=1 󵄩 The latter is the notion of the semivariation of a vector-valued measure for a normed space as given for example in Chapter IV.10.3 of [26]. The assumptions of Proposition 2.1.1 apply in this case, hence any such measure is strongly additive. (See Theorem 1 in IV.10.3 in [26].) The assumptions of Corollary 2.1.4 for its modulus can however not be guaranteed. The case that E = ℝ and that F is a vector space over ℝ is similar. (f) If E = F and 𝒫E = E, then every real- or complex-valued measure θ may be considered to be an L(E, E)-valued, hence L(E, 𝒬E )-valued measure in our sense since every z in ℝ or ℂ constitutes a continuous linear operator from E to E via a 󳨃→ za. We have θA (a)(μ) = Re μ(θA a) for A ∈ R, z ∈ ℂ and μ ∈ F ˚ . For C ∈ Conv(E), we calculate n n 󵄨󵄨 |θ|A (C)(μ) = sup{∑ Re μ(θAi ci ) 󵄨󵄨󵄨 ci ∈ C, Ai ∈ R disjoint, ⋃ Ai = A} 󵄨 i=1

i=1

for all μ ∈ E ˚ . If C is balanced, this expression simplifies to n

󵄨󵄨 |θ|A (C)(μ) = sup{∑ |θAi | 󵄨󵄨󵄨 Ai ∈ R disjoint, Ai ⊂ A}pC (μ), 󵄨 i=1

where

2.1 Operator-valued measures

� 59

󵄨 󵄨 pC (μ) = sup{󵄨󵄨󵄨μ(c)󵄨󵄨󵄨 | c ∈ C} ∈ ℝ+ . According to Proposition 2.1.1, the measure θ is strongly additive at all elements of E, and the above implies that |θ| is strongly additive at all neighborhoods in 𝒱 . Following Lemma 2.1.4, θ is therefore strongly additive at all bounded subsets of E. (g) If both E = F = ℂ, then L(E, F) = ℂ, F ˚ = ℂ and Conv(E) consists of all nonempty convex subsets of ℂ. Thus an L(E, F)-valued measure is indeed complexvalued, and the expressions from (f) read as n n 󵄨󵄨 |θ|A (C)(z) = sup{∑ Re(θAi ci z) 󵄨󵄨󵄨 ci ∈ C, Ai ∈ R disjoint, ⋃ Ai = A} 󵄨 i=1

i=1

for C ∈ Conv(ℂ), and for the unit ball 𝔹 of ℂ in particular n

󵄨󵄨 |θ|A (𝔹)(z) = sup{∑ |θAi | 󵄨󵄨󵄨 Ai ∈ R disjoint, Ai ⊂ A}|z| 󵄨 i=1

for all z ∈ ℂ. The positive real-valued measure [|θ|(𝔹)(1)], that is, the mapping A 󳨃→ |θ|A (𝔹)(1) : R → ℝ+ is the usual absolute value of a complex-valued measure (see, e. g., III.1.4 in [26]). The case that E = F = ℝ is similar. (h) For a counterexample to strong additivity of a measure consider the following. Let X = [0, 1] and let R be the σ-field of all Borel subsets of X. Let E = ℝ and F = 𝒞 ([0, 1]) endowed with the supremum norm. By the Riesz representation theorem, F ˚ consists of all regular Borel measures on X, and L(E, F ˚‚ ) may be identified with F ˚‚ (see Part (e)). We define an F ˚‚ -valued measure θ by θA (μ) = μ(A), that is, the value of the Borel measure μ at A, for all μ ∈ F ˚ and A ∈ R. The countable additivity of the measures μ ∈ F ˚ transfers and yields the countable additivity of θ, pointwise on F ˚ . Moreover, since n 󵄨 󵄨 󵄨󵄨 |θ|A (𝔹)(μ) = sup{∑󵄨󵄨󵄨μ(Ai )󵄨󵄨󵄨 󵄨󵄨󵄨 Ai ∈ R disjoint, Ai ⊂ A} ⩽ ‖μ‖, 󵄨 i=1

where 𝔹 = [−1, +1] is the unit ball of ℝ and ‖μ‖ is the dual norm of μ ∈ F ˚ , θ is seen to be bounded. It is however not countably additive with respect to the symmetric topology of 𝒬F , that is, the natural topology of F ˚‚ . Indeed, consider the sets An = (0, 1/n). Then limn→∞ θAn (μ) = 0 for all μ ∈ F ˚ , but this convergence is not uniform on the dual unit ball in F ˚ .

60 � 2 Integration Compact and weakly compact measures An L(𝒫E , 𝒬F )-valued measure θ is called compact (or weakly compact) if it is L(𝒫E , F)valued and for every A ∈ R and every bounded subset C ∈ Conv(E) the set n n 󵄨󵄨 {∑ θAi (ci ) 󵄨󵄨󵄨 ci ∈ C, Ai ∈ R disjoint, ⋃ Ai = A} 󵄨 i=1

i=1

is relatively compact (or relatively weakly compact) in F. This set was of course used in our definition of the modulus of θ and was shown to be bounded in F in Example 2.1.8(a). If F is semireflexive, then every bounded subset of F is relatively weakly compact (see IV.5.5 in [56]). Hence every bounded L(𝒫E , F)-valued measure is weakly compact in this case. Because the balanced convex hull of a bounded set in Conv(E) is again bounded, it suffices to verify the condition for compactness or weak compactness of a measure for balanced convex sets C ∈ Conv(E). Moreover, since the closures of a convex set in the given and in the weak topologies of F coincide, the latter implies that every compact measure is also weakly compact. Restriction of a measure The restriction θ|B of a measure θ to a set B ∈ A is the measure A 󳨃→ θA∩B : R → L(𝒫E , 𝒬F ). The following is immediate from the respective definitions. 2.1.9 Proposition. The restriction θ|B of a (bounded) L(𝒫E , 𝒬F )-valued measure θ to a set B ∈ A is again bounded and its modulus is the restriction of |θ| to B. If θ is compact (or weakly compact), then θ|B is compact (or weakly compact). Sets of measure zero A set Z ∈ A is of measure zero (with respect to an L(𝒫E , 𝒬F )-valued measure θ) if θA = 0 for all subsets A ∈ R of Z. 2.1.10 Proposition. Let θ be an L(𝒫E , 𝒬F )-valued measure. (a) A set B ∈ A is of measure zero if and only if all subsets A ∈ R of B are of measure zero. (b) If A ∈ R is of measure zero, then both |θA | = 0 and ‖θ‖(v,Π) = 0 for any choice of A v and Π. (c) If the sets Bi ∈ A, for i ∈ ℕ, are of measure zero, then their union is of measure zero.

2.1 Operator-valued measures

� 61

Proof. Parts (a) and (b) are clear from the definition. For (c), suppose that the sets Bi ∈ A for i ∈ ℕ are of measure zero and let A ∈ R such that A ⊂ ⋃i∈ℕ Bi . All the sets Ai = A ∩ Bi are of measure zero by (a) and there union is A. We set C1 = A1 and Cn+1 = An+1 \ ⋃ni=1 Ci . Again, all the sets Ci are of measure zero, they are disjoint and their union is A. Thus θA = ∑∞ i=1 θCi = 0. For the following result, we say that, given a measure θ, a subset Π of F ˚ is θdense in F ˚ if for all A ∈ R and a ∈ E we have θA (a) = 0 whenever θA (a)(μ) = 0 for all μ ∈ Π; that is to say, if the linear span of Π is dense in F ˚ in the weak topology of the dual pair (F ˚ , F θ ), where F θ is the linear span in F ˚‚ of the set {θA (a) | A ∈ R, a ∈ E}. 2.1.11 Proposition. Let θ be an L(𝒫E , 𝒬F )-valued measure, which coincides with its modulus on 𝒫E and let A ∈ R. (a) If there is V ∈ 𝒱 such that |θ|A (V ) = 0, then A is of measure zero. (b) If there is a function space neighborhood v and a θ-dense subset Π of F ˚ such that ‖θ‖(v,Π) = 0, then A is of measure zero. A Proof. Let A ∈ R. For Part (a), suppose that θ satisfies our assumptions and that there is V ∈ 𝒱 such that |θ|A (V ) = 0. For every a ∈ E, there is ε > 0 such that γa ∈ V for all |γ| ⩽ ε. Hence θB (a) = 0 for every subset B ∈ R of A. Inspecting the definition of the modulus, we conclude that θB (C) = |θ|B (C) = 0 holds for all C ∈ 𝒫E , that is, θB = 0, for all such sets B. Hence our claim follows. For Part (b), suppose that ‖θ‖(v,Π) = 0 for A A ∈ R and that Π is θ-dense. According to our definition of a function space neighborhood, there is V ∈ 𝒱 such that χA ⋅ V ⩽ v. Then |θ|A (V )(μ) = 0 holds for all μ ∈ Π by Lemma 2.1.5 and, therefore, θB (a) = 0 for all subsets B ∈ R of A and all a ∈ E by our assumption on Π. As in our argument for Part (a), we conclude that θB = 0 for all subsets B ∈ R of A. Properties that hold almost everywhere If Z ∈ A is of measure zero, the measure θ is said to be supported by its complement Y = X \ Z, since θA = θA∩Y holds for all A ∈ R. For a subset B of X, we shall say that a pointwise defined property of functions on X holds θ-almost everywhere on B if it holds on B \ Z with some set Z ∈ A of measure zero. In particular, we shall use the symbols ⩽ or = if the relations ⩽ or = hold θ-almost everywhere on the set B, a.e. B

a.e. B

respectively. Likewise, f = g, f ⩽ g or f ⩽ g + v for functions f , g ∈ ℱ (X, Conv(E)) a.e. B

a.e. B

a.e. B

means that χ(B\Z) ⋅f = χ(F\Z) ⋅g, χ(B\Z) ⋅f ⩽ χ(F\Z) ⋅g or χ(B\Z) ⋅f ⩽ χ(F\Z) ⋅g +v holds with some Z ∈ A of measure zero. These relations are of course transitive and compatible with the algebraic operations. Two measures are said to be mutually singular if they are supported by disjoint subsets of X.

62 � 2 Integration Point evaluation measures A point evaluation measure is a nonzero measure supported by a singleton set, that is, a measure δxT , for x ∈ X and 0 ≠ T ∈ L(𝒫E , 𝒬F ) such that for every A ∈ R it evaluates as (δxT )A = T if x ∈ A, and (δxT )A = 0, else. Regularity of measures We continue with some particular consideration in the topological case, that is, the case that X is a locally compact Hausdorff space and that R consists of all relatively compact Borel subsets of X. We shall assume this setting for the remainder of this section. The concept of regularity of a measure then arises naturally, and we shall approach it in the following way: An L(𝒫E , 𝒬F )-valued measure θ is said to be outer regular (or inner regular) for a set A ∈ R if for every μ ∈ F ˚ there is V ∈ 𝒱 such that |θ|A (V )(μ) < +∞ and |θ|A (V )(μ) = lim |θ|O (V )(μ) (or |θ|A (V )(μ) = lim |θ|K (V )(μ)). K⊂A

O⊃A

The limits are taken over the downward directed family of all open sets O ∈ O containing A for outer regularity, and over the upward directed family of all compact subsets K of A for inner regularity. We realize that the above conditions can be rewritten as lim |θ|O\A (V )(μ) = 0

O⊃A

and

lim |θ|A\K (V )(μ) = 0,

K⊂A

respectively. This follows from the additivity of the modulus, that is, |θ|O\A (V )(μ) = |θ|O (V )(μ) − |θ|A (V )(μ) and |θ|A\K (V )(μ) = |θ|A (V )(μ) − |θ|K (V )(μ), provided that |θ|A (V )(μ) < +∞. Consequently, the above limits are valid with all neighborhoods U ∈ 𝒱 in place of V whenever U ⊂ V . 2.1.12 Lemma. If θ is outer regular (or inner regular) for A ∈ R, then |θ|A (C) = lim |θ|O (C) O⊃A

(or |θ|A (C) = lim |θ|K (C)) K⊂A

holds pointwise on F ˚ for all bounded sets C ∈ Conv(E). Proof. Let C ∈ Conv(E) be a bounded subset of E and let μ ∈ F ˚ . Recall from Example 2.1.8(a) that |θ|A (C)(μ) < +∞ for all A ∈ R. There is V ∈ 𝒱 as in the definition

2.1 Operator-valued measures

� 63

of outer regularity. Since C ⊂ λV for some λ > 0, for every O ∈ O containing A we have 󵄨 󵄨 󵄨 󵄨󵄨 󵄨󵄨|θ|O (C)(μ) − |θ|A (C)(μ)󵄨󵄨󵄨 = 󵄨󵄨󵄨|θ|O\A (C)(μ)󵄨󵄨󵄨 ⩽ λ|θ|O\A (V )(μ). This yields limO⊃A |θ|O (a)(μ) = |θ|A (a)(μ) as claimed. The argument for the bracketed case is similar. Recall from Theorem 2.1.3 that θ and |θ| coincide on the elements of E. If the measure θ is indeed L(E, F)-valued (or L(E, F ˚˚ )-valued), then the statement of Lemma 2.1.12 implies convergence of the operators θO and θK to θA in the weak operator topology of L(E, F) (or in the weak* operator topology of L(E, F ˚˚ )). A measure that is both outer and inner regular for all sets in R is called regular. 2.1.13 Proposition. If θ is either inner regular for all sets in R or outer regular for all sets in R, then θ is regular. Proof. First, suppose that θ is inner regular for all sets in R. We shall demonstrate outer regularity for all sets in R. For this, let A ∈ R and let U ∈ O be an open set containing A. Since θ is inner regular for U \ A, given μ ∈ F ˚ there is V ∈ 𝒱 such that both |θ|U (V )(μ) ⩽ +∞ and for every ε > 0 there exists a compact subset K of U \ A such that |θ|(U\A)\K (V )(μ) ⩽ ε. Let O = U \ K. Obviously, O is open and contains A. Indeed, if x ∈ A, then x ∈ U and x ∉ K ⊂ U \ A. Because (U \ A) \ K = (U \ K) \ A = O \ A we compute that |θ|O\A (V )(μ) = |θ|(U\A)\K (V )(μ) ⩽ ε. Hence θ is outer regular for A. Now suppose that θ is outer regular for all sets in R. We shall demonstrate inner regularity for all sets in R. For this, let A ∈ R and let U ∈ O be an open set containing A, the closure of A. Since θ is outer regular for U \ A, given μ ∈ F ˚ there is V ∈ 𝒱 such that both |θ|U (V )(μ) < +∞ and for every ε > 0 there exists an open set O ∈ O containing U \ A such that |θ|O\(U\A) (V )(μ) ⩽ ε. We may assume that O ⊂ U. Let K = A \ O. We recognize that K is a compact subset of A as A \ O = A \ O. Indeed, if x ∈ A \ O, then x ∈ U, but x ∉ O, hence x ∉ U \ A. That is to say, x ∈ A. Next, we observe that O \ (U \ A) = O ∩ A = A \ (A \ O) = A \ K. This shows |θ|A\K (V )(μ) = |θ|O\(U\A) (V )(μ) ⩽ ε, and θ is seen to be inner regular for A. Our claim follows.

64 � 2 Integration Some of the earlier introduced concepts can be augmented for regular measures. 2.1.14 Proposition. Let A ∈ R. If θ is regular, then n 󵄨󵄨 |θ|A (C) = sup{∑ θKi (ci ) 󵄨󵄨󵄨 ci ∈ C, Ki disjoint compact, Ki ⊂ A} 󵄨 i=1

for C ∈ Conv(E) such that 0 ∈ C, and n 󵄨󵄨 K disjoint compact, K ⊂ A, 󵄨󵄨 i i ‖θ‖(v,Π) = sup { θ (a )(μ ) } ∑ 󵄨󵄨 K i i A i 󵄨󵄨 ai ∈ E, χK ⋅ ai ⩽ v and μi ∈ Π i i=1

for a function space neighborhood v and Π ⊂ F ˚ . Proof. For the first statement concerning the modulus of θ, since 0 ∈ C, the supremum on the right-hand side of the above expression is less or equal to |θ|A (C). For the reverse inequality, let Ai ∈ R be disjoint subsets of A, and let ci ∈ C for i = 1, . . . , n. Because θ is regular, we have θAi (ci ) = limKi ⊂Ai θAi (ci ), where Ki are compact subsets of Ai , for all k = 1, . . . , n. Hence n

n

∑ θAi (ci ) = lim ∑ θKi (ci ). Ki ⊂Ai

i=1

i=1

Thus n

n

i=1

i=1

󵄨󵄨 ∑ θAi (ci ) ⩽ sup{∑ θKi (ci ) 󵄨󵄨󵄨 ci ∈ C, Ki disjoint compact, Ki ⊂ A}, 󵄨 our claim. The argument for the variation of θ is similar. 2.1.15 Proposition. If θ is outer (or inner) regular for a set A ∈ R, then |θ| is also outer (or inner) regular for A. If θ is regular and C ∈ Conv(E), then |θ|A (C) = limK⊂A |θ|K (C) holds for all A ∈ R. If in addition |θ|A (C)(μ) < +∞ holds for μ ∈ F ˚˚ and all A ∈ R, then |θ|A (C)(μ) = limO⊃A |θ|O (C)(μ). Proof. The first statement is evident from the definition of regularity together with the finding from Theorem 2.1.3 that the modulus of the modulus |θ| of a measure θ is again |θ|. For the second statement, suppose that θ is regular and let A ∈ R. In a first step, we consider a set C ∈ Conv(E) such that C ⩾ 0, that is, 0 ∈ C. The inequality lim |θ|K (C) = sup |θ|K (C) ⩽ |θ|A (C)

K⊂A

K⊂A

is obvious in this case. For the reverse inequality, for any α < |θ|A (C), by Proposition 2.1.14 there are disjoint compact subsets Ki of A and ci ∈ C such that n

α ⩽ ∑ θKi (ci ) ⩽ |θ|K (C), i=1

� 65

2.1 Operator-valued measures

whereby K = ⋃ni=1 Ki ⊂ A. Hence |θ|A (C) ⩽ supK⊂A |θ|K (C). Now for any C ∈ Conv(E) we choose an element c0 ∈ C and set C0 = C −c0 ∈ Conv(E). Then C0 ⩾ 0 and we have lim |θ|K (C) = lim |θ|K (C0 ) + lim |θ|K (c0 )

K⊂A

K⊂A

K⊂A

= |θ|A (C0 ) + |θ|A (c0 ) = |θ|A (C) by our first step together with Lemma 2.1.12. If |θ|A (C)(μ) < +∞ for μ ∈ F ˚˚ and all A ∈ R, then the set function A 󳨃→ |θ|A (C)(μ) : R → ℝ is countably additive by Theorem 2.1.3 and defines a positive real-valued measure [|θ|A (C)(μ)] on R. This measure is inner regular for all sets in R and, therefore, regular by Proposition 2.1.13. Our claim follows. The transfer of regularity of a measure to its ℝ+ -valued variation is however less obvious. 2.1.16 Proposition. Let v be a function space neighborhood and Π ⊂ F ˚ . If θ is regular < +∞ for O ∈ O, then the ℝ+ -valued measure ‖θ‖(v,Π) is both outer and if ‖θ‖(v,Π) O and inner regular for all subsets A ∈ R of O. Proof. Suppose that θ is regular. Using Proposition 2.1.14, we realize that in the calculation of ‖θ‖(v,Π) it suffices to consider only disjoint compact subsets Ki of A. Let A O0 ∈ O such that ‖θ‖(u,Π) < +∞ and assume that for some subset A ∈ R of O0 we O 0

. Hence there is ε > 0 such that ‖θ‖(u,Π) > ε holds for all < infO⊃A ‖θ‖(u,Π) have ‖θ‖(u,Π) A O O\A O ∈ O such that O ⊃ A. Choose disjoint compact subsets Ki of O0 \ A, χKi ⋅ ai ⩽ u and μi ∈ Π for i = 1, . . . , n such that ∑ni=1 θKi (ai )(μi ) ⩾ ε. Now we set O1 = O0 \ ⋃ni=1 Ki ∈ O. Then O0 ⊃ O1 ⊃ A and ‖θ‖(u,Π) ⩾ ε. Repeating the preceding step with O1 in place of O \O 0

1

O0 , we find O2 ∈ O such that O1 ⊃ O2 ⊃ A and ‖θ‖(u,Π) ⩾ ε, that is, ‖θ‖(u,Π) ⩾ 2ε. O \O O \O 1

2

0

2

After n such steps, we have On ∈ O such that On−1 ⊃ On ⊃ A and ‖θ‖(u,Π) ⩾ nε. O \O 0

n

= +∞, thus contradicting our assumption Since O0 \ On ⊂ O0 \ A, this renders ‖θ‖(u,Π) O \A 0

that ‖θ‖(u,Π) < +∞. The argument for inner regularity is similar. O 0

2.1.17 Proposition. Let θ be an L(𝒫E , 𝒬F )-valued measure such that for every A ∈ R ˝ ) and W ∈ 𝒲 there is a function space neighborhood v such that ‖θ‖(v,W < +∞. If θ A is regular, then θA (C) = lim θO (C) = lim θK (C) O⊃A

K⊂A

for all bounded sets C ∈ 𝒫E and all A ∈ R in the symmetric topology of 𝒬F .

66 � 2 Integration Proof. Suppose that θ is regular and satisfies our assumption for its variation. Let A ∈ R, W ∈ 𝒲 and let C be a bounded element of 𝒫E . We choose an open set O0 ∈ O ˝ ) containing A and a function space neighborhood v such that ‖θ‖(v,W < +∞. There O0 is V ∈ 𝒱 such that χO0 ⋅ V ⩽ v and there is λ > 0 such that C ⊂ λV . According to (v,W Proposition 2.1.16, given ε > 0 there is O1 ∈ O such that O0 ⊃ O1 ⊃ A and ‖θ‖O\A ε/λ for all O ∈ O such that O1 ⊃ O ⊃ A. Thus

˝

)

⩽

θO\A (C)(μ) = |θ|O\A (C)(μ) ⩽ ε for all μ ∈ W ˝ by Lemma 2.1.5(a). Since −μ ∈ W ˝ whenever μ ∈ W ˝ and since θO\A (C)(μ) ⩾ −θO\A (C)(−μ) ⩾ −ε, we infer that both θO (C)(μ) = θA (C)(μ) + θO\A (C)(μ) ⩽ θA (C)(μ) + ε and θA (C)(μ) = θO (C)(μ) − θO\A (C)(μ) ⩽ θO (C)(μ) + ε holds for all μ ∈ W ˝ and all O ∈ O such that O1 ⊃ O ⊃ A. But this verifies that θO (C) is an element of the symmetric neighborhood W s (θA (C)) of θO (C) (see Section 1.4). The argument for the second equality regarding inner regularity is similar. 2.1.18 Proposition. The restriction θ|B of an L(𝒫E , 𝒬F )-valued measure θ to a set B ∈ A is outer (or inner) regular for a set A ∈ R whenever θ is outer (or inner) regular for A. Proof. If θ is outer regular for a set A ∈ R, then for every μ ∈ F ˚ there is V ∈ 𝒱 such that |θ|A (V )(μ) < +∞ and that limO⊃A |θ|O\A (V )(μ) = 0. Since |(θ|B )|O\A (V ) ⩽ |θ|O\A (V ), this yields limO⊃A |(θ|B )|O\A (V )(μ) = 0 as well. The argument for inner regularity is similar. 2.1.19 Proposition. Let θ be a regular L(𝒫E , 𝒬F )-valued measure, which coincides with its modulus on 𝒫E , and suppose that all sets in 𝒫E are bounded. If θK = 0 for every compact subset K of a set B ∈ A, then B is of measure zero. The union of a family of open sets of measure zero is of measure zero. Proof. Suppose that θ satisfies our assumptions, and let B ∈ A and let A ∈ R be a subset of B. Together with our assumptions, Lemma 2.1.12 yields that θA = 0, provided that θK = 0 for every compact subset K of A. Our first claim follows. Now let 𝒪 be a family of open sets of measure zero and let K be a compact subset of their union. There are finitely many sets Oi ∈ 𝒪 such that K ⊂ ⋃ni=1 Oi , and their union is of measure zero by Proposition 2.1.10(c). Hence K is of measure zero by 2.1.10(a), and our second statement follows from the first one.

2.1 Operator-valued measures

� 67

If θ satisfies the assumptions of Proposition 2.1.19, then there exists a largest open subset of measure zero in X. Its complement is the smallest closed subset of X supporting θ, called the support of θ and denoted as supp(θ). Composition measures If (G, 𝒰 ) and (H, ℛ) are additional locally convex topological vector spaces such that H is Hausdorff and if θ is an L(E, 𝒬F )-valued measure, then for any choice of operators S ∈ L(G, E) and T ∈ L(F, H) the formula (T ˝ θ ˝ S A (g))(ν) = (θA (S(g)))(T ˚ (ν)) defines an L(G, 𝒬H )-valued measure T ˝ θ ˝ S. In this expression, we have A ∈ R, g ∈ G, ν ∈ H ˚ , and T ˚ : H ˚ → F ˚ is the adjoint of the operator T ∈ L(F, H). We go forward to check he requirements for a measure for T ˝ θ ˝ S. First, for A ∈ R and g ∈ G the function ν 󳨃→ ((T ˝ θ ˝ S)A (g))(ν) is linear on H ˚ and bounded on the polars of the neighborhoods in ℛ. Indeed for R ∈ ℛ, there is W ∈ 𝒲 such that T(W ) ⊂ R, hence T ˚ (R˝ ) ⊂ W ˝ . There is V ∈ 𝒱 such that θA (a) ⩽ σW for all a ∈ V and there is U ∈ 𝒰 such that S(U) ⊂ V . Then for all u ∈ U we have θA (S(u)) ⩽ σW . There is λ ⩾ 0 such that g ∈ λU. Hence (θA (S(g)))(T ˚ (ν)) ⩽ λ holds for all ν ∈ R˝ . We infer that (T ˝ θ ˝ S)A (g) ∈ 𝒬H . Next, we realize that for every A ∈ R the operator g 󳨃→ (T ˝ θ ˝ S)A (g) is linear from G to 𝒬H and continuous, since for R ∈ ℛ we choose U ∈ 𝒰 as above and have θA (S(g)) ⩽ σW , hence (θA (S(g)))(T ˚ (ν)) ⩽ 1 for all g ∈ U and all ν ∈ R˝ . Summarizing, we have established that (T ˝ θ ˝ S)A ∈ L(G, 𝒬H ). Because the countable additivity of T ˝ θ ˝ S follows immediately from the countable additivity of θ, all that is left to verify is boundedness. For this, we shall work out the modulus of this measure. For A ∈ R, a set D ∈ Conv(G) and ν ∈ H ˚ we have S(D) ∈ Conv(E) and calculate n 󵄨󵄨 d ∈ D, A ∈ R disjoint, 󵄨 i |T ˝ θ ˝ S|A (D)(ν) = sup {∑(T ˝ θ ˝ S)Ai (di )(ν) 󵄨󵄨󵄨 i n } 󵄨󵄨 ⋃i=1 Ai = A i=1 n 󵄨󵄨󵄨 c ∈ S(D), Ai ∈ R disjoint, = sup {∑ θAi (ci )(T ˚ (ν)) 󵄨󵄨󵄨 i n } 󵄨󵄨 ⋃i=1 Ai = A i=1

= |θ|A (S(D))(T ˚ (ν)). Now for boundedness of the measure T ˝ θ ˝ S, given A ∈ R and R ∈ ℛ there is W ∈ 𝒲 such that T ˚ (R˝ ) ⊂ W ˝ and V ∈ 𝒱 such that |θ|A (V ) ⩽ σW . There is U ∈ 𝒰 such that S(U) ⊂ V , and using the above we realize that |T ˝ θ ˝ S|A (U)(ν) ⩽ |θ|A (V )(T ˚ (ν)) ⩽ 1

68 � 2 Integration for all ν ∈ R˝ . That is, |T ˝ θ ˝ S|A (U) ⩽ σR , as required. Regularity like countable additivity is determined pointwise on the elements of F ˚ and H ˚ , respectively. Thus regularity for θ implies regularity for T ˝ θ ˝ S. Moreover, if the measure θ is indeed L(E, F ˚‚ )-valued, that is, θA (αa)(μ) = θA (a)(αμ) holds for all A ∈ R, all a ∈ E, μ ∈ F ˚ and α in the common scalar field of the involved vector spaces, then this property transfers directly to the composition measure. That is, the measure T ˝ θ ˝ S is L(E, H ˚‚ )-valued in this case.

2.2 Measurable functions Measurability for vector-valued functions was introduced in various places (see, e. g., Dunford and Schwartz [26] III.2.10 or Diestel and Uhl [21] II.1). Different from most of these concepts, our forthcoming take on measurability will be quite restrictive and not refer to a particular measure. Rather than being a prerequisite for integrability, we shall establish a class of functions that are integrable with respect to any measure on a given σ-ring R. Integrability with respect to a specific measure will be developed in the following section and extend beyond this class. We shall use the same broad assumptions and notation as in the preceding sections. The set X is generally not required to carry a topological structure, and R is a σ-ring of subsets of X. Any measure, if referred to, is supposed to be defined on R, L(𝒫E , 𝒬F )valued and bounded. The topological scenario, where X is locally compact and R consists of all relatively compact Borel subsets of X, is included as a special case. We shall utilize step functions as introduced in Section 1.2, but it will turn out to be advantageous to have these available as not just E-valued, but indeed as Conv(E)-valued functions. Set-valued step functions We consider Conv(E)-valued step functions on X, that is, functions n

h = ∑ χAi ⋅ Ci i=1

for sets Ai ∈ R and elements Ci in Conv(E). The subcone of ℱ (X, Conv(E)) constituted by these functions is denoted by ℱ𝒮 (X, Conv(E)), the subspace of all E-valued step functions is denoted by ℱ𝒮 (X, E). (Recall that via the embedding a 󳨃→ {a} we consider E as a subspace of Conv(E).) The above representation of a step function in ℱ𝒮 (X, Conv(E)) is of course not unique, but we may obtain a representation where the sets Ai ∈ R are disjoint in the following way.

2.2 Measurable functions

� 69

2.2.1 Lemma. For every step function, h = ∑ni=1 χAi ⋅ Ci there exists a representation n ∑m k=1 χBk ⋅Dk such that the sets Bk ∈ R are pairwise disjoint and such that ∑i=1 |θ|Ai (Ci ) = m ∑k=1 |θ|Bk (Dk ) holds for any L(𝒫E , 𝒬F )-valued measure θ. Proof. Let h = ∑ni=1 χAi ⋅ Ci ∈ ℱ𝒮 (X, Conv(E)) For a nonempty subset I of {1, . . . , n}, set BI = ⋂ Ai i∈I

⋂ X \ Ai .

i∈{1,...,n}\I

The sets BI are contained in R, pairwise disjoint for distinct index sets I, and their union is the union of the sets Ai . Let ℐ be the family of all I such that BI ≠ H, and for every I ∈ ℐ set DI = ∑i∈I Ci . One easily verifies that h = ∑ χBI ⋅ DI . I∈ℐ

Moreover, for every i we have Ai = ⋃i∈I BI . Hence |θ|Ai (Ci ) = ∑i∈I |θ|BI (Ci ) holds for any L(𝒫E , 𝒬F )-valued measure θ and, therefore, n

n

∑ |θ|Ai (Ci ) = ∑ ∑ |θ|BI (Ci ) = ∑ |θ|BI (∑ Ci ) = ∑ |θ|BI (DI ), i=1

i=1 i∈I

I∈ℐ

i∈I

I∈ℐ

our claim. The representation of a step function h = ∑ni=1 χAi ⋅ Ci with pairwise disjoint sets in Ai ∈ R is unique, provided that the elements Ci ∈ Conv(E) are distinct and not zero. In this case, it is called the standard representation of h. We recall from Section 1.4 the definition of boundedness for elements and subsets of ℱ (X, Conv(E)). A function f ∈ ℱ (X, Conv(E)) is bounded above (or below) if for every function space neighborhood v there is λ ⩾ 0 such that f ⩽ λv (or 0 ⩽ f + λv). A subset of ℱ (X, Conv(E)) is bounded above (or below) if these conditions hold with the same λ ⩾ 0 for all functions in this subset. 2.2.2 Lemma. If f ∈ ℱ (X, Conv(E)) is bounded above (or bounded below), then there is A ∈ R such that f (x) = 0 (or f (x) ⩾ 0) for all x ∈ X \ A, and for every V ∈ 𝒱 there is λ ⩾ 0 such that f ⩽ λχA ⋅ V (or 0 ⩽ f + λχA ⋅ V ). The range of f is bounded above (or bounded below) in Conv(E). Proof. For our first statement, let us consider the case that f ∈ ℱ (X, Conv(E)) is bounded below. For every A ∈ R, choose a neighborhood VA ∈ 𝒱 . The convex hull 𝒩 of the neighborhood functions χA ⋅ VA defines a function space neighborhood v (see Example 1.2.7(f)). There is n ∈ 𝒩 and λ ⩾ 0 such that 0 ⩽ f + λn. Because all neighborhood functions in 𝒩 are supported by a set in R, 0 ⩽ f (x) + λn(x) for all x ∈ X implies that there is A ∈ R such that f (x) ⩾ 0 for all x ∈ X \ A. The case of upper boundedness is similar, taking into account that f (x) ⩽ 0 means that f (x) = 0. Now for

70 � 2 Integration V ∈ 𝒱 consider the function space neighborhood v defined by the constant neighborhood function x 󳨃→ V . If f is bounded above (or below), then there is λ ⩾ 0 such that f ⩽ λχX ⋅ V

(or 0 ⩽ f + λχX ⋅ V ).

f ⩽ λχA ⋅ V

(or 0 ⩽ f + λχA ⋅ V ).

That is to say,

The latter also implies that f (X) ⩽ λV (or that 0 ⩽ f (X) + λV ). Our claim follows. 2.2.3 Lemma. (a) Every step function is bounded below. (b) Let f ∈ ℱ (X, Conv(E)). If for every function space neighborhood v there is a bounded below function g ∈ ℱ (X, Conv(E)) such that g ⩽ f + v, then f is bounded below. Proof. (a) Let h = ∑ni=1 χAi ⋅ Ci be a Conv(E)-valued step function and let v be a function space neighborhood. We may assume that the sets Ai are disjoint and set A = ⋃ni=1 Ai ∈ R. There is V ∈ 𝒱 such that χA ⋅ V ⩽ v, and in turn there is λ ⩾ 0 such that 0 ⩽ Ci + λV for all i = 1, . . . , n. This shows 0 ⩽ h + λχA ⋅ V ⩽ h + λv, our claim in (a). Now suppose that the assumption of (b) holds for f ∈ ℱ (X, Conv(E)). Given a function space neighborhood v, there is a bounded below function g ∈ ℱ (X, Conv(E)) such that g ⩽ f + v and λ ⩾ 0 such that 0 ⩽ g + λv. Hence 0 ⩽ f + (λ + 1)v, our claim in Part (b). Measurable vector-valued functions We shall first define measurability for E-valued functions. A function f ∈ ℱ (X, E) is said to be measurable if for every function space neighborhood v there is a step function h ∈ ℱ𝒮 (X, E) such that h ⩽ f + v, that is, h − f ⩽ v. 2.2.4 Proposition. Every measurable function in ℱ (X, E) is bounded and is supported by a set in R. Its range is a bounded subset of E. The measurable functions form a subspace of ℱ (X, E). Proof. The first statements follow from Lemmas 2.2.3 and 2.2.2 and since for E-valued functions upper and lower boundedness coincide. The last statement is obvious, since h − f ⩽ v and l − g ⩽ v for step functions h, l ∈ ℱ𝒮 (X, E) and f , g ∈ ℱ (X, E) implies that (h + l) − (f + g) ∈ 2v and αh − αf ∈ |α|v for all α in ℝ or ℂ. 2.2.5 Proposition. A function f ∈ ℱ (X, E) is measurable if and only if it is supported by a set A ∈ R and for every V ∈ 𝒱 there is h ∈ ℱ𝒮 (X, E) such that h − f ⩽ χA ⋅ V . The step function h can be chosen such that all its values are in the range of f .

2.2 Measurable functions

� 71

Proof. Suppose that f ∈ ℱ (X, E) is measurable and let V ∈ 𝒱 . Using the function space neighborhood v defined by the constant neighborhood function x 󳨃→ (1/2)V in our definition of measurability we find a step function h ∈ ℱ𝒮 (X, E) such that h − f ⩽ (1/2) χX ⋅ V . According to Proposition 2.2.4, the function f is supported by a set A ∈ R and we may assume the same for the step function h. That is, h − f ⩽ (1/2) χA ⋅ V . We have h = ∑ni=1 χAi ⋅ ai for disjoint sets Ai ∈ R whose union is A and ai ∈ E. We choose xi ∈ Ai , set bi = f (xi ) and g = ∑ni=1 χAi ⋅ bi ∈ ℱ𝒮 (X, E). Then all values of g are in the range of f , and for all x ∈ Ai we notice that f (x) − g(x) = f (x) − f (xi ) = (f (x) − h(x)) + (h(xi ) − f (xi )) ∈ V since h(x) = h(xi ). Hence g − f ⩽ χA ⋅ V , our claim. Conversely, if our condition holds for f ∈ ℱ (X, E), there is A ∈ R such that f = χA ⋅ f and for every function space neighborhood v there is V ∈ 𝒱 such that χA ⋅ V ⩽ v. We find h ∈ ℱ𝒮 (X, E) such that h − f ⩽ χA ⋅ V and, therefore, h − f ⩽ v, as required. We shall consider two special cases of sufficient interest. 2.2.6 Proposition. If E is a normed space, then a function in ℱ (X, E) is measurable if and only if it is supported by a set in R and the uniform limit of a sequence of step functions in ℱ𝒮 (X, E). Proof. Let E be a normed space with unit ball 𝔹. According to Proposition 2.2.5, a function f ∈ ℱ (X, E) is measurable if and only if for every n ∈ ℕ there is a step function hn ∈ ℱ𝒮 (X, E) such that hn − f ⩽ χA ⋅ (1/n)𝔹, that is, ‖hn − f ‖ ⩽ 1/n. A real-valued function is measurable if it satisfies our condition of measurability with E = ℝ. We shall however also use the term A-measurable for an ℝ-valued function φ if it is measurable in the conventional sense with respect to the σ-field A, that is, if for every α ∈ ℝ the set {x ∈ X | φ(x) > α} is contained in A (see 11.2 in [53]). 2.2.7 Proposition. A real-valued function in ℱ (X) is measurable if and only if it is bounded, supported by a set in R and A-measurable. Proof. If φ ∈ ℱ (X) is measurable in our sense with E = ℝ, then by 2.2.4 it is bounded and supported by a set A ∈ R. Following Proposition 2.2.6, φ is the uniform limit of a sequence of real-valued step functions and, therefore, A-measurable (Theorem 6 in 11.2 of [53]). Conversely, suppose that φ is bounded, supported by a set in A ∈ R and A-measurable. Let ρ ⩾ 0 be a bound for |φ|. Given a function space neighborhood v for ℱ (X) there is ε > 0 such that χA ⋅(ε𝔹) ⩽ v, where 𝔹 = [−1, +1]. Choose n > (1/ε)ρ. Since φ is A-measurable, the sets Ai = {x ∈ A | (i − 1)ε ⩽ φ(x) < iε}

72 � 2 Integration for i = −n, . . . , n are contained in R and their union is all of A. We choose xi ∈ Ai , provided that Ai ≠ H, and consider the real-valued step function. h = ∑ni=−n χAi ⋅ φ(xi ). Then for every i, we have 󵄨 󵄨 󵄨 󵄨󵄨 󵄨󵄨φ(x) − h(x)󵄨󵄨󵄨 = 󵄨󵄨󵄨φ(x) − φ(xi )󵄨󵄨󵄨 ⩽ ε for all x ∈ Ai . This demonstrates that h − f ⩽ χA ⋅ (ε𝔹) ⩽ v, as required. 2.2.8 Proposition. If φ ∈ ℱ (X) is measurable, then the function φ ⋅ a ∈ ℱ (X, E) is measurable for every a ∈ E. Proof. Suppose that φ ∈ ℱ (X) is measurable and let a ∈ E. There is A ∈ R such that φ is supported by A, and given a function space neighborhood v there is V ∈ 𝒱 such that χA ⋅ V ⩽ v, and there is λ > 0 such that a ∈ λV . There is a real-valued step function ψ such that |ψ(x) − φ(x)| ⩽ (1/λ) for all x ∈ X. We may assume that ψ is also supported by A and set h = ψ ⋅ a ∈ ℱ𝒮 (X, E). Then h(x) − f (x) = (ψ(x) − φ(x))a ∈ V for all x ∈ A and h(x) = f (x) = 0 for all x ∈ X \ A. Therefore, h − f ⩽ χA ⋅ V ⩽ v, as required. The topological case offers the following observation, which is an immediate consequence of Proposition 1.2.5(ii) applied with the finest function space topology V˚ (see Example 1.2.7(f)). 2.2.9 Proposition. Suppose that X is a locally compact Hausdorff space and that R consists of all relatively compact Borel subsets of X. Then for every f ∈ 𝒞 (X, E) and every A ∈ R the function χA ⋅ f ∈ ℱ (X, E) is measurable. In particular, all functions in 𝒞𝒦 (X, E) are measurable. Proof. For every function f ∈ 𝒞 (X, E) and A ∈ R, there is a function g ∈ 𝒞𝒦 (X, E), which coincides with f on A. Because 𝒞𝒦 (X, E) ⊂ 𝒞V˚ (X, E), according to Proposition 1.2.5(ii) for every function space neighborhood v there is a step function h ∈ ℱ𝒮 (X, E) such that h − g ⩽ v and, therefore, χA ⋅ h − χA ⋅ g = χA ⋅ h − χA ⋅ f ⩽ v. Since χA ⋅ h ∈ ℱ𝒮 (X, E), the definition of measurability is satisfied for the function χA ⋅ f . Our next proposition prepares the extension of the concept of measurability to setvalued functions. 2.2.10 Proposition. A function f ∈ ℱ (X, E) is measurable if and only if for every function space neighborhood v there is a Conv(E)-valued step function h such that f ⩽ h ⩽ f +v.

2.2 Measurable functions

� 73

Proof. Suppose that f ∈ ℱ (X, E) is measurable and let v be a function space neighborhood. According to Proposition 2.2.5, f is supported by a set A ∈ R. There is V ∈ 𝒱 such that χA ⋅ V ⩽ (1/2)v, and again using Proposition 2.2.5 we find an E-valued step function g ∈ ℱ𝒮 (X, E) such that both f ⩽ g + χA ⋅ V and g ⩽ f + χA ⋅ V . We choose the Conv(E)-valued step function h = g + χA ⋅ V ∈ ℱ𝒮 (X, Conv(E)) and realize that f ⩽ h. Moreover, since both g ⩽ f +χA ⋅V ⩽ f +(1/2)v and χA ⋅V ⩽ (1/2)v, we have h ⩽ f + v. Thus, the condition in our proposition holds for f with the step function h ∈ ℱ𝒮 (X, Conv(E)). Conversely, let f ∈ ℱ (X, E) and suppose that for every function space neighborhood v there is a Conv(E)-valued step function h such that f ⩽ h ⩽ f + v. For any choice of v, since step functions are supported by sets in R, the relation f ⩽ h implies that f is supported by a set A ∈ R. Given a function space neighborhood v, there is V ∈ 𝒱 such that χA ⋅V ⩽ v. Now we use our assumption with the neighborhood function vV generated by the single neighborhood function of uniform convergence x 󳨃→ V (Example 1.2.7(a)). There is a Conv(E)-valued step function n

h = ∑ χAi ⋅ Ci ∈ ℱ𝒮 (X, Conv(E)) i=1

with Ai ∈ R and Ci ∈ Conv(E) such that f ⩽ h ⩽ f + vV . We may assume that h is also supported by A. We choose ai ∈ Ci and set g = ∑ni=1 χAi ⋅ ai ∈ ℱ𝒮 (X, E). Then g ⩽ h ⩽ f + χA ⋅ V ⩽ f + v, that is, g − f ⩽ v. Hence f is measurable. Measurable set-valued functions We shall utilize the criterion from Proposition 2.2.10 in order to extend the notion of measurability to Conv(E)-valued functions. This is however less than straightforward. Just applying the definition for E-valued functions to Conv(E)-valued ones with Conv(E)valued step functions would prove unduly restrictive. For an unbounded set C ∈ Conv(E) and a measurable function φ ∈ ℱ (X, E), for example, not even the mapping x 󳨃→ φ(x)C : ℝ → Conv(E) is guaranteed to be measurable if we would use this criterion (see Example 2.2.22(c) below). As a remedy, we shall employ sequences of step functions rather than single ones and implicitly use the relative topologies of the locally convex cone Conv(E). These were introduced and discussed in detail in Section I.4 of [48] to which we refer for details, motivation and clarification. We shall properly define and employ these topologies in a later section. We shall say that a function f ∈ ℱ (X, Conv(E)) is measurable if for every A ∈ R, every function space neighborhood v and ε > 0 there is 1 ⩽ γ ⩽ 1 + ε and a countable

74 � 2 Integration set ℋ of step functions in ℱ𝒮 (X, Conv(E)) such that h ⩽ f + v for all h ∈ ℋ, and for every x ∈ A there is h ∈ ℋ such that f (x) ⩽ γh(x). We need to verify that this definition of measurability for Conv(E)-valued functions is consistent with the preceding one for E-valued functions. In fact, if f ∈ ℱ (X, E) is measurable with respect to the earlier definition, then it satisfies the condition from above for every A ∈ R, and ε > 0 with the step function h from Proposition 2.2.10 and the singleton set ℋ = {h}. Conversely, if the above condition holds for a function f ∈ ℱ (X, E), then for H ∈ R, for every function space neighborhood v there is a step function n

h = ∑ χAi ⋅ Ci ∈ ℱ𝒮 (X, Conv(E)) i=1

such that h ⩽ f + v. We choose elements ci ∈ Ci and set g = ∑ni=1 χAi ⋅ ci . Then g ∈ ℱ𝒮 (X, E) and g ⩽ h ⩽ f + v. Hence the E-valued function f is measurable in the manner that was specified earlier in this section. The presence of the only imprecisely specified factor γ in this definition of measurability is the cause for many of the technical difficulties in some of the upcoming arguments. Its involvement in dealing with set-valued functions appears however to be indispensable as simple examples (see 2.2.22(c)) can demonstrate. 2.2.11 Proposition. Every measurable function f ∈ ℱ (X, Conv(E)) is bounded below, and there is A ∈ R such that f (x) ⩾ 0 for all x ∈ X \ A. The measurable functions form a subcone of ℱ (X, Conv(E)). Proof. Let f ∈ ℱ (X, Conv(E)) be measurable. Using the above definition for H ∈ R and any ε > 0 concedes that there is a step function h ∈ ℱ𝒮 (X, Conv(E)) such that h ⩽ f + v. Lemmas 2.2.3 and 2.2.2 then yield the first and second statements. For the final statement, it is obvious that λf is measurable for all λ ⩾ 0 whenever f is measurable. Suppose that both f and g are measurable, let A ∈ R, let v be a function space neighborhood and let ε > 0. There is λ ⩾ 0 such that both 0 ⩽ f + λv and 0 ⩽ g + λv. We set δ = min{ε, 1/(2 + 2λ)}. There are countable sets of step functions ℋ and ℒ of step functions satisfying the condition for measurability with A, the function space neighborhood δv and δ > 0 for f and g, respectively. In particular, there are 1 ⩽ γ, ρ ⩽ 1+δ such that for every x ∈ A there are h ∈ ℋ and l ∈ ℒ such that f (x) ⩽ γh(x) and g(x) ⩽ ρ l(x). We may assume that γ ⩾ ρ and observe that 0 ⩽ (1 − ρ/γ)g + (1 − ρ/γ)λv ⩽ (1 − ρ/γ)g + δλv. The last inequality uses that 1 − ρ/γ = (γ − ρ)/γ ⩽ δ. We continue to calculate that (ρ/γ)l ⩽ (ρ/γ)g + (ρ/γ)δv ⩽ (ρ/γ)g + δv + ((1 − ρ/γ)g + δλv) = g + δ(1 + λ)v

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⩽ g + (1/2)v for all l ∈ ℒ. This argument uses that ρ/γ ⩽ 1 and δ ⩽ 1/(2 + 2λ). Because δ ⩽ (1/2) we also have h ⩽ f + (1/2)v for all h ∈ ℋ. Now we set 𝒮 = ℋ + (ρ/γ)ℒ

and claim that this is a countable family of step functions hat fulfils the requirements of measurability for the function f + g. Indeed, we have s = h + (ρ/γ)l ⩽ (f + (1/2)v) + (g + (1/2)v) = (f + g) + v for all s ∈ 𝒮 by the above, and because δ ⩽ ε we have 1 ⩽ γ ⩽ 1 + δ ⩽ 1 + ε. Moreover, for every x ∈ A there is s = h + (ρ/γ)l ∈ 𝒮 for h ∈ ℋ and l ∈ ℒ such that both f (x) ⩽ γh(x) and g(x) ⩽ ρ l(x). That is to say, (f + g)(x) ⩽ γh(x) + γ(ρ/γ)l(x) = γs(x). This demonstrates that the function f + g is also measurable. For the following, recall from Section 1.4 that for f ∈ ℱ (X, Conv(E)) and an element α of the scalar field of E the function αf ∈ ℱ (X, Conv(E)) is defined as the mapping x 󳨃→ αf (x) ∈ Conv(E). 2.2.12 Proposition. Let f ∈ ℱ (X, Conv(E)) and C ∈ Conv(E). (a) If f is measurable, then αf is measurable for every α in ℝ or ℂ. (b) If f is measurable, then χB ⋅ f is measurable for all B ∈ A. (c) The function χA ⋅ C is measurable for all A ∈ R. (d) If B ∈ A \ R, then χB ⋅ C is measurable if and only if C ⩾ 0. Proof. For (a), if the countable set ℋ of step functions satisfies the conditions for the measurability of f and if α is in the scalar field of E, then the family 𝒢 = {αh | h ∈ ℋ} of step functions satisfies the conditions for the measurability of αf . For (b), if the countable set ℋ of step functions satisfies the conditions for the measurability of f , then multiplying each of its elements by χB returns a countable set 𝒢 of step functions that satisfies the conditions for the measurability of χB ⋅ f . For (c), we note that the conditions for the measurability of the function χA ⋅ C are satisfied with the singleton set of step functions ℋ = {χA ⋅ C}. For (d), suppose that B ∈ A \ R. If C ⩾ 0, then for every A ∈ R every function space neighborhood v and ε > 0 we use the singleton set ℋ = {χB∩A ⋅C} that satisfies the requirements for the measurability of χB ⋅C. Conversely, if the function χB ⋅ C is measurable, then according to Proposition 2.2.11 there is A ∈ R such that χB ⋅ C(x) ⩾ 0 for all x ∈ X \ A. Since B ∉ R, we infer that B ⊄ A, and there is a point x ∈ B ∩ (X \ A). Thus (χB ⋅ C)(x) = C ⩾ 0, our claim.

76 � 2 Integration The next proposition establishes that the condition for measurability can be formulated in a stronger way, which will prove to be useful in our upcoming explorations. 2.2.13 Proposition. If f ∈ ℱ (X, Conv(E)) is measurable, then for every A ∈ R, every function space neighborhood v and ε > 0 there is 1 ⩽ γ ⩽ 1 + ε and an increasing sequence (hn )n∈ℕ of step functions in ℱ𝒮 (X, Conv(E)) such that hn ⩽ f + v for all n ∈ ℕ and for every x ∈ A there is n ∈ ℕ such that f (x) ⩽ γhn (x.) Proof. Suppose that f ∈ ℱ (X, Conv(E)) is measurable. Let A ∈ R, let v be a function space neighborhood and let ε > 0. According to Proposition 2.2.11, there is B ∈ R such that A ⊂ B and such that f (x) ⩾ 0 for all x ∈ X\B. In turn, we find V ∈ 𝒱 such that χB ⋅ V ⩽ v. We apply the above definition of measurability with the function space neighborhood w generated by the single neighborhood function of uniform convergence x 󳨃→ V in place of v. There is 1 ⩽ γ ⩽ 1 + ε and a countable bounded below set ℋ = {hn }n∈ℕ of step functions such that hn ⩽ f + χX ⋅ V , and for every x ∈ A there is hn ∈ ℋ such that f (x) ⩽ γhn (x). Next, for every n ∈ ℕ we define a step function gn such that gn (x) is the convex hull in E of the set ⋃ni=1 hi (x) for every x ∈ X. Clearly, gn ∈ ℱ𝒮 (X, Conv(E)) and hi (x) ⊂ f (x) + V for all i ∈ ℕ implies that gn (x) ⊂ f (x) + V , thus gn ⩽ f +χX ⋅V . We set ln = χB ⋅gn and proceed to establish that the sequence (ln )n∈ℕ in ℱ𝒮 (X, Conv(E)) answers our claim. Indeed, we have both ln = χB ⋅ gn ⩽ χB ⋅ f + χB ⋅ V and 0 ⩽ χX\B ⋅ f and, therefore, ln ⩽ (χB ⋅ f + χB ⋅ V ) + χX\B ⋅ f = f + χB ⋅ V ⩽ f + v for all n ∈ ℕ. Moreover, our construction yields that ln ⩽ ln+1 and χB ⋅ hn ⩽ ln . All that is left to verify is that the sequence (ln )n∈ℕ is bounded below. Let w be a function space neighborhood. There is ρ ⩾ 0 such that 0 ⩽ h1 + ρw for all n ∈ ℕ. This shows 0 ⩽ χB ⋅ h1 + ρw ⩽ χB ⋅ gn + ρw = ln + ρw for all n ∈ ℕ. 2.2.14 Proposition. Let f ∈ ℱ (X, Conv(E)). If for every A ∈ R, every function space neighborhood v and ε > 0 there is 1 ⩽ γ ⩽ 1 + ε and a countable set 𝒢 of measurable functions in ℱ (X, Conv(E)) such that g ⩽ f + v for all g ∈ 𝒢 and for every x ∈ A there is g ∈ 𝒢 such that f (x) ⩽ γg(x), then f is measurable. Proof. Suppose that f ∈ ℱ (X, Conv(E)) satisfies our assumptions. Let A ∈ R, let v be a function space neighborhood and ε > 0. According to Lemma 2.2.3(b) f is bounded below, hence there is λ > 0 such that 0 ⩽ f + λv. We set δ = min{1/(2λ), √1 + ε − 1} and find a countable family 𝒢 = {gn }n∈ℕ of measurable functions and 1 ⩽ γ ⩽ 1 + δ such that gn ⩽ f + (1/4)v for all n ∈ ℕ and for every x ∈ A there is gn ∈ 𝒢 such that f (x) ⩽ γgn (x). In turn, using the definition of measurability for each of the functions gn with the function space neighborhood (1/4)v, for every n ∈ ℕ there is

2.2 Measurable functions

� 77

1 ⩽ γn ⩽ 1 + δ and a countable set {h(n,m) }m∈ℕ of step functions in ℱ𝒮 (X, Conv(E)) such that h(n,m) ⩽ gn + (1/4)v, hence h(n,m) ⩽ f + (1/2)v for all m ∈ ℕ, and for every x ∈ A there is h(n,m) such that gn (x) ⩽ γn h(n,m) (x). We set l(n,m) = (γn /(1 + δ))h(n,m) and collect the following observations. Since 1/(1 + δ) ⩽ γn /(1 + δ) ⩽ 1 and 0 ⩽ (1 − γn /(1 + δ)) ⩽ δ/(1 + δ) ⩽ δ, we infer that 0 ⩽ (1 − γn /(1 + δ))(f + λv) ⩽ (1 − γn /(1 + δ))f + δλv. By our choice of δ this renders l(n,m) ⩽ (γn /(1 + δ))(f + (1/2)v) + (1 − γn /(1 + δ))f + δλv ⩽ f + ((1/2)γn /(1 + δ) + δλ)v ⩽ f + ((1/2) + (1/2))v =f +v for all n, m ∈ ℕ. For every x ∈ A on the other hand, by the above there is l(n,m) such that f (x) ⩽ γgn (x) ⩽ γγn h(n,m) (x) = (1 + δ)γ l(n,m) (x). Since 1 ⩽ (1+δ)γ ⩽ (1+δ)2 ⩽ 1+ε by our choice of δ, the countable set ℒ = {l(n,m) }n,m∈ℕ of step functions meets the requirements for the measurability of the function f with (1 + δ)γ in place of γ. 2.2.15 Proposition. If f ∈ ℱ (X, Conv(E)) is measurable, then for every A ∈ R, V ∈ 𝒱 and ε > 0 there are 1 ⩽ γ ⩽ 1 + ε and disjoint sets An ∈ R for n ∈ ℕ such that ⋃n∈ℕ An = A and such that f (x) ⩽ γf (y) + V whenever x, y ∈ An for the same n ∈ ℕ. Proof. Suppose that the function f ∈ ℱ (X, Conv(E)) is measurable. For A ∈ R, V ∈ 𝒱 and ε > 0, we consider the function space neighborhood v generated by the single neighborhood function of uniform convergence x 󳨃→ (1/2)V . We set δ = min{1, √1 + ε− 1} and use the definition of measurability for f with the neighborhood v and δ in place of ε. There is 1 ⩽ γ ⩽ 1 + δ and a countable set ℋ = {hn }n∈ℕ of step functions in ℱ𝒮 (X, Conv(E)) as in the definition of measurability. In particular, we have m hn (x) ⩽ f (x) + (1/2)V for all x ∈ X. For every n ∈ ℕ, we express hn = ∑i=1n χAin ⋅ Cni

78 � 2 Integration with disjoint sets Ain ∈ R whose union contains A (for this we may of course always add terms χAin ⋅ {0}) and Cni ∈ Conv(E). For every n ∈ ℕ and i = 1, . . . , mn , the sets Bni = {x ∈ Ain ∩ A | hm (x) ⩽ γhn (x) + (1/2)V , for all m ∈ ℕ} m

are in R, and we have ⋃n∈ℕ ⋃i=1n Bni = A. Indeed, for every x ∈ A there is n ∈ ℕ such that f (x) ⩽ γhn (x), and i ∈ {1, . . . , mi } such that x ∈ Ain . Thus hm (x) ⩽ f (x) + (1/2)V ⩽ γhn (x) + (1/2)V for all m ∈ ℕ. Hence x ∈ Bni . Moreover, because for any x ∈ A we have f (x) ⩽ γhm (x) for some m ∈ ℕ, and since the step function hn is constant on Ain , for any choice of x, y in the same set Bni we calculate f (x) ⩽ γhm (x)

⩽ γ2 hn (x) + (γ/2)V = γ2 hn (y) + (γ/2)V

⩽ γ2 f (y) + (1/2)(1 + γ2 )V . We have 1 ⩽ γ2 ⩽ (1 + δ)2 ⩽ 1 + ε and 1 + γ2 ⩽ 2 by our choice of δ. The family of the sets Bni is countable. Thus after renumbering, we obtain sets Bk ∈ R for k ∈ ℕ such that f (x) ⩽ γ2 f (y) + V whenever x, y ∈ Bk for some k ∈ ℕ. Finally, we set A1 = B1 and An = Bn \ ⋃n−1 i=1 Ai for n ⩾ 2. The sets An are disjoint, we have An ⊂ Bn and ⋃n∈ℕ An = ⋃n∈ℕ Bn = A. The family of the sets An then fulfills our claim. Corresponding to a neighborhood V ∈ 𝒱 for sets C, D ∈ Conv(E) we denote that C ≼V D if C ≤ D + εV for all ε > 0. That is, C is contained in the closure of D with respect to the topology on E generated by the neighborhood system {λV | λ > 0}. 2.2.16 Corollary. For measurable functions f , g ∈ ℱ (X, Conv(E)) and a neighborhood V ∈ 𝒱 , the set {x ∈ X | f (x) ≼V g(x)} is contained in A. Proof. Let f , g ∈ ℱ (X, Conv(E)) be measurable and let V ∈ 𝒱 . We denote B = {x ∈ X | f (x) ≼V g(x)} and proceed to verify that B ∈ A, that is, B ∩ A ∈ R for every set A ∈ R. For this, let A ∈ R, and for ε > 0 set BεA = {x ∈ A | f (x) ⩽ γg(x) + εV for some 1 ⩽ γ ⩽ 1 + ε}. According to Proposition 2.2.15, for any choice of 0 < ε ⩽ 1 there are 1 ⩽ γ1 , γ2 ⩽ 1 + ε and disjoint sets Ai ∈ R for i ∈ ℕ such that ⋃n∈ℕ An = A and f (x) ⩽ γ1 f (y) + εV as well as g(x) ⩽ γ2 g(y) + εV whenever x, y ∈ Ai for the same i ∈ ℕ. Set

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CεA = ⋃ { Ai | f (xi ) ⩽ γ3 g(xi ) + εV for some xi ∈ Ai and 1 ⩽ γ3 ⩽ 1 + ε}. Clearly, CεA ∈ R and BεA ⊂ CεA . If x ∈ CεA , on the other hand, then x ∈ Ai for one of the sets Ai forming this union, hence f (x) ⩽ γ1 f (xi ) + εV ,

f (xi ) ⩽ γ3 g(xi ) + εV

and

g(xi ) ⩽ γ2 g(x) + εV .

Combining these inequalities, we obtain that f (x) ⩽ (γ1 γ2 γ3 )g(x) + (1 + γ1 + γ1 γ2 )εV ⩽ (γ1 γ2 γ3 )g(x) + 7εV . The latter follows since 0 < ε ⩽ 1, hence 1 ⩽ γ1 , γ2 ⩽ 2. Moreover, since 1 ⩽ (γ1 γ2 γ3 ) ⩽ (1 + ε)3 = 1 + (3 + 3ε + ε2 )ε ⩽ 1 + 7ε, we infer that A BεA ⊂ CεA ⊂ B7ε .

This yields A A = ⋂ B(1/n) ∈ R. ⋂ C(1/n)

n∈ℕ

n∈ℕ

A A All that is left to show is that ⋂n∈ℕ B(1/n) = B ∩ A. The inclusion B ∩ A ⊂ ⋂n∈ℕ B(1/n) A is obvious. For the reverse, let x ∈ ⋂n∈ℕ B(1/n) and let ε > 0. For a ∈ f (x), there is λ ⩾ 0 such that a ∈ λV . Then for any choice of n ⩾ (1 + λ)/ε we have f (x) ⩽ γg(x) + (1/n)V with some 1 ⩽ γ ⩽ 1 + (1/n). Hence a ∈ γg(x) + (1/n)V and, therefore, (1/γ)a ∈ g(x) + (1/n)V . From 1 − (1/γ) = (γ − 1)/γ ⩽ 1/n, we deduce that

a = (1/γ)a + (1 − (1/γ))a ∈ (g(x) + (1/n)V ) + (λ/n)V ⊂ g(x) + εV . This yields f (x) ⩽ g(x) + εV for all ε > 0 and we infer that f (x) ≼V g(x). Therefore, x ∈ B, and our claim follows. The reverse conclusion of Proposition 2.2.15 holds true under an additional assumption. 2.2.17 Proposition. Let f ∈ ℱ (X, Conv(E)) such that for every function space neighborhood v there is a step function h ∈ ℱ𝒮 (X, Conv(E)) such that h ⩽ f + v. If for every A ∈ R, V ∈ 𝒱 and ε > 0 there are 1 ⩽ γ ⩽ 1 + ε and sets An ∈ R for n ∈ ℕ such that ⋃n∈ℕ An = A and f (x) ⩽ γf (y) + V whenever x, y ∈ An for the same n ∈ ℕ, then f is measurable.

80 � 2 Integration Proof. Suppose that f ∈ ℱ (X, Conv(E)) satisfies our assumptions. Given A ∈ R, a function space neighborhood v and ε > 0, there is h ∈ ℱ𝒮 (X, Conv(E)) such that h ⩽ f + (1/3)v and there is V ∈ 𝒱 such that χA ⋅ V ⩽ (1/3)v. We set δ = √1 + ε − 1. There are 1 ⩽ γ ⩽ 1 + δ and sets An ∈ R as in our assumption. We may assume that the sets An are disjoint and not empty and choose points xn ∈ An . For every n ∈ ℕ, we set hn = (1/γ)χAn ⋅ (f (xn ) + V ) + χX\An ⋅ h. Then hn ∈ ℱ𝒮 (X, Conv(E)), and for every x ∈ An we have f (x) ⩽ γf (xn ) + V ⩽ γ(f (xn ) + V ) = γ2 hn (x) and hn (x) = (1/γ)(f (xn ) + V ) ⩽ (1/γ)(γf (x) + 2V ) ⩽ f (x) + 2V . The former shows that for every x ∈ A there is n ∈ ℕ such that f (x) ⩽ γ2 hn (x), and our choice of δ guarantees that 1 ⩽ γ2 ⩽ 1 + ε. The latter inequality yields that χAn ⋅ hn ⩽ χχAn ⋅ f + χAn ⋅ 2V ⩽ χAn ⋅ f + (2/3)v. Since χX\An ⋅ h ⩽ χX\An ⋅ f + (1/3)v by Lemma 1.2.1(a), we infer that hn = χAn ⋅ hn + χX\An ⋅ hn ⩽ f + v. The family ℋ = {hn }n∈ℕ of step functions then satisfies the criterion for the measurability of the function f . For a positive function in ℱ (X, Conv(E)), the existence of a step function h as required in Proposition 2.2.17, that is, h = 0, is of course guaranteed. Moreover, we may choose γ = 1 + ε in the respective criteria of Propositions 2.2.15 and 2.2.17. We obtain the following. 2.2.18 Corollary. A positive function f ∈ ℱ (X, Conv(E)) is measurable if and only if for every A ∈ R, V ∈ 𝒱 and ε > 0 there are sets An ∈ R for n ∈ ℕ such that ⋃n∈ℕ An = A and f (x) ⩽ (1 + ε)f (y) + V whenever x, y ∈ An for the same n ∈ ℕ. 2.2.19 Proposition. Let f ∈ ℱ (X, Conv(E)) be measurable and let φ be a nonnegative real-valued A-measurable function on X. If either f is positive or if φ is bounded on every set in R, then φ ⋅ f is measurable. Proof. It is obvious from our definition of measurability for Conv(E)-valued functions that their restrictions to sets in R, that is, products with the characteristic functions of these sets, are also measurable (Proposition 2.2.12(b)). Suppose that f ∈ ℱ (X, Conv(E))

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� 81

is measurable and that φ is a positive real-valued A-measurable function on X with the stated properties. Let A ∈ R, let v be a function space neighborhood and let ε > 0. We may assume that f (x) ⩾ 0 for all x ∈ X \ A. There is V ∈ 𝒱 such that χA ⋅ V ⩽ v, and since f is bounded below, there is λ ⩾ 0 such that 0 ⩽ f + λχX ⋅ V . We shall verify the criterion of Proposition 2.2.14 in order to prove that the function φ⋅f is measurable. For any choice of γ > 1, the sets An = {x ∈ A | γn ⩽ φ(x) < γn+1 } for all n ∈ ℤ are contained in R by the A-measurability of φ. In our first case, if f is positive, we choose γ = 1+ε and consider the countable set 𝒢 of measurable functions gn = γn χAn ⋅ f , for all n ∈ ℤ. Following Proposition 2.2.12(b), these functions are measurable. For all x ∈ An , since γn ⩽ φ(x), we have gn (x) = γn f (x) ⩽ φ(x)f (x) and 0 = gn (x) ⩽ φ(x)f (x) for all x ∈ X \ An . Thus gn ⩽ φ ⋅ f ⩽ φ ⋅ f + v for all n ∈ ℤ. Let x ∈ A. If φ(x) = 0, then φ(x)f (x) ⩽ γgn (x) for all n ∈ ℤ. If x ∈ An for some n ∈ ℤ, then φ(x)f (x) ⩽ γn+1 f (x) = γgn (x). Thus the criterion of Proposition 2.2.14 applies for the function φ ⋅ f with the countable set 𝒢 = {gn }n∈ℕ of measurable functions. In our second case, if φ is bounded on A but f is not necessarily positive, there is ρ > 0 such that 0 ⩽ φ(x) ⩽ ρ for all x ∈ A. We shall first argue that there is a step function h ∈ ℱ𝒮 (X, Conv(E)) such that h ⩽ φ ⋅ f + χx ⋅ V . Surely, since f is measurable there is l ∈ ℱ𝒮 (X, Conv(E)) such that l ⩽ f +(1/2ρ)χX ⋅V , and since φ is A-measurable and bounded, there is a real-valued nonnegative step function ψ on X such that ψ(x) ⩽ φ(x) ⩽ ψ(x) +

1 2λ

for all x ∈ X. Then h = ψ ⋅ l ∈ ℱ𝒮 (X, Conv(E)), and indeed 1 ψ⋅V 2ρ 1 ⩽ ψ ⋅ f + χX ⋅ V + (φ − ψ) ⋅ (f + λV ) 2 1 1 ⩽ φ ⋅ f + χX ⋅ V + χX ⋅ V = φ ⋅ f + χX ⋅ V , 2 2

h⩽ψ⋅f +

our claim. Now we select γ = 1 + 1/(2ρλ) and use the sets An as specified for our first case. Since φ is bounded by ρ on A, the sets An are empty whenever γn > ρ. Hence for a nonempty set An , we have γn+1 −γn = γn (γ−1) ⩽ 1/(2λ). We consider the countable set 𝒢 of measurable functions gn = χAn ⋅ (γn f + (1/2)V ) + χA\An ⋅ h,

82 � 2 Integration for all n ∈ ℤ such that An ≠ H. First, we shall verify that gn ⩽ φ ⋅ f + v. Indeed, for x ∈ An we have gn (x) = γn f (x) + (1/2)V

⩽ γn f (x) + (1/2)V + (φ(x) − γn )(f (x) + λV )

⩽ φ(x)f (x) + V . The latter uses that An ≠ H and φ(x) − γn ⩽ 1/(2λ) by the above. For x ∈ A \ An , we have gn (x) = h(x) ⩽ φ(x)f (x) + V by our choice of the step function h, and 0 = gn (x) ⩽ φ(x)f (x) for all x ∈ X \ A. Hence indeed, gn ⩽ f + χA ⋅ V ⩽ f + v for all n ∈ ℤ such that An ≠ H. Finally, let x ∈ A. If φ(x) = 0, then φ(x)f (x) ⩽ γgn (x) for all n ∈ ℤ. If x ∈ An for some n ∈ ℤ, then φ(x)f (x) ⩽ φ(x)f (x) + (γn+1 − φ(x))(f (x) + λV ) ⩽ γn+1 f (x) + (1/2)V ⩽ γgn (x). Therefore, Proposition 2.2.14 also applies in this case for the function φ ⋅ f with the family 𝒢 = {gn }n∈ℕ . Our first corollary broadens the statement of Proposition 2.2.7. 2.2.20 Corollary. If φ ∈ ℱ (X) is nonnegative and measurable, then the function φ ⋅ C ∈ ℱ (X, E) is measurable for every C ∈ Conv(E). Proof. Let φ be as stated. According to Proposition 2.2.7, then φ is bounded, supported by a set A ∈ R and A-measurable. Hence φ ⋅ C = φ ⋅ (χA ⋅ C) for every C ∈ Conv(E), and our claim follows from Propositions 2.2.19 and 2.2.12(c). For our second corollary recall that in Section 1.1, the product of a positive set C ∈ Conv(E) with +∞ ∈ ℝ was defined to be the subcone of E spanned by C. If C is absorbing, then (+∞)C = E. 2.2.21 Corollary. If φ is an ℝ+ -valued A-measurable function on X and if C ∈ Conv(E) is positive, then the function φ ⋅ C is measurable. Proof. Because φ is nonnegative and A-measurable, the set B = {x ∈ X | φ(x) < +∞} is contained in A and the real-valued function χB ⋅ φ is A-measurable. Let D be the conic hull of C in E, that is, D = (+∞)C. According to Proposition 2.2.12(d), since C, D ⩾ 0, the mappings χX ⋅ C and χX\B ⋅ D are both measurable. Proposition 2.2.19 applies and renders that (χB ⋅ φ) ⋅ (χX ⋅ C) = χB ⋅ (φ ⋅ C) is also measurable. Summarizing, the function φ ⋅ C = χB ⋅ (φ ⋅ C) + χX\B ⋅ (φ ⋅ C) = χB ⋅ (φ ⋅ C) + χX\B ⋅ D is measurable, our claim.

2.2 Measurable functions

� 83

Corollary 2.2.21 applies in particular to neighborhood functions φ ⋅ V where φ is an ℝ+ -valued A-measurable function on X and V ∈ 𝒱c , in which 𝒱c denotes the extended neighborhood system of E, that is, the subcone of Conv(E) spanned by 𝒱 ∪ {E} (see Section 1.2). All function space neighborhoods involved in our Examples 1.2.7(a) to (d) in Section 1.2 are convex combinations of functions of this type and, therefore, are measurable. 2.2.22 Examples. (a) There are various well-established definitions of measurability in case that E is a Banach space. Most involve a specific measure and as a precondition for integrability are considerably more generous than ours. (See, e. g., Chapter II.1 in [21].) Given a measure θ, an E-valued function f is said to be strongly θ-measurable if there exists a sequence (hn )n∈ℕ of E-valued step functions such that the sequence (‖hn (x) − f (x)‖)n∈𝔹 converges θ-almost everywhere. This is clearly implied by our requirement for measurability for E-valued functions from the beginning of this section, which calls for the existence of such a sequence of step functions, which converges uniformly toward f . A function f ∈ ℱ (X, E) is said to be weakly θ-measurable if for every linear functional μ ∈ E ˚ the real- or complex-valued function x 󳨃→ μ(f (x)) is strongly θ-measurable. This notion is of course weaker than strong measurability and also implied by our requirements. We note that strong θ-measurability of a function f ∈ ℱ (X, E) implies that the real-valued step functions ‖hn ‖ converge pointwise θ-almost everywhere to the real-valued function ‖f ‖. Since the pointwise limit of A-measurable functions is again A-measurable (see Theorem 6 in 11.2 of [53]), there is a set Z ∈ A of measure zero such that the function χX\Z ⋅ ‖f ‖ is A-measurable. (b) Let us consider the case that E = ℝ. According to Proposition 2.2.7, a real-valued function in ℱ (X) is measurable if and only if it is bounded, supported by a set in R and A-measurable. For Conv(ℝ)-valued, that is, interval-valued functions we observe the following. Let φ and ψ be ℝ+ -valued A-measurable functions on X. We claim that the function f ∈ ℱ (X, Conv(ℝ)) such that f (x) = [−φ(x), ψ(x)] for all x ∈ X is measurable. Indeed, let A ∈ R and let v be a function space neighborhood for ℱ (X). According to the definition of a function space neighborhood, there is ρ > 0 such that χA ⋅ (ρ𝔹) ⩽ v, where 𝔹 = [−1, +1] denotes the unit ball in ℝ. For all n ∈ ℕ, the sets An = {x ∈ A | (n − 1)ρ ⩽ φ(x) < nρ} and Bn = {x ∈ A | (n − 1)ρ ⩽ ψ(x) < nρ}

84 � 2 Integration are contained in R by the A-measurability of the functions φ and ψ. So, are the sets A+∞ = {x ∈ A | φ(x) = +∞} and B+∞ = {x ∈ A | ψ(x) = +∞}. We consider the countable set 𝒢 of Conv(ℝ)-valued step functions hnm = χAn ∩Bm ⋅ [−nρ, mρ] for n, m ∈ ℕ ∪ {+∞}. The sets An ∩ Bm cover all of A and for every x ∈ An ∩ Bm we have f (x) ⩽ hnm (x) ⩽ f (x) + ρ𝔹. Since hnm (x) = 0 ⩽ f (x) for all x ∈ X \ (An ∩ Bm ), the right-hand side of this inequality shows that hnm ⩽ f +χA ⋅(ρ𝔹) ⩽ f +v, while its left-hand side demonstrates that for every x ∈ A there is hnm ∈ ℋ such that f (x) ⩽ hnm (x). Hence f is measurable according to our definition. (c) Let us illustrate our earlier remark that the inclusion of the unspecified factor γ in the definition of measurability for a Conv(E)-valued function is essential and that its omission would exclude some basic and indispensable cases. We consider the following simple example. Let X = [0, 1] and let E = ℝ2 with the neighborhood system 𝒱 = {ρ𝔹 | ρ > 0}, where 𝔹 denotes the Euclidean unit ball in ℝ2 . Let φ ∈ ℱ ([0, 1]) such that φ(x) = x and let C = {(x, y) ∈ ℝ2 | y ⩾ x 2 } ∈ Conv(ℝ2 ). We shall see that the function φ⋅C ∈ ℱ (X, Conv(ℝ2 )), which according to Corollary 2.2.21 is measurable and does however not satisfy the criterion of measurability with the fixed factor γ = 1, that is, ε = 0. Indeed, let A = [0, 1] ∈ R and let v𝔹 be the function space neighborhood of uniform convergence defined by the constant neighborhood function x 󳨃→ 𝔹 and assume that there is a countable set ℋ of Conv(ℝ2 )-valued step functions such that h ⩽ φ ⋅ C + v𝔹 for all h ∈ ℋ and for every x ∈ [0, 1] there is h ∈ ℋ such that (φ ⋅ C)(x) ⩽ h(x). Each of the step function in ℋ is defined using finitely many disjoint sets in R, so only countably many are involved in the build of all the functions in ℋ. We find an element x ∈ [0, 1], which is not the infimum of any of these sets. There is h ∈ ℋ such that xC ⩽ h(x). The point x is contained in one of the disjoint sets in R used for the definition of h, and there is a point y < x contained in the same set. Thus xC ⩽ h(x) = h(y) ⩽ yC + 𝔹. But a brief inspection of the geometry of the set C ⊂ ℝ2 reveals that this can not hold true.

2.2 Measurable functions

� 85

(d) A modification of the settings in (c) leads to the construction of a nonmeasurable function. With the same choices for X, E and V, we probe the function f ∈ ℱ (X, Conv(ℝ2 )) defined as f (t) = {(x, y) ∈ ℝ2 | y ⩾ |x|t+1 } for t ∈ [0, 1]. It is again self-evident that for s, t ∈ [0, 1] such that s < t there is no choice of γ ⩾ 1 such that f (s) ⩽ γf (t) + 𝔹. The measurability criterion in Proposition 2.2.15 can therefore not be satisfied.

Composition of measurable functions If (G, 𝒰 ) is an additional locally convex topological vector space and S ∈ L(G, E), then for a function f ∈ ℱ (X, Conv(G)) we define the function S ˝ f ∈ ℱ (X, Conv(E)) by (S ˝ f )(x) = S(f (x)) for all x ∈ X. We shall demonstrate that S ˝ f is measurable whenever f is measurable. Indeed, given A ∈ R, a function space neighborhood v and ε > 0 there is B ∈ R such that A ⊂ B and f (x) ⩾ 0 for all x ∈ X \ B. There is V ∈ 𝒱 such that χB ⋅ V ⩽ v and we find U ∈ 𝒰 such that S(U) ⊂ V . Using the function space neighborhood u for ℱ (X, G) defined by the single neighborhood function χX ⋅ U and the measurability of f there is 1 ⩽ γ ⩽ 1 + ε and a countable set ℋ of step functions in ℱ𝒮 (X, Conv(G)) such that h ⩽ f + u for all h ∈ ℋ, and for every x ∈ A there is h ∈ ℋ such that f (x) ⩽ γh(x). We may assume that all functions in ℋ are supported by B. Set S ˝ ℋ = {S ˝ h | h ∈ ℋ} ⊂ ℱ (X, Conv(E)). We understand that h ⩽ f + u implies that S ˝ h ⩽ S ˝ f + v. Indeed, we have (S ˝ h)(x) = 0 ⩽ (S ˝ f )(x) for all x ∈ X \ B and h(x) ⩽ f (x) + U, hence (S ˝ h)(x) ⩽ S(f (x) + U) = S(f (x)) + S(U) ⩽ (S ˝ f )(x) + V for all x ∈ B. Thus S ˝ h ⩽ S ˝ f + χB ⋅ V ⩽ S ˝ f + v as claimed. Moreover, for every x ∈ A there is h ∈ ℋ such that f (x) ⩽ γh(x). Then (S ˝ f )(x) ⩽ γ(S ˝ h)(x). The countable set S ˝ ℋ of Conv(E)-valued step functions therefore satisfies the criterion for measurability of the function S ˝ f ∈ ℱ (X, Conv(E)).

86 � 2 Integration

2.3 Integrals of set-valued functions We continue to use the general assumptions and notation from the preceding sections. We suppose that R is a σ-ring of subsets of a set X, that 𝒫E = E and that θ is a bounded L(E, F ˚‚ )-valued measure defined on the sets in R. Recall from Sections 1.4 and 2.1 that the latter requires that θA (αa)(μ) = θA (a)(αμ) holds for all A ∈ R, all a ∈ E, all α in the common scalar field of E and F and all μ ∈ F ˚ . This condition needs to be verified only in the complex case and then only for α = i. The topological scenario, where X is locally compact, R consists of all relatively compact Borel subsets of X and θ is a regular measure, will be treated as a special case. In a first step, we define integrals for step functions, then for measurable functions and finally for a wider class of θ-integrable functions. Having these integrals available for not just E-valued, but also for Conv(E)-valued functions is of interest not only for their own sake, but essential for the development of our approach to integration theory. Integrals for set-valued functions had been established by Aumann [3] and others using integrable selections. They arise more naturally in the framework of our methodology. The values of the integrals of E-valued functions can be interpreted to be elements of F, F ˚˚ or F ˚‚ , those of Conv(E)-valued functions to be convex subsets of these spaces (Propositions 2.3.10 and 2.3.23 below). We start out with functions in ℱ𝒮 (X, Conv(E)), that is, Conv(E)-valued step functions, and shall utilize the modulus |θ| of θ as its extension to an L(Conv(E), 𝒬F )valued measure. The values of these integrals will be elements of 𝒬F . In preparation, we observe the following. 2.3.1 Lemma. Let Ai , Bk ∈ R and Ci , Dk ∈ Conv(E) for i = 1, . . . , n and k = 1, . . . , m. If n

m

i=1

k=1

∑ χAi ⋅ Ci ⩽ ∑ χBk ⋅ Dk ,

then

n

m

i=1

k=1

∑ |θ|Ai (Ci ) ⩽ ∑ |θ|Bk (Dk ).

Proof. Following Lemma 2.2.1, we may assume that both families of sets Ai and Bk are pairwise disjoint and suppose that ∑ni=1 χAi ⋅ Ci ⩽ ∑m k=1 χBk ⋅ Dk . By adding suitable terms χA′ ⋅ 0 and χB′ ⋅ 0 on the left- and right-hand sides, we may assume in addition that ⋃ni=1 Ai = ⋃m k=1 Bk . Under these assumptions, the sets Ai ∩ Bk form a disjoint partition of this union and we have either Ai ∩ Bk = H or Ci ⩽ Dk . This yields n

n

m

∑ |θ|Ai (Ci ) = ∑ ∑ |θ|(Ai ∩Bk ) (Ci ) i=1

i=1 k=1 m n

m

k=1 i=1

k=1

⩽ ∑ ∑ |θ|(Ai ∩Bk ) (Dk ) = ∑ |θ|Bk (Dk ), as claimed.

2.3 Integrals of set-valued functions

� 87

Integrals of step functions We are now in a position to define the integral for a Conv(E)-valued step function n

h = ∑ χAi ⋅ Ci ∈ ℱ𝒮 (X, Conv(E)) i=1

over a set B ∈ A with respect to θ as an element of 𝒬F by n

∫ h dθ = ∑ |θ|(Ai ∩B) (Ci ). i=1

B

Lemma 2.3.1 implies that the sum on the right-hand side is independent of the particular representation for the step function h. The integral constitutes a monotone linear operator from ℱ𝒮 (X, Conv(E)) into 𝒬F . 2.3.2 Proposition. Let B ∈ A, let g, h ∈ ℱ𝒮 (X, Conv(E)) and α ⩾ 0. Then (a) ∫B αg dθ = α ∫B g dθ. (b) ∫B (g + h) dθ = ∫B g dθ + ∫B h dθ.

(c) ∫B g dθ = ∫X χB ⋅ g dθ.

(d) ∫B g dθ ⩽ ∫B h dθ whenever g ⩽ h. a.e. B

Proof. Properties (a), (b) and (c) are obvious from the above definition of the integral. For (d), suppose that g ⩽ h for g, h ∈ ℱ𝒮 (X, Conv(E)). That is, χB\Z ⋅ g ⩽ χB\Z ⋅ h with a.e. B

some set Z ∈ A of measure zero. It then follows immediately from the definition of the integral for step functions that ∫ g dθ ⩽ ∫ h dθ. B\Z

B\Z

As |θ|(A∩Z) = 0 holds for all A ∈ R by Proposition 2.1.10(a) and (b), we infer that ∫B∩Z l dθ = 0 for all step functions l ∈ ℱ𝒮 (X, Conv(E)). Moreover, parts (b) and (c) yield that ∫ l dθ = ∫ dθ + ∫ dθ = ∫ l dθ. B

B∩Z

B\Z

B\Z

Combining all of the above, we obtain ∫ g dθ = ∫ g dθ ⩽ ∫ h dθ = ∫ h dθ. B

B\Z

B\Z

B

The integral of an E-valued step function is directly determined by the values of θ rather than those of its modulus because both measures coincide on the elements

88 � 2 Integration of E. The following lemma demonstrates how this observation relates to the integrals of Conv(E)-valued step functions. 2.3.3 Lemma. Let B ∈ A and g ∈ ℱ𝒮 (X, Conv(E)). Then 󵄨 ∫ g dθ = sup{∫ h dθ 󵄨󵄨󵄨 h ∈ ℱ𝒮 (X, E), h ⩽ g}. B

B

Proof. Following 2.3.2(c), we may assume that B = X. Let g = ∑ni=1 χAi ⋅ Ci ∈ m ℱ𝒮 (X, Conv(E)) and let h = ∑k=1 χBk ⋅ ak ∈ ℱ𝒮 (X, E) such that h ⩽ g. Then m

n

k=1

i=1

∫ h dθ = ∑ θBk (ak ) ⩽ ∑ |θ|Ai (Ci ) = ∫ g dθ

X

X

by Lemma 2.3.1. Conversely, for each of the sets Ai suppose that the sets Aki ∈ R for k = 1, . . . , ni form a disjoint partition of Ai and let cik ∈ Ci . Then n

ni

∑ ∑ θAk (cik ) = ∫ h dθ, i=1 k=1

where h =

ni χAk ∑ni=1 ∑k=1 i n

i

h

⋅ (cik ). Then h ∈ ℱ𝒮 (X, E) and h ⩽ g. Thus

ni

󵄨 ∑ ∑ θAk (cik ) ⩽ sup{∫ h dθ 󵄨󵄨󵄨 h ∈ ℱ𝒮 (X, E), h ⩽ g} i

i=1 k=1

B

Taking the supremum over all choices for partitions of the sets Ai on the left-hand side returns n

󵄨 ∫ g dθ = ∑ |θ|Ai (Ci ) ⩽ sup{∫ h dθ 󵄨󵄨󵄨 h ∈ ℱ𝒮 (X, E), h ⩽ g}, i=1

B

B

hence our claim. Integrals of measurable functions Now in a second step, for a measurable function f ∈ ℱ (X, Conv(E)), a set B ∈ A and a function space neighborhood v we define the expression (v)

󵄨 ∫ f dθ = sup{∫ h dθ 󵄨󵄨󵄨 h ∈ ℱ𝒮 (X, Conv(E)), h ⩽ f + v} B

B

󵄨 = sup{∫ h dθ 󵄨󵄨󵄨 h ∈ ℱ𝒮 (X, E), h ⩽ f + v} B

2.3 Integrals of set-valued functions

� 89

as a precursor of the integral. The supremum is meant pointwise on F ˚ and according to Proposition 1.4.7(a) defines an element of 𝒬F . The equality of both expressions on the right-hand side of this definition results from Lemma 2.3.3. For our further investigations, with every neighborhood W ∈ 𝒲 we associate the function space neighborhood vW , defined by the family 𝒩vW of all neighborhood-valued step functions n = ∑ni=1 χAi ⋅ Vi , where Ai ∈ R and Vi ∈ 𝒱 such that ∫X n dθ ⩽ σW . This clearly is a function space neighborhood in the sense of Section 1.2, as 𝒩vW is convex and for every set A ∈ R there is V ∈ 𝒱 such that |θ|A (V ) ⩽ σW by our premise that θ is bounded. Hence χA ⋅ V ∈ 𝒩W , that is, χA ⋅ V ⩽ vW , thus satisfying the requirements. We note that all neighborhood functions in 𝒩vW are supported by some set A ∈ R. We continue with a simple observation for step functions in ℱ𝒮 (X, Conv(E)). 2.3.4 Lemma. Let B ∈ A, h ∈ ℱ𝒮 (X, Conv(E)) and W ∈ 𝒲 . Then (vW )

∫ h dθ ⩽ ∫ h dθ ⩽ ∫ h dθ + σW . B

B

B

Proof. The first part of this inequality is trivial. For the second part, let g ⩽ h + vW for g ∈ ℱ𝒮 (X, Conv(E)). That is to say, g ⩽ h + n for some neighborhood-valued function n ∈ 𝒩vW . Following Proposition 2.3.2(b) and (c), this implies ∫ g dθ ⩽ ∫ h dθ + ∫ n dθ ⩽ ∫ h dθ + σW B

B

B

B (vW )

for each such step function g ∈ ℱ𝒮 (X, Conv(E)), hence ∫B claimed.

h dθ ⩽ ∫B h dθ + σW as

For the following, we recall the definition of boundedness for subsets of ℱ (X, Conv(E)) from Section 2.2. 2.3.5 Proposition. Let A ∈ R, let f ∈ ℱ (X, Conv(E)) be measurable, and let (hn )n∈ℕ be a bounded below sequence of step functions in ℱ𝒮 (X, Conv(E)) such that for every x ∈ A there is n0 ∈ ℕ such that f (x) ⩽ hn (x) for all n ⩾ n0 . Then (vW )

∫ f dθ ⩽ lim inf ∫ hn dθ + σW A

n→∞

A

for every W ∈ 𝒲 . The limit is taken pointwise on F ˚ . Proof. Let A ∈ R, let f ∈ ℱ (X, Conv(E)) be measurable and let (hn )n∈ℕ be a sequence of step functions in ℱ𝒮 (X, Conv(E)) satisfying our assumptions. For W ∈ 𝒲 , let l ∈ ℱ𝒮 (X, Conv(E)) such that l ⩽ f + vW , that is, l ⩽ f + n for some n ∈ 𝒩vw . Now we set An = {x ∈ A | l(x) ⩽ hm (x) + n(x) for all m ⩾ n}.

90 � 2 Integration Since all the functions involved are step functions, all the sets An are in R. We have An ⊂ An+1 and A = ⋃n∈ℕ An by our assumption. There is V ∈ 𝒱 such that |θ|A (V ) ⩽ σW , and as the sequence (hn )n∈ℕ is bounded below, there is λ ⩾ 0 such that 0 ⩽ hn + λχX ⋅ V for all n ∈ ℕ. Thus χAn ⋅ l ⩽ χAn ⋅ (hn + n) + χA\An (hn + λχX ⋅ V ) ⩽ χA ⋅ hn + χAn ⋅ n + λχA\An ⋅ V .

Hence by Proposition 2.3.2, and as ∫ χAn n dθ ⩽ ∫ n dθ ⩽ σW ,

X

and

X

∫ χA\An ⋅ V = |θ|(A\An ) (V ),

X

when taking the integrals over X in the above inequality, we obtain ∫ l dθ ⩽ ∫ hn dθ + λ|θ|A\An (V ) + σW .

An

A

Because An ⊂ An+1 and ⋃n∈ℕ An = A, Theorem 2.1.3 together with Lemma 2.1.2(a) yields that |θ|(B∩A) (C) = limn→∞ |θ|(B∩An ) (C) for all B ∈ R and C ∈ Conv(E). Considering the definition of the integral of a step function, this renders lim ∫ l dθ = ∫ l dθ,

n→∞

An

A

and Lemma 2.1.2(b) yields that limn→∞ |θ|(A\An ) (V )(μ) = 0 for all μ ∈ W ˝ . Thus we obtain ∫ l dθ ⩽ lim inf ∫ hn dθ + σW . n→∞

A

A

Our claim follows, since the above inequality holds true for all step functions l ∈ ℱ𝒮 (X, Conv(E)) such that l ⩽ f + vW . 2.3.6 Corollary. Let B ∈ A, let f ∈ ℱ (X, Conv(E)) be measurable, let W ∈ 𝒲 and let v be a function space neighborhood. Then (vW )

(v)

∫ f dθ ⩽ ∫ f dθ + σW . B

B

Proof. Let B ∈ A, let f ∈ ℱ (X, Conv(E)) be measurable, let W ∈ 𝒲 and let v be a function space neighborhood. Let l ∈ ℱ𝒮 (X, Conv(E)) such that l ⩽ f + vW , and according to Proposition 2.2.11 we choose A ∈ R such that l is supported by A and χX\A ⋅ f ⩾ 0. For the set B ∩ A ∈ R, let (hn )n∈ℕ be a sequence of step functions in

2.3 Integrals of set-valued functions

� 91

ℱ𝒮 (X, Conv(E)) approaching f as in Proposition 2.2.13 with the function space neigh-

borhood v and ε > 0. We may assume that the functions hn are supported by A, because we may otherwise replace them by their product with the characteristic function of this set. Proposition 2.3.5 renders that there is 1 ⩽ γ ⩽ 1 + ε such that (vW )

∫ f dθ ⩽ γ lim inf ∫ hn dθ + σW . n→∞

B∩A

B∩A

On the other hand, we have (vW )

∫ l dθ = ∫ l dθ ⩽ ∫ f dθ, B

B∩A

B∩A

since the step function l is supported by A. Similarly, for the functions hn ⩽ f + v we observe that (v)

∫ hn dθ = ∫ hn dθ ⩽ ∫ f dθ. B

(B∩A)

B

Now combining all of the above returns (vW )

(v)

∫ l dθ ⩽ ∫ f dθ ⩽ γ lim inf ∫ hn dθ + σW ⩽ γ ∫ f dθ + σW . B

n→∞

B∩A

B

B∩A

Our claim follows because this last inequality holds true, pointwise on F ˚ , for all ε > 0, and, consequently, we have (v)

∫ l dθ ⩽ ∫ f dθ + σW B

B

for all step functions l ∈ ℱ𝒮 (X, Conv(E)) such that h ⩽ f + vW . We are now in a position to define the integral over a set B ∈ A for a measurable function f ∈ ℱ (X, Conv(E)). Let V˚ denote the collection of all function space neighborhoods, downward directed in the order introduced in Section 1.2. Corollary 2.3.6 demonstrates that for a measurable function f ∈ ℱ (X, Conv(E)) and a set B ∈ A the (v) net (∫B f dθ)v∈V˚ forms a Cauchy net in the symmetric topology of 𝒬F . Indeed, given W ∈ 𝒲 we have (u)

s

(v)

∫ f dθ ∈ W ( ∫ f dθ), B

B

92 � 2 Integration that is, (u)

(v)

(v)

∫ f dθ ⩽ ∫ f dθ + σW B

and

B

(u)

∫ f dθ ⩽ ∫ f dθ + σW , B

B

for all u, v ⩽ vW . Thus according to Proposition 1.4.5 this net is convergent in the symmetric topology of 𝒬F , and we set (v)

(v)

∫ f dθ = lim ∫ f dθ = inf ∫ f dθ B

v∈V˚

v∈V˚

B

B

󵄨 = lim sup{∫ h dθ 󵄨󵄨󵄨 h ∈ ℱ𝒮 (X, Conv(E)), h ⩽ f + v} v∈V ˚

B

󵄨 = lim sup{∫ h dθ 󵄨󵄨󵄨 h ∈ ℱ𝒮 (X, E), h ⩽ f + v}. v∈V˚

B

The limits in these expressions refer to the symmetric topology of 𝒬F and the index set V˚ , whereas the infima and suprema are taken pointwise on F ˚ . Our earlier observation in Lemma 2.3.4 justifies that the above definition of the integral is consistent with the preceding one for step functions. Corollary 2.3.6 yields that (vW )

∫ f dθ ⩽ ∫ f dθ ⩽ ∫ f dθ + σW B

B

B

holds for all W ∈ 𝒲 and that the integral of a measurable function f is already determined using only the function space neighborhoods vW . That is, (vW )

∫ f dθ = lim ∫ f dθ. B

W ∈𝒲

B

Obviously, the integral is monotone, and we shall go forward to verify that it determines a linear operator from the subcone of measurable functions in ℱ (X, Conv(E)) into 𝒬F . We begin with an obvious, but useful consequence of Proposition 2.3.5 in the light of our definition of the integral of a measurable function. 2.3.7 Corollary. Let A ∈ R, let f ∈ ℱ (X, Conv(E)) be measurable and let (hn )n∈ℕ be a bounded below sequence of step functions in ℱ𝒮 (X, Conv(E)) such that for every x ∈ A there is n0 ∈ ℕ such that f (x) ⩽ hn (x) for all n ⩾ n0 . Then ∫ f dθ ⩽ lim inf ∫ hn dθ. A

n→∞

A

2.3 Integrals of set-valued functions

� 93

In Part (e) of the following theorem, we consider R as the index set of a net, directed upward by set inclusion. 2.3.8 Theorem. Let B ∈ A, let f , g ∈ ℱ (X, Conv(E)) be measurable functions and let α ⩾ 0. Then (a) ∫B αf dθ = α ∫B f dθ. (b) ∫B (f + g) dθ = ∫B f dθ + ∫B g dθ.

(c) ∫B f dθ ⩽ ∫B g dθ whenever f ⩽ g. a.e. B

(d) ∫B f dθ = ∫X χB ⋅ f dθ.

(e) ∫B f dθ = limA∈R ∫A∩B f dθ. Proof. Part (a) follows straightaway from our definition of the integral of measurable functions. We defer the proof of Part (b). For Part (c), suppose that f ⩽ g for measurable a.e. B

functions f , g ∈ ℱ𝒮 (X, Conv(E)). That is to say, χB\Z ⋅ f ⩽ χB\Z ⋅ g holds with some set Z ∈ A of measure zero. Let v ∈ V˚ . The inequality (v)

(v)

∫ f dθ ⩽ ∫ g dθ B\Z

B\Z

follows immediately from our definition. Proposition 2.3.2(d) renders ∫B h dθ ∫(B\Z) h dθ for all step functions h ∈ ℱ𝒮 (X, Conv(E)) and, therefore, (v)

=

(v)

∫ l dθ = ∫ l dθ B

B\Z

holds for all measurable functions l ∈ ℱ (X, Conv(E)). Combining with the above, we obtain (v)

(v)

(v)

(v)

∫ f dθ = ∫ f dθ ⩽ ∫ g dθ = ∫ g dθ. B

B\Z

B\Z

B

Our claim now follows since the last inequality holds true for every v ∈ V˚ . For Part (d), we recall from Proposition 2.2.12(b) that χB ⋅ f is measurable. Let v ∈ V˚ . We have (v)

󵄨 ∫ f dθ = sup{∫ h dθ 󵄨󵄨󵄨 h ∈ ℱ𝒮 (X, Conv(E)), h ⩽ f + v} B

and

B

94 � 2 Integration (v)

󵄨 ∫ χB ⋅ f dθ = sup{∫ h′ dθ 󵄨󵄨󵄨 h′ ∈ ℱ𝒮 (X, Conv(E)), h′ ⩽ χB ⋅ f + v} X

X

First, let h ∈ ℱ𝒮 (X, Conv(E)) such that h ⩽ f + v. Then h′ = χB ⋅ h ⩽ χB ⋅ f + v, and ∫X h′ dθ = ∫B h dθ by Proposition 2.3.2(c). This shows (v)

(v)

∫ f dθ ⩽ ∫ χB ⋅ f dθ. B

X

Taking the infima over all v ∈ V˚ on both sides of this inequality then yields ∫ f dθ ⩽ ∫ χB ⋅ f dθ. B

X

For the converse inequality, let W ∈ 𝒲 and let v ∈ V˚ such that v ⩽ vW . There is h0 ∈ ℱ𝒮 (X, Conv(E)) such that h0 ⩽ f + v and, therefore, χ(X\B) ⋅ h0 ⩽ χ(X\B) ⋅ f + v. For any h′ ∈ ℱ𝒮 (X, Conv(E)) such that h′ ⩽ χB ⋅ f + v, we have both χB ⋅ h′ ⩽ χB ⋅ f + v and χ(X\B) ⋅ h′ ⩽ v. We set h = χB ⋅ h′ + χ(X\B) ⋅ h0 . Then h ∈ ℱ𝒮 (X, Conv(E)) and h ⩽ (χB ⋅ f + v) + (χ(X\B) ⋅ f + v) ⩽ f + 2v. We have ∫B h dθ = ∫B h′ dθ by Proposition 2.3.2(d) and ∫(X\B) h′ dθ ⩽ σW , as χ(X\B) ⋅ h′ ⩽ v ⩽ vW . Hence using both 2.3.2(b) and 2.3.2(d), we obtain (2v) ′

′

′

∫ h dθ = ∫ h dθ + ∫ h dθ ⩽ ∫ h dθ + σW ⩽ ∫ f dθ + σW X

B

B

X\B

B

for all step functions h′ ∈ ℱ𝒮 (X, Conv(E)) such that h′ ⩽ χB ⋅ f + v and, therefore, (v)

(2v)

∫ χB ⋅ f dθ ⩽ ∫ χB ⋅ f dθ ⩽ ∫ f dθ + σW . X

X

B

Taking the infima first over all v ⩽ wW , then over all W ∈ 𝒲 on the right-hand side of this inequality yields ∫ χB ⋅ f dθ ⩽ ∫ f dθ. X

B

2.3 Integrals of set-valued functions

� 95

Part (d) follows. For Part (e), it is therefore sufficient to consider the case that B = X, because the function f may be replaced by its product with the characteristic function χB . Let A0 ∈ R such that f (x) ⩾ 0 for all x ∈ X \ A0 . Then χA1 ⋅ f ⩽ χA2 ⋅ f ⩽ f whenever A0 ⊂ A1 ⊂ A2 for A1 , A2 ∈ R, hence ∫A f dθ ⩽ ∫A f dθ ⩽ ∫X f dθ by parts (c) and (d). 1 2 This shows lim ∫ f dθ = sup ∫ f dθ ⩽ ∫ f dθ.

A∈R

A0 ⊂A∈R

A

X

A

For the converse inequality, let W ∈ 𝒲 and h ⩽ f + vW for h ∈ ℱ𝒮 (X, Conv(E)). Because h is supported by a set in R, there is A0 ⊂ A1 ∈ R such that ∫X h dθ = ∫A h dθ ⩽ (v )

(vW )

∫A W f dθ for all A1 ⊂ A ∈ R. Moreover, Corollary 2.3.6 shows that ∫A ∫A f dθ + σW . Thus

f dθ ⩽

∫ h dθ ⩽ sup ∫ f dθ + σW = lim ∫ f dθ + σW . A1 ⊂A∈R

X

A∈R

A

A

This shows (vW )

∫ f dθ ⩽ ∫ f dθ ⩽ lim ∫ f dθ + σW . X

A∈R

X

A

Since W ∈ 𝒲 was arbitrarily chosen, our claim follows. Finally, in the light of Part (e) it suffices to verify the statement in (b) for the sets in R. Let A ∈ R, let v ∈ V˚ and let h ⩽ f + v and l ⩽ g + v for h, l ∈ ℱ𝒮 (X, E). Then (2v)

(h + l) ⩽ (f + g) + 2v ⩽ ∫ (f + g) dθ. A

From this, we infer that (v)

(v)

(2v)

∫ f dθ + ∫ g dθ ⩽ ∫ (f + g) dθ. A

A

A

Taking the infima over all v ∈ V˚ renders ∫ f dθ + ∫ g dθ ⩽ ∫(f + g) dθ. A

A

A

For the reverse inequality, let ε > 0, let v ∈ 𝒱˚ and let (fn )n∈ℕ and (gn )n∈ℕ be increasing sequences of step functions in ℱ𝒮 (X, Conv(E)) as in Proposition 2.2.13 for the measurable functions f and g, respectively. Let 1 ⩽ γ, σ ⩽ 1 + ε be the respective

96 � 2 Integration constants involved in this statement and set hn = γfn + σgn ∈ ℱ𝒮 (X, Conv(E)) Then for every x ∈ A there is n0 ∈ ℕ such that (f + g)(x) ⩽ hn (x) holds for all n ⩾ n0 . Thus ∫(f + g) dθ ⩽ lim inf ∫ hn dθ n→∞

A

A

by Corollary 2.3.7. The sequences (f )n∈ℕ , (g)n∈ℕ and (h)n∈ℕ are increasing, so are the respective sequences of their integrals, and we conclude that lim inf ∫ hn dθ = lim ∫ hn dθ = γ lim ∫ fn dθ + σ lim ∫ gn dθ. n→∞

n→∞

A

n→∞

A

n→∞

A

A

(v)

Moreover, fn ⩽ f + v and gn ⩽ g + v implies that ∫A fn dθ ⩽ ∫A f dθ and ∫A gn dθ ⩽ (v)

∫A g dθ for all n ∈ ℕ. Combining all of the above then yields (v)

(v)

∫(f + g) dθ ⩽ γ ∫ f dθ + σ ∫ g dθ. A

A

A

The latter holds for all v ∈ V˚ and ε > 0 with γ and σ depending on both. First, with v ∈ V˚ fixed, passing to the limit over ε tending to 0, and then to the downward directed limits over v ∈ V˚ , we obtain ∫ (f + g) dθ ⩽ ∫ f dθ + ∫ g dθ. A

A

A

This completes our argument. In the expression below, for the calculation of the integral of a measurable function f ∈ ℱ (X, Conv(E)) we consider only step functions with values in the range of f . This will turn out to be advantageous for our interpretation of the value of such an integral. 2.3.9 Proposition. Let B ∈ A and let f ∈ ℱ (X, Conv(E)) be a measurable function. Then 󵄨󵄨 h ∈ ℱ (X, E), h ⩽ f + v } { 󵄨 𝒮 W ∫ f dθ = lim sup { ∫ h dθ 󵄨󵄨󵄨 }. 󵄨󵄨 h(B) ⊂ ⋃x∈B f (x) W ∈𝒲 B {B } Proof. Let B ∈ A and let f ∈ ℱ (X, Conv(E)) be a measurable function. Given W ∈ 𝒲 , for the sake of our argument we abbreviate 󵄨󵄨 h ∈ ℱ (X, E), h ⩽ f + v } { 󵄨 𝒮 W CW = { ∫ h dθ 󵄨󵄨󵄨 }. 󵄨󵄨 h(B) ⊂ ⋃x∈B f (x) {B } For the corresponding element φCW = sup CW ∈ 𝒬F , we have

2.3 Integrals of set-valued functions

� 97

(vw )

φCW ⩽ ∫ f dθ ⩽ ∫ f dθ + σW . B

B

Conversely, let h = ∑ni=1 χAi ⋅ ai ∈ ℱ𝒮 (X, E) such that h ⩽ f + v(1/2)W . That is to say, h ⩽ f + n for a neighborhood function n = ∑m k=1 χCk ⋅ Vk with disjoint sets Ck ∈ R and Vk ∈ 𝒱 such that ∫X n dθ = ∑m |θ| (V ) ⩽ σ Ck k (1/2)W . We may also assume that the sets k=1 Ai ∈ R are disjoint, either contained in B or in X \ B, and that their union contains the union C of the sets Ck . The sets Cik = Ai ∩ Ck form a disjoint cover of C. If Cik ≠ H, we choose xik ∈ Cik and have ai = h(xik ) ⩽ f (xik ) + Vk hence there is cik ∈ f (xik ) such that ai − cik ∈ Vk . If Cik = H, we assign cik = 0. Using these choices, we set h̃ = ∑ χC k ⋅ cik + χX\C ⋅ h ∈ ℱ𝒮 (X, E) i

i,k

and observe the following. For every x ∈ B ∩ C, we have x ∈ Cik for exactly one choice of i and k and ̃ h(x) = cik ∈ f (xik ) ⊂ ⋃ f (x). x∈B

For x ∈ B ∩ (X \ C), on the other hand, we have n(x) = 0 and ̃ h(x) = h(x) ⩽ f (x) + n(x) = f (x). ̃ ̃ Thus h(x) ∈ ⋃x∈B f (x) also holds. We conclude that h(B) ⊂ ⋃x∈B f (x). Moreover, for k x ∈ C we have x ∈ Ci for some i and k. Thus ̃ h(x) = cik ⩽ ai + Vk = h(x) + n(x) ⩽ f (x) + 2n(x) and ̃ + n(x). h(x) = ai ⩽ cik + Vk = h(x) ̃ Since h(x) = h(x) for all x ∈ X \ C, we infer that h̃ ⩽ f + 2n, that is, h̃ ⩽ f + vW and ̃ h ⩽ h + n. Summarizing, we have h̃ ∈ CW and ∫ h dθ ⩽ ∫ h̃ dθ + ∫ n dθ ⩽ ∫ h̃ dθ + σ(1/2)W ⩽ φCW + σW . B

B

B

B

The latter holds true for all h ∈ ℱ𝒮 (X, E) such that h ⩽ f + v(1/2)W , and we conclude that

98 � 2 Integration (v(1/2)W )

∫ f dθ ⩽

∫

B

B

f dθ ⩽ φCW + σW .

Together with (vW )

φCW ⩽ ∫ f dθ ⩽ ∫ f dθ + σW B

B

this yields φCW ∈ W s (∫B f dθ), and our claim follows. The following result allows us to interpret the value of the integral of a measurable Conv(E)-valued function as an element of Conv(F ˚˚ ) or of Conv(F). Proposition 1.4.3(a) establishes a correspondence between elements of 𝒬F that are σ(F ˚ , F ˚˚ ) -lower semicontinuous and convex subsets of E ˚˚ , and Proposition 1.4.3(d) states that an element of 𝒬F represents a weakly compact convex subset of F if and only it is continuous in the Mackey topology τ(F ˚ , F). 2.3.10 Proposition. (a) Suppose that F is a normed space. If f ∈ ℱ (X, Conv(E)) is measurable, then for every B ∈ A the integral ∫B f dθ represents a convex subset of F ˚˚ . (b) Suppose that the measure θ is L(E, F)-valued and weakly compact. Let f ∈ ℱ (X, Conv(E)) be a measurable function and let A ∈ R such that ⋃x∈A f (x) is bounded in E. Then ∫A f dθ represents a weakly compact convex subset of F. Proof. (a) If F is a normed space, then F ˚‚ = F ˚˚ , and our measures are L(E, F ˚˚ )valued. For a set B ∈ A, the integral of an E-valued step function over B is an element of F ˚˚ ⊂ 𝒬F , and the integral of a Conv(E)-valued step function evaluates as a σ(F ˚ , F ˚˚ )-lower semicontinuous element of 𝒬F . The same observation holds (v)

true for the expression ∫B f dθ for a measurable function f ∈ ℱ (X, Conv(E)), a set B ∈ A and a function space neighborhood v ∈ V˚ . Moreover, the integral ∫B f dθ of a measurable function is the limit of these expressions in the symmetric topology of 𝒬F , hence also σ(F ˚ , F ˚˚ )-lower semicontinuous by Proposition 1.4.6(b). For the latter, we recall from 1.4.4 that a sublinear functional on F ˚ is σ(F ˚ , F ˚˚ )-lower semicontinuous if and only if it is β(F ˚ , F)-lower semicontinuous, provided that F is a normed space. Thus in the light of Proposition 1.4.3(a), which states a correspondence between σ(F ˚ , F ˚˚ )-lower semicontinuous sublinear functionals on F ˚ and convex subsets of F ˚˚ , the integral of ∫B f dθ may be considered to be an element of Conv(F ˚˚ ). For Part (b), let θ and f ∈ ℱ (X, Conv(E)) be as stated. For A ∈ R, let n

n

i=1

i=1

󵄨󵄨 D = {∑ θAi (ci ) 󵄨󵄨󵄨 ci ∈ ⋃ f (x), Ai ∈ R disjoint, ⋃ Ai = A} 󵄨 x∈A

2.3 Integrals of set-valued functions

� 99

As ⋃x∈A f (x) is bounded and θ is weakly compact, this set is relatively weakly compact in F. For every W ∈ 𝒲 , we have 󵄨󵄨 h ∈ ℱ (X, E), h ⩽ f + v } { 󵄨 𝒮 W CW = { ∫ h dθ 󵄨󵄨󵄨 } ⊂ D. 󵄨󵄨 h(B) ⊂ ⋃x∈A f (x) } {A Hence the sets CW are also relatively weakly compact and the corresponding sublinear functionals φCW ∈ 𝒬F are τ(F ˚ , F)-continuous. According to Proposition 2.3.9, we have ∫ f dθ = lim φCW = inf φCW . W ∈𝒲

A

W ∈𝒲

Thus following Proposition 1.4.3(c), the functional ∫A f dθ ∈ 𝒬F is also τ(F ˚ , F)-continuous and in the light of 1.4.3(d) may be interpreted to be a weakly compact convex subset of F. For the following, recall from Section 1.2 that a single neighborhood function n defines a function space neighborhood if and only if for every A ∈ R there is V ∈ 𝒱 such that χA ⋅ V ⩽ n. For an L(E, F ˚‚ )-valued measure, a fixed neighborhood V ∈ 𝒱 and μ ∈ F ˚ we introduce the positive ℝ-valued measure [|θ|(V )(μ)], that is, the countably additive set-function A 󳨃→ |θ|A (V )(μ) : R → ℝ+ . This measure is easily extended to the σ-field A if we set $[|θ|(V )(μ)]B = sup [|θ|(V )(μ)]A A∈R

for all B ∈ A. For an ℝ+ -valued A-measurable function ρ and B ∈ A, we have n 󵄨󵄨 α ⩾ 0, A ∈ R, A ⊂ B, 󵄨 i i }. ∫ ρ d[|θ|(V )(μ)] = sup {∑ αi [|θ|(V )(μ)]A 󵄨󵄨󵄨 in i 󵄨 ∑ 󵄨 i=1 αi χBi ⩽ ρ i=1 B

Using this, we formulate the next result. 2.3.11 Proposition. Suppose that the function space neighborhood v is defined by a single measurable neighborhood function n. Then (v,{μ})

(∫ n dθ)(μ) = sup ‖θ‖A∩B B

A∈R

for every μ ∈ F ˚ and B ∈ A. If n = ρ ⋅ V for V ∈ 𝒱 and an ℝ+ -valued A-measurable function ρ such that inf{ρ(x) | x ∈ A} > 0 for every A ∈ R, then the above expressions equal ∫B ρ d[|θ|(V )(μ)].

100 � 2 Integration Proof. Let v and n be as stated and let B ∈ A and μ ∈ F ˚ . According to our definition of the variation of a measure in Section 2.1, we have (v,{μ})

‖θ‖A

n 󵄨󵄨 A ∈ R disjoint, A ⊂ A, 󵄨 i } = sup {∑ θAi (ai )(μ) 󵄨󵄨󵄨 i 󵄨󵄨 ai ∈ E, χA ⋅ ai ⩽ n i i=1

for every A ∈ R. Using Lemma 1.2.1(b), we realize that χAi ⋅ ai ⩽ n for disjoint sets Ai ∈ R implies that ∑ni=1 χAi ⋅ ai ⩽ n and, therefore, ∑ni=1 θAi (ai ) ⩽ ∫B n dθ, provided that all Ai ⊂ B. The inequality (v,{μ})

sup ‖θ‖A∩B

A∈R

⩽ (∫ n dθ)(μ) B

is therefore evident. For the reverse relation, let ε > 0. For any step function h ∈ ℱ𝒮 (X, E) such that h ⩽ n + εv, we have h ⩽ (1 + ε)n. This yields (εv)

󵄨 ∫ n dθ ⩽ (1 + ε) sup{∫ h dθ 󵄨󵄨󵄨 h ∈ ℱ𝒮 (X, E), h ⩽ n} B

B

for all ε > 0. Thus 󵄨 ∫ n dθ ⩽ sup{∫ h dθ 󵄨󵄨󵄨 h ∈ ℱ𝒮 (X, E), h ⩽ n}. B

B

Our claim follows, because for every step function h ∈ ℱ𝒮 (X, E) that is included in the supremum on the right-hand side of the last inequality, its product χB ⋅ h with χB is (v,{μ}) involved in the definition of ‖θ‖A∩B for some set A ∈ R. This completes the argument for the first part of our claim. Now suppose that n = ρ ⋅ V for an ℝ+ -valued A -measurable function ρ such that inf{ρ(x) | x ∈ A} > 0 for every A ∈ R, and V ∈ 𝒱 . We calculate that n 󵄨󵄨 A ∈ R, A ⊂ B, α ⩾ 0, 󵄨 i i } ∫ ρ d[|θ|(V )(μ)] = sup {∑ αi [|θ|(V )(μ)]A 󵄨󵄨󵄨 in i 󵄨 ∑ 󵄨 i=1 αi χAi ⩽ ρ i=1 B

n 󵄨󵄨 A ∈ R, A ⊂ B, α ⩾ 0, 󵄨 i i = sup {∑ |θ|Ai (αi V )(μ) 󵄨󵄨󵄨 in } 󵄨󵄨 ∑i=1 χA ⋅ (αi V ) ⩽ n i i=1

⩽ (∫ n dθ)(μ). B

For the reverse inequality, let γ > 1, let Ai ∈ R be disjoint subsets of B and ai ∈ E such that χAi ⋅ ai ⩽ n and set αi = inf{ρ(x) | x ∈ Ai } > 0. Then ai ∈ ρ(x)V for all x ∈ Ai implies that ai ∈ γαi V and, therefore,

2.3 Integrals of set-valued functions n

n

i=1

i=1

�

101

∑ χAi ⋅ ai ⩽ ∑ χAi ⋅ (γαi V ) ⩽ γn. Hence n

n

i=1

i=1

∑ θAi (ai )(μ) ⩽ ∑ |θ|Ai (γαi V )(μ) ⩽ γ ∫ ρ d[|θ|(V )(μ)]. B

Taking the supremum over all such choices for Ai ∈ R and ai ∈ E yields (v,{μ})

(∫ n dθ)(μ) = sup ‖θ‖A∩B A∈R

B

⩽ γ ∫ ρ d[|θ|(V )(μ)] B

for all γ > 1. Hence our claim. If the measure θ is L(E, G)-valued for a subspace G of F ˚‚ , then the integral of a neighborhood function n satisfying the assumptions of Proposition 2.3.11 may be considered to be an element of Conv(G). Indeed, it follows immediately from the definition of the variation of a measure that the mapping (v,{μ})

μ 󳨃→ sup ‖θ‖A∩B A∈R

: F˚ → ℝ

is sublinear and lower semicontinuous in the topology σ(F ˚ , G). Hence it represents a closed convex subset of G. (See Sections 1.4 and 3.1 below.)

Integrals of set-valued functions We are now in a position to extend our notion of integrability to Conv(E)-valued functions that are not necessarily measurable in the sense of Section 2.2. First, for a set A ∈ R we shall say that a function f ∈ ℱ (X, Conv(E)) is integrable over A with respect to an L(E, F ˚‚ )-valued measure θ if for every W ∈ 𝒲 there are a measurable function fW and a measurable neighborhood function nW in ℱ (X, Conv(E)) such that f ⩽ fW ⩽ f + nW a.e. A

a.e. A

and

∫ nW dθ ⩽ σW . A

Recall from Section 1.2 that neighborhood functions are 𝒱c -valued, where 𝒱c denotes the extended function space neighborhood system of E, that is, the convex cone generated by 𝒱 ∪ {E} in Conv(E). We consider nets (fW )W ∈𝒲 of measurable functions as in this definition. The index set 𝒲 is endowed with the reverse set inclusion as its order. For the following, we recall from Section 1.4 that convergence of a net in the symmetric topology of 𝒬F implies

102 � 2 Integration pointwise convergence on the elements of F ˚ with respect to the symmetric topology of ℝ. 2.3.12 Lemma. For a function f ∈ ℱ (X, Conv(E)) that is integrable over A ∈ R and measurable functions fW as in the definition of integrability, the net (∫A fW dθ)W ∈𝒲 converges in the symmetric topology of 𝒬F and its limit is independent of the particular choice for the functions fW . Proof. Suppose that f ∈ ℱ (X, Conv(E)) is integrable over A ∈ R. Let W ∈ 𝒲 . For any choice of U, V ∈ 𝒲 such that U, V ⊂ W and any choice of the measurable functions fU and fV from our definition of integrability, we have fU ⩽ f + nU ⩽ fV + nU ⩽ fV + nW . Hence using the statements of Theorem 2.3.8, we obtain ∫ fU dθ ⩽ ∫ fV dθ + σW . A

A

The net (∫A fW dθ)W ∈𝒲 is therefore a Cauchy net in the symmetric topology of 𝒬F , hence convergent according to Proposition 1.4.5. Our argument also implies that its limit is independent of the particular choice for the functions fW . Integrals over sets in R Consequently, for a function f ∈ ℱ (X, Conv(E)) that is integrable over a set A ∈ R we define its integral over A as ∫ f dθ = lim ∫ fW dθ, A

W ∈𝒲

A

where (fW )W ∈𝒲 is a net of measurable functions in ℱ (X, Conv(E)) as in the above definition of integrability for the function f , and the limit refers to the symmetric topology of 𝒬F . This notion is of course consistent with the earlier definition of the integral of a measurable function. 2.3.13 Proposition. If f ∈ ℱ (X, Conv(E)) is integrable over a set A ∈ R, then for every W ∈ 𝒲 there is a measurable neighborhood function n and λ ⩾ 0 such that 0 ⩽ f + n and ∫A n dθ ⩽ λσW . Proof. Suppose that f ∈ ℱ (X, Conv(E)) is integrable over A ∈ R and let W ∈ 𝒲 . According to the definition of integrability, there are measurable functions g ∈ ℱ (X, Conv(E)) and a measurable neighborhood function n such that f ⩽ g ⩽ f +n a.e. A

a.e. A

2.3 Integrals of set-valued functions

� 103

and ∫A n dθ ⩽ σW . We choose V ∈ 𝒱 such that |θ|A (V ) ⩽ σW . Following Proposition 2.2.11, there is λ ⩾ 0 such that 0 ⩽ g + λχX ⋅ V . There is a set Z ∈ A of measure zero such that g ⩽ f + n + χ(X\A)∪Z ⋅ E. The combination of the above renders 0 ⩽ g + λχX ⋅ V ⩽ f + (n + λχX ⋅ V + χ(X\A)∪Z ⋅ E). As m = n + λχX ⋅ V + χ(X\A)∪Z ⋅ E is a measurable neighborhood function, and as ∫ m dθ = ∫ n dθ + λ|θ|A (V ) + ∫ χ ⋅ E dθ ⩽ (1 + λ)σW A

A

Z

holds by 2.3.8, our claim follows.

Integrals over sets in A Obviously, integrability for a function f ∈ ℱ (X, Conv(E)) over a set A ∈ R implies integrability over all subsets B ∈ R of A. This observation, together with Theorem 2.3.8(e) shows that we may consistently define integrability over sets in the σ-field A in the following way: We shall say that a function f ∈ ℱ (X, Conv(E)) is integrable over B ∈ A with respect to θ if f is integrable over the sets A ∩ B for all A ∈ R and if the limit ∫ f dθ = lim ∫ f dθ B

A∈R

A∩B

exists, pointwise on F ˚ , in 𝒬F . The set of all functions in ℱ (X, Conv(E)) or in ℱ (X, E) that are integrable over B with respect to θ shall be denoted by ℱ(θ,B) (X, Conv(E)) or ℱ(θ,B) (X, E), respectively. Obviously, if f ∈ ℱ(θ,B) (X, Conv(E)) and f = g for g ∈ a.e. B ℱ (X, Conv(E)), then g ∈ ℱ(θ,B) (X, Conv(E)). Theorem 2.3.8(e) implies that all measurable functions in ℱ (X, Conv(E)) are contained in ℱ(θ,B) (X, Conv(E)) for every set B ∈ A and every L(E, F ˚‚ )-valued measure θ on R. We recall from Section 1.4 that for f ∈ ℱ (X, Conv(E)) and an element α of the scalar field of E the function αf ∈ ℱ (X, Conv(E)) is defined as the mapping x 󳨃→ αf (x) ∈ Conv(E). An element of 𝒬F represents a convex subset of F, F ˚˚ or F ˚‚ , if it is lower semicontinuous as a sublinear functional on F ˚ in a topology that is consistent with the duality (F ˚ , F), (F ˚ , F ˚˚ ) or (F ˚ , F ˚‚ ), respectively. 2.3.14 Proposition. If f ∈ ℱ(θ,B) (X, Conv(E)) for B ∈ A, then αf ∈ ℱ(θ,B) (X, Conv(E)) for all α in ℝ or ℂ and (∫ αf dθ)(μ) = (∫ f dθ)(αμ) B

B

104 � 2 Integration holds for all μ ∈ F ˚ . If ∫B f dθ represents a convex subset C of F, F ˚˚ or F ˚‚ , then ∫B αf dθ represents αC. Proof. We recollect that αf ⩽ αg holds whenever f ⩽ g, and αf ⩽ αg + |α|v holds whenever f ⩽ g + v for f , g ∈ ℱ (X, Conv(E)), an element α of the common scalar field of E and F and a function space neighborhood v. Moreover, the function αf is measurable whenever f is measurable (Proposition 2.2.12(a)). Because θ is supposed to be L(E, F ˚‚ )-valued, we have θA (αa)(μ) = θA (a)(αμ) for all A ∈ R, all a ∈ E, all α in ℝ or ℂ and all μ ∈ F ˚ . The latter yields immediately that our claim holds for integrals of step functions in ℱ𝒮 (X, Conv(E)), and since they are pointwise suprema of integrals of step functions on F ˚ , also for the precurv sory expressions ∫A f dθ for measurable functions. Because they are pointwise limits of these expressions, this property then further transfers to integrals of measurable functions. We continue to use the same argument for a function f ∈ ℱ (X, Conv(E)) that is integrable over a set A ∈ R. Indeed, if (fW )W ∈𝒲 is a net of measurable functions in ℱ (X, Conv(E)) approaching f as in our definition of integrability over A, then since αnW = |α|nW , the net (αfW )W ∈𝒲 approaches αf in the prescribed way with the corresponding neighborhood functions |α|nW . Pointwise convergence of the integrals on F ˚ then yields our claim. The extension to integrals over a set B ∈ A is argued in the same way. Finally, if ∫B f dθ represents a convex subset C of F, F ˚˚ or F ˚‚ , then (∫ f dθ)(μ) = sup{Re μ(c) | c ∈ C} B

for all μ ∈ F ˚ , and (∫ αf dθ)(μ) = (∫ αf dθ)(αμ) = sup{Re μ(c) | c ∈ αC} B

B

Hence ∫B αf dθ represents αC. 2.3.15 Theorem. Let B ∈ A. Then ℱ(θ,B) (X, Conv(E)) is a subcone of the cone ℱ (X, Conv(E)) that contains all measurable functions, and for f , g ∈ ℱ(θ,B) (X, Conv(E)) and 0 ⩽ α ∈ ℝ we have (a) ∫B (αf ) dθ = α ∫B f dθ for all α ⩾ 0.

(b) ∫B (f + g) dθ = ∫B f dθ + ∫B g dθ.

(c) ∫B f dθ ⩽ ∫B g dθ whenever f ⩽ g. a.e. B

2.3 Integrals of set-valued functions

� 105

Proof. Part (a) follows immediately from Proposition 2.3.14 and the fact that the integral is a sublinear functional on F ˚ . Thus for f ∈ ℱ(θ,B) (X, Conv(E)) and α ⩾ 0, we calculate that (∫ αf dθ)(μ) = (∫ f dθ)(αμ) = (α ∫ f dθ)(μ) B

B

B

holds for all μ ∈ F ˚ . Hence our claim in (a). We shall argue for the remaining parts in steps. First, we shall verify our statements for integrals over sets in R. For this, let A ∈ R and let f , g ∈ ℱ(θ,A) (X, Conv(E)). Let (fW )W ∈𝒲 , (nW )W ∈𝒲 and (gW )W ∈𝒲 , (mW )W ∈𝒲 be nets of measurable functions as in the definition of the integral for the functions f and g over A, respectively. For Part (b), we set hW = fW + gW

and sW = nW + mW

and observe that (f + g) ⩽ hW ⩽ (f + g) + sW a.e. A

a.e. A

and ∫ sW dθ = ∫ nW dθ + ∫ mW dθ ⩽ 2σW A

A

A

holds for all W ∈ 𝒲 by Theorem 2.3.8(b). The measurable functions hW and sW therefore satisfy the definition of integrability over A with 2W in place of W for the function f + g. Now the corresponding statement for measurable functions in 2.3.8(b) and the usual limit rules yield ∫(f + g) dθ = lim ∫ hW dθ W ∈𝒲

A

A

= lim ∫ fW dθ + lim ∫ gW dθ W ∈𝒲

W ∈𝒲

A

A

= ∫ f dθ + ∫ g dθ. A

A

For Part (c), suppose that f ⩽ g. Then a.e. A

fW ⩽ f + nW ⩽ g + nW ⩽ gW + mW + nW a.e. A

a.e. A

a.e. A

holds for all W ∈ 𝒲 . Hence using Theorem 2.3.8(a), (b) and (c) for measurable functions, we infer that

106 � 2 Integration ∫ fW dθ ⩽ ∫ gW dθ + 2σW A

A

for all W ∈ 𝒲 . According to the limit rules, this yields ∫ f dθ = lim ∫ fW dθ ⩽ lim ∫ gW dθ = ∫ g dθ. W ∈𝒲

A

W ∈𝒲

A

A

A

For the general case, let B ∈ A and f , g ∈ ℱ(θ,B) (X, Conv(E)). Then the claims in parts (b) and (c) hold for integrals over all sets A∩B for A ∈ R. The definition of the respective integrals over B together with the validity of our claims for integrals over sets in R and the usual limit rules yield their validity for the integrals over B ∈ A as well. Parts (a) and (b) imply that ℱ(θ,B) (X, Conv(E)) forms a subcone of ℱ (X, Conv(E)). For the following corollaries, recall the definition of the order ≼V in Conv(E) from Section 2.2. That is to say, for a neighborhood V ∈ 𝒱 and C, D ∈ Conv(E) we denote C ≼V D if C ⩽ D + εV for all ε > 0. 2.3.16 Corollary. Let f , g ∈ ℱ(θ,B) (X, Conv(E)) for B ∈ A and let V ∈ 𝒱 . If f (x) ≼V g(x) θ-almost everywhere on B, then (∫B f dθ)(μ) ⩽ (∫B g dθ)(μ) for all μ ∈ F ˚ such that (∫B χX ⋅ V dθ)(μ) < +∞. Proof. Let f , g ∈ ℱ(θ,B) (X, Conv(E)) for B ∈ A and let V ∈ 𝒱 . If f (x) ≼V g(x) holds θ-almost everywhere on B, then f ⩽ g + εχX ⋅ V for all ε > 0. Thus a.e. B

∫ f dθ ⩽ ∫ g dθ + ε ∫ χX ⋅ V dθ B

B

B

by Proposition 2.3.15(c). Hence our claim. We note that ∫B χX ⋅ V dθ = supA∈R |θ|A∩B (V ) for V ∈ 𝒱 and B ∈ A. Other than for measurable functions f , g ∈ ℱ (X, Conv(E)) and a neighborhood V ∈ 𝒱 (see Corollary 2.2.16), for integrable functions it can no longer be guaranteed that the set {x ∈ X | f (x) ≼V g(x)} is contained in A. However, we have the following. 2.3.17 Corollary. Let f , g ∈ ℱ(θ,A) (X, Conv(E)) for A ∈ R and let W ∈ 𝒲 and V ∈ 𝒱 such that |θ|A (V ) ⩽ σW . There is a subset B ∈ R of A that contains all points x ∈ A such that f (x) ≼V g(x), and (∫C f dθ)(μ) ⩽ (∫C g dθ)(μ) holds for all μ ∈ F ˚ such that σW (μ) < +∞ and every subset C ∈ R of A. Proof. Let f , g ∈ ℱ(θ,A) (X, Conv(E)) for A ∈ R and let W ∈ 𝒲 and V ∈ 𝒱 such that |θ|A (V ) ⩽ σW . For every n ∈ ℕ, there are measurable functions fn , gn ∈ ℱ (X, Conv(E)), measurable 𝒱c -valued functions nn and mn such that f ⩽ fn ⩽ f + nn a.e. A

a.e. A

and g ⩽ gn ⩽ g + mn , a.e. A

a.e. A

2.3 Integrals of set-valued functions

�

107

and both ∫A nn dθ ⩽ (1/n)σW and ∫A mn dθ ⩽ (1/n)σW . We may assume that the above order relations hold everywhere on A instead of θ-almost everywhere. Indeed, the relations for each of the functions fn , gn , nn and mn hold for all x ∈ A \ Zn with some set Zn ∈ R of measure zero. According to Proposition 2.1.10(c), the set Z = ⋃n∈ℕ Zn ∈ R is again of measure zero, and the products of all of the functions involved with χX\Z are again integrable and satisfy the above order relations everywhere on A. Their respective integrals coincide with those of the given functions. Let x ∈ A. If f (x) ≼V g(x), that is, f (x) ⩽ g(x) + εV holds for all ε > 0, then for every choice of n, m ∈ ℕ, we compute fn (x) ⩽ f (x) + nn (x)

⩽ g(x) + (1/m)V + nn (x)

⩽ gn (x) + nn (x) + (2/m)V . According to Corollary 2.2.16, all the sets Bnm = {x ∈ A | fn (x) ≼V gn (x) + nn (x) + (2/m)V } are contained in R and so is their intersection B = ⋂n,m∈ℕ Bnm . Moreover, B contains all points x ∈ B such that f (x) ≼V g(x), and following Corollary 2.3.16 for every subset C ∈ R of B, we have ∫ fn dθ ⩽ ∫ gn dθ + ∫ nn dθ + (2/m)σW C

C

C

⩽ ∫ gn dθ + (1/n + 2/m)σW . C

Therefore, ∫ f dθ ⩽ ∫ gn dθ + (1/n + 2/m)σW C

C

⩽ ∫(g + mn ) dθ + (1/n + 2/m)σW C

⩽ ∫ g dθ + (2/n + 2/m)σW C

holds for all n, m ∈ ℕ. Let μ ∈ F ˚ such that σW (μ) < +∞. Evaluating the above inequality at μ and taking the limit over n and m now returns our claim. According to our definition, integrability for a function in ℱ (X, Conv(E)) over a set in B ∈ A with respect to a measure θ requires integrability over all subsets A ∈ R of B. However, the following simple example shows that this claim does not generally extend to all subsets C ∈ A of B. For this, let X = [0, +∞), let R be the collection of

108 � 2 Integration all relatively compact Borel subsets of X and let E = F = ℝ. Let θ be the Lebesgue measure on X. Consider the function f on X such that f (x) = (−1)n+1 /n for x ∈ [n − 1, n) and n ∈ ℕ. Then f is integrable over X in the sense of our definition, and n+1 /n = ln 2. It is however not integrable over the set C = ⋃n∈ℕ [2n − ∫X f dθ = ∑∞ n=1 (−1) 1, 2n) ∈ A, since limA∈R ∫A∩C f dθ = −∞. 2.3.18 Theorem. Let f , g ∈ ℱ (X, Conv(E)) and B, C ∈ A. (a) The function f is integrable over B ∩ C if and only if χC ⋅ f is integrable over B, if and only if χB ⋅ f is integrable over C. In this case, we have ∫B∩C f dθ = ∫B χC ⋅ f dθ = ∫C χB ⋅ f dθ. (b) If B and C are disjoint and f is integrable over B and C, then f is integrable over B ∪ C and ∫B∪C f dθ = ∫B f dθ + ∫C f dθ. (c) If B ⊂ C, if f and g are integrable over B and C and if f ⩽ g, then a.e. C

∫C f dθ + ∫B g dθ ⩽ ∫B f dθ + ∫C g dθ.

Proof. Let f ∈ ℱ (X, Conv(E)). For Part (a), in a first step we shall verify that f is integrable over a set A ∈ R if and only χA ⋅ f is integrable over X and that ∫A f dθ = ∫X χA ⋅ f dθ holds in this case. Indeed, if χA ⋅ f is integrable over X, our definition of integrability over in X ∈ A yields that χA ⋅ f is integrable over all sets in R. But it is immediate from our definition of integrability over a set in R that χA ⋅ f is integrable over A if and only if f is integrable over A and that ∫A f dθ = ∫A χA ⋅ f dθ. Conversely, assume that f is integrable over A. We shall verify that χA ⋅f is integrable over X. For this, let D ∈ R. Given W ∈ 𝒲 , there are measurable functions fW and nW as in our definition of integrability over A, that is, that f ⩽ fW ⩽ f + nW and ∫A nW dθ ⩽ σW . a.e. A

Then we have

a.e. A

χA ⋅ f ⩽ χA ⋅ fW ⩽ χA ⋅ f + χA ⋅ nW a.e. D

a.e. D

and ∫D χA ⋅ nW dθ ⩽ σW as well. Because the functions χA ⋅ fW and χA ⋅ nW are also measurable (Proposition 2.2.12(b)), we conclude that the function χA ⋅ f is integrable over D and using Theorem 2.3.8(d) for measurable functions, that ∫ χA ⋅ f dθ = lim ∫ χA ⋅ fW dθ = lim ∫ χD∩A ⋅ fW dθ. W ∈𝒲

D

W ∈𝒲

D

X

The above holds for all sets D ∈ R, hence using our definition for the integrability over a set in A, we realize that the function χA ⋅ f is indeed integrable over X and that ∫ χA ⋅ f dθ = lim ∫ χA ⋅ f dθ X

D∈R

D

= lim lim ∫ χD∩A ⋅ fW dθ D∈R W ∈𝒲

X

2.3 Integrals of set-valued functions

� 109

= lim ∫ χA ⋅ fW dθ W ∈𝒲

X

= lim ∫ fW dθ W ∈𝒲

A

= ∫ f dθ. A

This demonstrates our claim in the first step. Now in a second step we shall verify the statement of our first step for the sets in A. For this, let B ∈ A. By our first step, the function f is integrable over all sets A ∩ B for A ∈ R, if and only if all the functions χ(A∩B) ⋅ f = χA ⋅ (χB ⋅ f ) are integrable over X. In this case, ∫ f dθ = ∫ χA∩B ⋅ f dθ = ∫ χA ⋅ (χB ⋅ f ) dθ = ∫ χB ⋅ f dθ A∩B

X

X

A

holds by our first step. According to our definition of integrability over sets in A, therefore f is integrable over B if and only χB ⋅ f is integrable over X, and in this case we have ∫ f dθ = lim ∫ f dθ = lim ∫ χB ⋅ f dθ = ∫ χB ⋅ f dθ. A∈R

B

A∈R

A∩B

A

X

In a third and final step for Part (a), let B, C ∈ A. From the preceding, we conclude that χC ⋅ f is integrable over B if and only if χB ⋅ (χC ⋅ f ) = χ(B∩C) ⋅ f is integrable over X, that is, f is integrable over B ∩ C, and all the integrals coincide. For Part (b), suppose that B, C ∈ A, that B ∩ C = H and that f is integrable over both B and C. Then both functions χB ⋅ f and χC ⋅ f are integrable over X by Part (a), hence χ(B∪C) ⋅ f = χB ⋅ f + χC ⋅ f is also integrable over X by Theorem 2.3.15(b). Thus f is indeed integrable over B ∪ C and ∫ f dθ = ∫ χB∪C ⋅ f = ∫ χB ⋅ f + ∫ χC ⋅ f = ∫ f dθ + ∫ f dθ B∪C

X

X

X

B

C

by 2.3.15(b). Part (c) is obvious if both B, C ∈ R such that B ⊂ C, since both f and g are integrable over C\B ∈ R in this case and Proposition 2.3.15(b) yields that ∫C\B f dθ ⩽

∫C\B g dθ. Adding ∫B f dθ+∫B g dθ to both sides of this inequality renders our claim with Part (b). For the general case, that is, B, C ∈ A, let A ∈ R. Then both A ∩ B, A ∩ C ∈ R and A ∩ B ⊂ A ∩ C. Both functions f and g are integrable over these sets, and we have ∫ f dθ + ∫ g dθ ⩽ ∫ f dθ + ∫ g dθ A∩C

A∩B

A∩B

A∩C

by the prior argument for sets in R. Taking the limit over all sets A ∈ R renders our claim by the definition of the integral over B and C, respectively.

110 � 2 Integration We take note that Part (b) of Theorem 2.3.18 implies that a function f in ℱ (X, Conv(E)) that is integrable over each of two not necessarily disjoint sets B, C ∈ R is also integrable over their union. Indeed, since the sets B \ C, C \ B and B ∩ C are in R and subsets of B and C, respectively, f is integrable over each of these sets and, therefore, over their disjoint union B ∪ C. For the following observation, recall the definition of the restriction θ|W of a measure θ to a set B ∈ A from Section 2.1 together with the results from Proposition 2.1.9. 2.3.19 Proposition. A function f ∈ ℱ (X, Conv(E)) is integrable over B ∈ A with respect to θ if and only if it is integrable over X with respect to θ|B . In this case, we have ∫B f dθ = ∫X f dθ|B . Proof. Our claim is immediate for step functions and, therefore, also for measurable functions. Since the relations ⩽ for θ and ⩽ for θ|B coincide, the respective defia.e. B

a.e. X

nitions of integrability agree and the integrals are equal.

2.3.20 Proposition. Let f ∈ ℱ(θ,A) (X, Conv(E)) for A ∈ R. For every W ∈ 𝒲 and every ε > 0, there is an increasing sequence (hn )n∈ℕ of step functions in ℱ𝒮 (X, Conv(E)), 1 ⩽ γ ⩽ 1 + ε and a measurable neighborhood function n such that: (i) hn ⩽ f + n for all n ∈ ℕ and ∫A n dθ ⩽ σW . a.e. A

(ii) θ-almost everywhere on A, for x ∈ A there is n ∈ ℕ such that f (x) ⩽ γhn (x). (iii) ∫B f dθ ⩽ γ lim infn→∞ ∫B hn dθ, pointwise on F ˚ , for all B ∈ R such that B ⊂ A.

Proof. Suppose that f ∈ ℱ(θ,A) (X, Conv(E)) for A ∈ R. Let W ∈ 𝒲 and ε > 0. Following our definition of integrability, there are a measurable function g and a measurable neighborhood function m such that f ⩽ g ⩽ f + m and ∫A m dθ ⩽ (1/2)σW . There is a.e. A

a.e. A

V ∈ 𝒱 such that |θ|A (V ) ⩽ (1/2)σW . We use Proposition 2.2.13 for the measurable function g with the function space neighborhood v generated by the single neighborhood function χX ⋅V . There is 1 ⩽ γ ⩽ 1+ε and an increasing sequence (hn )n∈ℕ of step functions in ℱ𝒮 (X, Conv(E)) such that hn ⩽ g + χX ⋅ V for all n ∈ ℕ, and for every x ∈ A there is n ∈ ℕ such that g(x) ⩽ γ hn (x). We proceed to establish that the sequence (hn )n∈ℕ of step functions and the measurable neighborhood function n = m + χX ⋅ V fulfill our claim. Indeed, we have ∫A n dθ = ∫A m dθ + |θ|A (V ) ⩽ σW by Theorem 2.3.8(b), and for all n ∈ ℕ we calculate that hn ⩽ g + χX ⋅ V ⩽ f + (m + χX ⋅ V ) = f + n a.e. A

holds, as required in (i). Moreover, θ-almost everywhere on A, for x ∈ A there is n ∈ ℕ such that f (x) ⩽ g(x) ⩽ γhn (x) as stated in (ii). Finally, since g is measurable, Corollary 2.3.7 yields that ∫ f dθ ⩽ ∫ g dθ ⩽ γ lim inf ∫ hn dθ B

B

holds for all B ∈ R such that B ⊂ A.

n→∞

B

2.3 Integrals of set-valued functions

�

111

The following observation extends the result from Proposition 2.3.10(a) from measurable to integrable functions, but restricts the domain to sets in R. 2.3.21 Proposition. Suppose that F is a normed space and let A ∈ R. For every f ∈ ℱ(θ,A) (X, Conv(E)), the integral ∫A f dθ represents a convex subset of F ˚˚ . Proof. Let f ∈ ℱ(θ,A) (X, Conv(E)) for A ∈ R. According to Lemma 2.3.12, the integral ∫A f dθ is the symmetric limit in 𝒬F of a net of integrals of measurable functions, which according to Proposition 2.3.10(a) are σ(F ˚ , F ˚˚ )-lower semicontinuous elements of 𝒬F . Hence ∫A f dθ itself is also σ(F ˚ , F ˚˚ )-lower semicontinuous by Proposition 1.4.6(b). Thus in the light of Proposition 1.4.3(a), which states a correspondence between σ(F ˚ , F ˚˚ )-lower semicontinuous sublinear functionals on F ˚ and convex subsets of E ˚˚ , the integral ∫A f dθ represents an element of Conv(F ˚˚ ). Integrals of vector-valued functions For E-valued functions, the last proposition leads to a straightforward characterization of integrability. 2.3.22 Corollary. A function f ∈ ℱ (X, E) is integrable over a set A ∈ R if and only if for every W ∈ 𝒲 there are a step function h ∈ ℱ𝒮 (X, E) and a measurable neighborhood function n such that h − f ⩽ n and ∫A n dθ ⩽ σW . a.e. A

Proof. Suppose that f ∈ ℱ (X, E) is integrable over A ∈ R. Given W ∈ 𝒲 and any ε > 0 according to Proposition 2.3.20(i), there is a step function h ∈ ℱ𝒮 (X, Conv(E)) and the neighborhood function n such that h ⩽ f + n and ∫A n dθ ⩽ σW . That is, a.e. A

h = ∑ni=1 χAi ⋅ Ci for disjoint sets Ai ∈ R and Ci ∈ Conv(E). We choose elements ci ∈ Ci and set g = ∑ni=1 χAi ⋅ ci ∈ ℱ𝒮 (X, E). Then g ⩽ h ⩽ f + n, that is, g − f ⩽ n, a.e. A

a.e. A

our claim. Conversely, if for every W ∈ 𝒲 there is h ∈ ℱ𝒮 (X, E) and a measurable neighborhood function n such that h − f ⩽ n and ∫A n dθ ⩽ σW , the function g = a.e. A

h + n ∈ ℱ (X, Conv(E)) is measurable and we have both f ⩽ g and g ⩽ f + 2n. Hence a.e. A

the definition of integrability over A is satisfied for the function f .

a.e. A

Corollary 2.3.22 implies that for a function f ∈ ℱ(θ,A) (X, E) there exists a net (hW )W ∈𝒲 in ℱ𝒮 (X, E) such that ∫A f dθ = limW ∈𝒲 ∫A hW dθ in the symmetric topology of 𝒬F . We continue with some observations about integrals of E-valued functions. For the following, recall the definition of a weakly compact measure from Section 2.1. 2.3.23 Proposition. Let B ∈ A. (a) ℱ(θ,B) (X, E) is a real- or complex-linear subspace of ℱ (X, E). For all f ∈ ℱ(θ,B) (X, E), the integral ∫B f dθ ∈ 𝒬F is a linear functional on F ˚ , that is, an element of F ˚‚ . If F is a normed space, then ∫B f dθ ∈ F ˚˚ . (b) The mapping f 󳨃→ ∫B f dθ : ℱ(θ,B) (X, E) → F ˚‚ is real or complex linear.

112 � 2 Integration (c) If θ is L(E, F)-valued, and if for a function f ∈ ℱ(θ,B) (X, E), the net (∫A∩B f dθ)A∈R ̂ the comconverges in the symmetric topology of 𝒬F to ∫B f dθ, then ∫B f dθ ∈ F, pletion of F. (d) If θ is L(E, F)-valued and weakly compact and if f ∈ ℱ (X, E) is measurable, then ∫B f dθ ∈ F. Proof. Let B ∈ R and let f ∈ ℱ(θ,B) (X, E). Then αf ∈ ℱ(θ,B) (X, E) for all α in the scalar field of E by Proposition 2.3.14. Hence ℱ(θ,B) (X, E), which by Proposition 2.3.15 is known to be a subcone, is indeed a subspace of ℱ (X, E). We have ∫B (−f ) dθ = − ∫B f dθ ∈ 𝒬F by 2.3.15(b). The latter implies that ∫B f dθ is a real-linear functional on F ˚ , hence an element of the embedding of F ˚‚ into 𝒬F . If F is a normed space, then F ˚‚ = F ˚˚ (see Section 1.4). Hence our claim in (a). Our claim in Part (b) is obvious in the real case, since we already established that ∫B (−f ) dθ = − ∫B f dθ. In the complex case, we recover the integral as an element ωI(f ) of F ˚‚ from its embedding in 𝒬F as ωI(f ) (μ) = (∫ f dθ)(μ) − i(∫ f dθ)(iμ) B

B

for all μ ∈ F ˚ (see Section 1.4). Thus ωI(f ) is a complex-valued complex-linear functional on F ˚ and for α ∈ ℂ, using Proposition 2.3.14, we calculate that ωI(αf ) (μ) = (∫ f dθ)(αμ) − i(∫ f dθ)(iαμ) = ωI(f ) (αμ) = αωI(f ) (μ) B

B

holds for all μ ∈ F ˚ . Hence ωI(αf ) = αωI(f ) , our claim in (b). For Part (c), suppose that θ is L(E, F)-valued. In a first step, we consider a set A ∈ R such that f ∈ ℱ(θ,A) (X, E). Following Corollary 2.3.22, there exists a net (hW )W ∈𝒲 of step functions in ℱ𝒮 (X, E) such that ∫A f dθ = limW ∈𝒲 ∫A hW dθ in the symmetric topology of 𝒬F . The integrals of the functions hW are of course F-valued. The net (∫A hW dθ)W ∈𝒲 is therefore a Cauchy net in F and its limit ∫A f dθ is contained inits completion F̂ ⊂ F ˚‚ . Now in a second step, let B ∈ A and suppose that the net (∫A∩B f dθ)A∈R converges in the symmetric topology of 𝒬F to ∫B f dθ. All integrals ∫A∩B f dθ are elements of F̂ by our ̂ For first step, and since (∫A∩B f dθ)A∈R is a Cauchy net, its limit is also contained in F. Part (d), suppose that θ is L(E, F)-valued and weakly compact and assume that the function f ∈ ℱ (X, E) is measurable. Then f is supported by a set D ∈ R, and f (X) is a bounded subset of E (Proposition 2.2.4). Thus ∫ f dθ = ∫ f dθ B

B∩D

and Proposition 2.3.10(b) applies. The integral corresponds to a weakly compact subset of F, that is, a singleton subset by Part (a), that is, an element of F.

2.3 Integrals of set-valued functions

� 113

If θ is L(E, F)-valued, Part (c) of Proposition 2.3.23 applies of course to all functions in ℱ(θ,A) (X, E) when A is a set in R. It also applies to all measurable functions in ℱ (X, E) and all sets B ∈ A because they are integrable over all sets in A and are supported by a set in R. The topological scenario, where X is locally compact and R consists of all relatively compact Borel subsets of X allows for some additional observations. 2.3.24 Proposition. Suppose that X is a locally compact Hausdorff space and that R consists of all relatively compact Borel subsets of X. Then every function f ∈ 𝒞 (X, E) is integrable over every set A ∈ R with respect to θ. If θ is regular, then ∫ f dθ = lim ∫ f dθ = lim ∫ f dθ. O⊃A

A

O

K⊂A

K

The limits are meant pointwise on F ˚ and are taken over all open sets O ∈ O containing A and all compact subsets K of A, respectively. If θ is L(E, F)-valued and weakly compact, then the integral is an element of F. Proof. Let f ∈ 𝒞 (X, E) and A ∈ R. Following Proposition 2.2.9, the function χA ⋅ f is measurable, hence f is integrable over A. Now suppose that θ is regular. For every μ ∈ F ˚ , there is V ∈ 𝒱 such that limK⊂A |θ|A\K (V )(μ) = 0. Since A is relatively compact in X and f is continuous, the set f (A) is bounded in E. Hence there is λ > 0 such that f (A) ⊂ λV . Given ε > 0, we find a compact subset K0 of A such that |θ|A\K0 (V )(μ) ⩽ ε/λ. Thus for every compact set K0 ⊂ K ⊂ A, this renders 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 󵄨 󵄨 󵄨󵄨(∫ f dθ)(μ) − (∫ f dθ)(μ)󵄨󵄨󵄨 = 󵄨󵄨󵄨( ∫ f dθ)(μ)󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 A

K

A\K

⩽ |θ|A\K (λV )(μ)

⩽ λ|θ|A\K0 (V )(μ) ⩽ ε.

The second part of the equality in our claim, which involves outer regularity and relatively compact open sets containing A is verified in a similar fashion. Since χA ⋅ f is measurable, if θ is L(E, F)-valued and weakly compact, the last statement follows from Proposition 2.3.23(d). For the following, recall from Section 1.2 that a single neighborhood function n defines a function space neighborhood if and only if for every A ∈ R there is V ∈ 𝒱 such that χA ⋅ V ⩽ n. 2.3.25 Proposition. Suppose that X is a locally compact Hausdorff space and that R consists of all relatively compact Borel subsets of X. Let n be a measurable weakly lower continuous neighborhood function that defines a function space neighborhood. If θ is regular, then

114 � 2 Integration 󵄨 ∫ n dθ = sup{∫ f dθ 󵄨󵄨󵄨 f ∈ 𝒞𝒦 (X, E), supp(f ) ⊂ O, f ⩽ n} B

B

for every B ∈ A, whereby O is any open subset of X containing B. Proof. Let v be the function space neighborhood defined by 𝒩v = {n}. By Proposition 2.3.11, we have (v,{μ})

(∫ n dθ)(μ) = sup ‖θ‖A∩B A∈R

B

for every μ ∈ F

˚

and all B ∈ A. In detail, and using Proposition 2.1.14 this reads as

n 󵄨󵄨 K disjoint compact, K ⊂ B, 󵄨 i (∫ n dθ)(μ) = sup {∑ θKi (ai )(μ) 󵄨󵄨󵄨 i }. 󵄨󵄨 ai ∈ E, χK ⋅ ai ⩽ n i i=1 B

Let O be an open subset of X such that B ⊂ O. The inequality 󵄨 sup{∫ f dθ 󵄨󵄨󵄨 f ∈ 𝒞𝒦 (X, E), supp(f ) ⊂ O, f ⩽ n} ⩽ ∫ n dθ B

B

is obvious. For the converse, let μ ∈ F ˚ and ε > 0. For i = 1, . . . , n let Ki be disjoint compact subsets of B and let ai ∈ E such that χKi ⋅ ai ⩽ n. There is V ∈ 𝒱 such that limO⊃Ki |θ|O\Ki (V )(μ) = 0, and there is λ ⩾ 0 such that ai ∈ λV for all i = 1, . . . , n. Next, we find disjoint open sets Oi ∈ O such that Ki ⊂ Oi ⊂ O, |θ|Oi \Ki (V )(μ) ⩽ ε/(nλ) and χOi ⋅ ai ⩽ (1 + ε)n. The last mentioned property uses Lemma 1.2.4(a). There are functions φi ∈ 𝒞𝒦 (X) such that 0 ⩽ φi ⩽ 1, supp(φi ) ⊂ Oi and φi (x) = 1 for all x ∈ Ki (see Proposition 9.16 in [53]). We set n

f = ∑ φi ⋅ ai i=1

and observe that f ∈ 𝒞𝒦 (X, E), supp(f ) ⊂ O and f ⩽ (1 + ε)n. The latter is easily established pointwise on X. We calculate that n

n

n

∫ f dθ = ∑ ∫ φi ⋅ ai dθ = ∑ ∫ φi ⋅ ai dθ + ∑ ∫ φi ⋅ ai dθ. B

i=1 O i

i=1 K i

i=1 O \K i i

We have ∑ni=1 ∫K φi ⋅ ai dθ = ∑ni=1 θKi (ai ). Moreover, for each i = 1, . . . , n, we observe i that γφi ⋅ ai ⩽ λχX ⋅ V for all |γ| ⩽ 1 and, therefore, 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 󵄨󵄨( ∫ φi ⋅ ai dθ)(μ)󵄨󵄨󵄨 ⩽ λ( ∫ χX ⋅ V dθ)(μ) = λ|θ|Oi \Ki (V )(μ) ⩽ ε/n. 󵄨󵄨 󵄨󵄨 Oi \Ki

Oi \Ki

󵄨 󵄨 Thus 󵄨󵄨󵄨( ∑ni=1 ∫O \K φi ⋅ ai dθ)(μ)󵄨󵄨󵄨 ⩽ ε, and consequently, i i

2.3 Integrals of set-valued functions

� 115

n

(∑ θKi (ai ))(μ) ⩽ (∫ f dθ)(μ) + ε. i=1

B

Furthermore, because 󵄨󵄨 f ∈ 𝒞 (X, E), } { 󵄨 𝒦 (∫ f dθ)(μ) ⩽ sup {(∫ f dθ)(μ) 󵄨󵄨󵄨 󵄨󵄨 supp(f ) ⊂ O, f ⩽ (1 + ε)n} B } { B 󵄨󵄨 f ∈ 𝒞 (X, E), { } 󵄨 𝒦 = (1 + ε) sup {(∫ f dθ)(μ) 󵄨󵄨󵄨 , 󵄨󵄨 supp(f ) ⊂ O, f ⩽ n} { B } we have n 󵄨󵄨 f ∈ 𝒞 (X, E), { } 󵄨 𝒦 (∑ θKi (ai ))(μ) ⩽ (1 + ε) sup {(∫ f dθ)(μ) 󵄨󵄨󵄨 + ε. 󵄨󵄨 supp(f ) ⊂ O, f ⩽ n} i=1 { B }

This argument holds true for all such choices of disjoint compact sets Ki ⊂ B, all such ai ∈ E and also for all ε > 0 and, therefore, yields that (v,{μ})

(∫ n dθ)(μ) = sup ‖θ‖A∩B B

A∈R

󵄨󵄨 f ∈ 𝒞 (X, E), { } 󵄨 𝒦 ⩽ sup {(∫ f dθ)(μ) 󵄨󵄨󵄨 . 󵄨󵄨 supp(f ) ⊂ O, f ⩽ n} { B }

Our claim follows. 2.3.26 Examples. (a) In case that E = F is a Banach space over ℝ or ℂ, real- or complex-valued measures are indeed L(E, E)-valued, and there is a variety of established concepts for the integration of vector-valued functions with respect to scalarvalued measures. The most prominent is the following (see Chapter II.2 in [21]). A function f ∈ ℱ (X, E) is called Bochner integrable over a set A ∈ R if it is strongly θ-measurable (see Example 2.2.22(a)) and if there exists a sequence (hn )n∈ℕ of Evalued step functions such that limn→∞ ∫A ‖hn − f ‖ d|θ| = 0. In this context, |θ| is the usual modulus of a real- or complex-valued measure. (See Example 2.1.8(f).) We shall verify that a Bochner-integrable function f ∈ ℱ (X, E) is integrable in our sense. Indeed, for every ε > 0, there is a step function h ∈ ℱ𝒮 (X, E) such that ∫A ‖h − f ‖ d|θ| ⩽ ε. The strong θ-measurability of the function h − f ∈ ℱ (X, E) implies that there is a set Z ∈ A of measure zero such that the real-valued function χA\Z ⋅ ‖h − f ‖ is A-measurable (see Example 2.2.22), and according to Corollary 2.2.21 the neighborhood function n = (χA\Z ⋅ ‖h − f ‖) ⋅ 𝔹, where 𝔹 denotes the unit ball of E, is measurable in the vein of our definition. We have h − f ⩽ h + n, and since we shall argue below a.e. A

that ∫A (2n) dθ ⩽ 2εσ𝔹 , following Corollary 2.3.22 the function f is integrable over A. Conversely, if a function f ∈ ℱ (X, E) is strongly θ-measurable and integrable over a

116 � 2 Integration set A ∈ R in our sense, then for every ε > 0 according to Corollary 2.3.22 there is a step function h ∈ ℱ𝒮 (X, E) and a measurable neighborhood function n such that h − f ⩽ n and ∫A n dθ ⩽ εσ𝔹 . Hence ‖h − f ‖ ⋅ 𝔹 ⩽ n and ∫A ‖h − f ‖ d|θ| ⩽ ε. Hence f a.e. A

is Bochner integrable over A. In order to complete this argument, let us consider a function φ⋅C ∈ ℱ (X, Conv(E)), where φ is an ℝ+ -valued A-measurable function on X and C is a balanced set in Conv(E). According to 2.2.21 this function is measurable, hence integrable over every set in A, and using an approximation of φ by ℝ+ -valued step functions, its integral over a set B ∈ A as an element of 𝒬F can be evaluated as (∫ φ ⋅ C dθ)(μ) = lim ( ∫ φ d|θ|) pC (μ), A∈R

B

A∩B

for μ ∈ F ˚ , where |θ| denotes the usual modulus of a real- or complex-valued measure and pC (μ) = sup{|μ(c)| | c ∈ C} ∈ ℝ (see Example 2.1.8(f)). If C = 𝔹, that is, the unit ball of E, then φ ⋅ 𝔹 is a neighborhood function and p𝔹 (μ) = ‖μ‖, the dual norm of the functional μ ∈ F ˚ . This observation validates the preceding argument. Moreover, Proposition 2.3.23(c) guarantees that the values of the integrals of E-valued functions are indeed elements of E. (b) In case that E = F = ℝ, integrals of real-valued functions are linear functionals on ℝ˚ = ℝ, that is, real numbers via their evaluations at +1, whereas the modulus of a real-valued measure and integrals of Conv(ℝ)-valued functions are ℝ-valued sublinear functionals on ℝ and determined by their evaluations at the elements +1 and −1 of ℝ˚ . A real-valued measure θ defined on the σ-field A associated with R is the difference of two positive measures θ+ and θ− , which are supported by disjoint sets (see Proposition 21 in Section 11.5 of [53]). For a set C = [a, b] ∈ Conv(ℝ), where a ⩽ b we have |θ|A (C)(+1) = b (θ+ )A − a (θ− )A and |θ|A (C)(−1) = b (θ− )A − a (θ+ )A . For nonnegative ℝ-valued A-measurable functions φ and ψ on X, the set-valued function f ∈ ℱ (X, Conv(ℝ)) such that x 󳨃→ [−φ(x), ψ(x)] is measurable (Example 2.2.22(b)), hence integrable over all sets B ∈ R. We calculate that (∫ f dθ)(+1) = lim ∫ ψ dθ+ + lim ∫ φ dθ− B

and

A∈R

A∩B

A∈R

A∩B

2.3 Integrals of set-valued functions

�

117

(∫ f dθ)(−1) = lim ∫ φ dθ+ + lim ∫ ψ dθ− . A∈R

B

A∈R

A∩B

A∩B

In other words, the integral ∫B f dθ represents the interval [(∫ f dθ)(+1), −(∫ f dθ)(−1)] B

B

in ℱ (X, Conv(ℝ)). Because the function −f , that is, the mapping x 󳨃→ [−ψ(x), φ(x)], is defined in the same way by the functions φ and ψ in reverse order, we verify that (∫B (−f ) dθ)(+1) = (∫B f dθ)(−1) and (∫B (−f ) dθ)(−1) = (∫B f dθ)(+1) holds, as stated in Proposition 2.3.14. We note that the integral of a set-valued function may take the value +∞ on ℝ˚ = ℝ. Further into this example, let φ be an ℝ+ -valued A-measurable function on X such that both ∫ φ dθ+ , ∫ φ dθ− < +∞ A

A

for A ∈ R. A simple argument returns that for every ε > 0 there is ρ ⩾ 0 such that both ∫(φ ∨ ρ − φ) dθ+ ⩽ ε

and

A

∫(φ ∨ ρ − φ) dθ− ⩽ ε. A

The real-valued functions χA ⋅ (φ ∨ ρ) and χA ⋅ (φ ∧ ρ) are measurable in our sense by Proposition 2.2.7 and so by the above is the Conv(ℝ)-valued neighborhood function n = (φ ∨ ρ − φ) ⋅ 𝔹, where 𝔹 = [−1, +1] is the unit ball of ℝ. We set g = χA ⋅ (φ ∧ ρ) + n. By Proposition 2.2.11, the function g ∈ ℱ (X, Conv(ℝ)) is also measurable, and because φ = (φ ∧ ρ) + (φ ∨ ρ − φ), we have φ ⩽ g ⩽ φ + 2n. a.e. A

a.e. A

Moreover, using the above we compute that both (∫A n dθ)(+1) ⩽ 2ε and (∫A n dθ)(−1) ⩽ 2ε. This observation can be reformulated as ∫A n dθ ⩽ 2εσ𝔹 , where σ𝔹 is the sublinear functional α 󳨃→ |α| on ℝ˚ = ℝ. The function φ is therefore integrable over every A ∈ R in the understanding of our definition. Integrals with respect to composition measures In the last paragraph of Section 2.1, we considered the following situation. If (G, 𝒰 ) and (H, ℛ) are additional locally convex topological vector spaces such that H is Hausdorff and if S ∈ L(G, E) and T ∈ L(F, H), then the formula

118 � 2 Integration ((T ˝ θ ˝ S)A (g))(ν) = (θA (S(g)))(T ˚ (ν)), where A ∈ R, g ∈ G and ν ∈ H ˚ , defines an L(G, H ˚‚ )-valued measure. In Section 2.2, we established that the function S ˝ f ∈ ℱ (X, Conv(E)) is measurable provided that f ∈ ℱ (X, Conv(G)) is measurable. We shall now investigate integrability. We note that the measures θ ˝ S and T ˝ θ are L(G, F ˚‚ )-valued and L(E, H ˚‚ )-valued, respectively. In preparation, we take note of the following. 2.3.27 Lemma. Let G be a subspace of E and let B ∈ R. If θA (a) = 0 for all subsets A ∈ R of B and all a ∈ G, then ∫B f dθ = 0 for all f ∈ ℱ(θ,B) (X, Conv(E)) such that f (x) ⊂ G for all x ∈ A. Proof. Let G, θ and B ∈ A be as stated. Because |θ|A (C) = 0 whenever A ⊂ B for A ∈ R and C ⊂ G, our claim is obvious for step functions. Suppose that f ∈ ℱ (X, Conv(E)) is measurable and Conv(G)-valued on B. Given W ∈ 𝒲 , let h = ∑ni=1 χAi ⋅ Ci be a step function in ℱ𝒮 (X, Conv(E)) such that h ⩽ f + vW . Then h ⩽ f + n with some neighborhood-valued step function n ∈ 𝒩vW . Thus h ⩽ ∑ni=1 χAi ⋅ G + n and, therefore, (v )

∫B h dθ ⩽ ∫B n dθ ⩽ σW . Hence ∫B f dθ ⩽ ∫B W f dθ ⩽ σW . This holds for all W ∈ 𝒲 . We infer that ∫B f dθ ⩽ 0 and, therefore, ∫B f dθ = 0 since ∫B f dθ is a sublinear functional on F ˚ . Now suppose that f ∈ ℱ(θ,A) (X, Conv(E)) for A ∈ R such that A ⊂ B and that f is Conv(G)-valued on A. For W ∈ 𝒲 , let fW and nW be measurable functions in ℱ (X, Conv(E)) as in the definition of integrability for f . Then fW ⩽ f + nW implies that fW ⩽ χA ⋅ G + nW and ∫A fW dθ ⩽ ∫A nW dθ ⩽ σW . Hence

a.e. A

a.e. A

∫ f dθ = lim ∫ fW dθ = 0. A

W ∈𝒲

A

Finally, if f ∈ ℱ(θ,B) (X, Conv(E)) for B ∈ A such that f is Conv(G)-valued on B, we recognize that ∫ f dθ = lim ∫ f dθ = 0, B

A∈R

A∩B

our claim. 2.3.28 Theorem. Let S ∈ L(G, E) and T ∈ L(F, H). Let f ∈ ℱ (X, Conv(G)) and B ∈ A. If both f and S ˝ f are integrable over B with respect to θ ˝ S and θ, respectively, then f is integrable over B with respect to T ˝ θ ˝ S and (∫ f d(T ˝ θ ˝ S))(ν) = (∫(S ˝ f ) dθ)(T ˚ (ν)) B

holds for all ν ∈ H ˚ .

B

2.3 Integrals of set-valued functions

� 119

Proof. Let h = ∑ni=1 χAi ⋅ Gi be a Conv(G)-valued step function. Using the evaluation of the modulus of the involved measures from Section 2.1, for every B ∈ A and ν ∈ H ˚ we calculate n

(∫ h d(T ˝ θ ˝ S))(ν) = ∑ |T ˝ θ ˝ S|Ai ∩B (Gi )(ν) i=1

B

n

= ∑ |θ|Ai ∩B (S(Gi ))(T ˚ (ν)) i=1

= (∫(S ˝ h) dθ)(T ˚ (ν)). B

Thus our statement holds for G-valued step functions. Now let f ∈ ℱ (X, Conv(G)) be measurable and let A ∈ R. Then the function S ˝ f ∈ ℱ (X, Conv(E)) was also understood to be measurable in the preceding section. Given ν ∈ H ˚ and ε > 0, there is R ∈ ℛ such that ν ∈ εR˚ . In turn, there is W ∈ 𝒲 such that T ˚ (R˝ ) ⊂ W ˝ and V ∈ 𝒱 such that |θ|A (V ) ⩽ σW . Then there is U ∈ 𝒰 such that S(U) ⊂ V . Hence both ∫ χX ⋅ U d(T ˝ θ ˝ S) ⩽ σR

and

A

∫ χX ⋅ V dθ ⩽ σW A

by the above. We choose the function space neighborhoods v for ℱ (X, E) and u for ℱ (X, G) defined by the single neighborhood functions χX ⋅ V and χX ⋅ U, respectively. There is an increasing sequence (hn )n∈ℕ of step functions in ℱS (X, Conv(G)) for the function f as in Proposition 2.2.13. That is, hn ⩽ f + u for all n ∈ ℕ, and there is 1 ⩽ γ ⩽ 1 + ε such that for every x ∈ A there is n ∈ ℕ such that f (x) ⩽ γhn (x.) The sequence (S ˝ hn )n∈ℕ in ℱ𝒮 (X, Conv(E)) has the same properties for the function S ˝ f ∈ Conv(E) with v in place of u. According to Corollary 2.3.7, this implies both ∫ f d(T ˝ θ ˝ S) ⩽ γ lim inf ∫ hn d(T ˝ θ ˝ S) n→∞

A

A

and ∫(S ˝ f ) dθ ⩽ γ lim inf ∫(S ˝ hn ) dθ, A

n→∞

A

pointwise on F and G , respectively. Recall that the measures T ˝ θ ˝ S and θ are L(G, H ˚‚ )-valued and L(E, F ˚‚ )-valued, respectively. On the other hand, we have both ˚

˚

∫ hn d(T ˝ θ ˝ S) ⩽ ∫ f d(T ˝ θ ˝ S) + σR A

and

A

120 � 2 Integration ∫(S ˝ hn ) dθ ⩽ ∫(S ˝ f ) dθ + σW A

A

for all n ∈ ℕ. Because (∫ hn d(T ˝ θ ˝ S))(ν) = (∫(S ˝ hn ) dθ)(T ˚ (ν)) A

A

by the above, and since ν ∈ εR˝ and T ˚ (ν) ∈ εW ˝ , we conclude that (∫ f d(T ˝ θ ˝ S))(ν) ⩽ γ lim inf(∫(S ˝ hn ) dθ)(T ˚ (ν)) n→∞

A

A

⩽ γ(∫(S ˝ f ) dθ)(T ˚ (ν)) + ε A

and, likewise, (∫(S ˝ f ) dθ)(T ˚ (ν)) ⩽ γ lim inf(∫ hn d(S ˝ θ ˝ S))(ν) n→∞

A

A

⩽ γ(∫ f d(T ˝ θ ˝ S))(ν) + ε. A

These inequalities hold true for all ε > 0 and, therefore, validate that (∫ f d(T ˝ θ ˝ S))(ν) = (∫(S ˝ f ) dθ)(T ˚ (ν)) A

B

holds for every measurable function f ∈ ℱ (X, Conv(G)) and all A ∈ R. In case that H = F and T is the identity operator, this reads ∫A f d(θ ˝ S) = ∫B (S ˝ f ) dθ. Next, we shall argue that every function f ∈ ℱ (X, Conv(G)) that is integrable with respect to θ ˝ S over a set A ∈ R is also integrable over A with respect to T ˝ θ ˝ S. Recall that the measures θ ˝ S and T ˝ θ ˝ S are L(G, F ˚‚ )-valued and L(G, H ˚‚ )valued, respectively. Given R ∈ ℛ, there is W ∈ 𝒲 such that T(W ) ⊂ R, and by the integrability of f with respect to θ ˝ S there are measurable functions fW and nW in ℱ (X, Conv(G)) such that nW is 𝒰c -valued and f ⩽ fW ⩽ f + nW a.e. A

a.e. A

and

∫ nW d(θ ˝ S) = ∫(S ˝ nW ) dθ ⩽ σW . A

A

Any set Z ∈ A of measure zero with respect to θ ˝ S is also of measure zero with respect to T ˝ θ ˝ S, and we have (∫ nW d(T ˝ θ ˝ S))(ν) = (∫(S ˝ nW ) dθ)(T ˚ (ν)) ⩽ 1 A

B

2.3 Integrals of set-valued functions

�

121

for all ν ∈ R˝ by our prior argument for measurable functions. Hence ∫ nW d(T ˝ θ ˝ S) ⩽ σR . A

As a consequence, the same measurable functions fR = fW and nR = nW from the definition of integrability of f with respect to θ ˝ S can also be used to demonstrate its integrability with respect to T ˝ θ ˝ S. Next, we assume that f ∈ ℱ (X, Conv(G)) is integrable over A ∈ R with respect to θ ˝ S, hence with respect to T ˝ θ ˝ S by the above, and that S ˝ f is integrable over A with respect to θ. Given W ∈ 𝒲 , we choose the functions fW and nW as in the definition of integrability with respect to θ ˝ S. That is, f ⩽ fW ⩽ f + nW a.e. A

a.e. A

and

S ˝ f ⩽ S ˝ fW ⩽ S ˝ f + S ˝ nW . a.e. A

a.e. A

We find a subset Z ∈ R of A of measure zero with respect to θ ˝ S such that the stated inequalities hold everywhere on A \ Z. By our assumption, the function S ˝ f is integrable over A with respect to θ and so are the measurable functions S ˝ fW and S ˝ nW . Then the preceding shows ∫ (S ˝ f ) dθ ⩽ ∫ (S ˝ fW ) dθ ⩽ ∫ (S ˝ f ) dθ + ∫ (S ˝ nW ) dθ. A\Z

A\Z

A\Z

A\Z

The values of all functions involved in these integrals are contained in the subspace S(G) of E, and Z being of measure zero with respect to θ ˝ S implies and θC (a) = 0 for all C ∈ R such that C ⊂ Z and all a ∈ G. Thus Lemma 2.3.27 applies and we infer that the integrals over Z with respect to θ of all the functions involved equals zero. We therefore conclude that ∫(S ˝ f ) dθ ⩽ ∫(S ˝ fW ) dθ ⩽ ∫(S ˝ f ) dθ + ∫(S ˝ nW ) dθ. A

A

A

A

Because ∫A (S ˝ nW ) dθ = ∫A nW d(S ˝ θ) ⩽ σW and because for every R ∈ ℛ there is W ∈ 𝒲 such that T ˚ (R) ⊂ W , for every ν ∈ H ˚ passing to the limit over W ∈ 𝒲 in this inequality leads to (∫(S ˝ f ) dθ)(T ˚ (ν)) = lim (∫(S ˝ fW ) dθ)(T ˚ (ν)) A

W ∈𝒲

A

= lim (∫ fW d(T ˝ θ ˝ S))(ν) W ∈𝒲

A

= (∫ f d(T ˝ θ ˝ S))(ν). A

122 � 2 Integration Finally, if f ∈ ℱ (X, Conv(G)) is integrable over a set B ∈ A with respect to θ ˝ S and S ˝ f is integrable over B with respect to θ. Then the limit ∫(S ˝ f ) dθ = lim ∫ (S ˝ f ) dθ B

A∈R

A∩B

exists pointwise on F ˚ . Hence for every ν ∈ H ˚ the limit lim ( ∫ (S ˝ f ) dθ)(T ˚ (ν)) = lim ( ∫ f d(T ˝ θ ˝ S))(ν)

A∈R

A∈R

A∩B

A∩B

also exists and equals (∫ f d(T ˝ θ ˝ S))(ν) = (∫(S ˝ f ) dθ)(T ˚ (ν)). B

B

This observation completes our argument. We shall consider a few special cases of interest. First, suppose that G = E with the identity operator for S. Theorem 2.3.28 states that if a function f ∈ ℱ (X, Conv(E)) is integrable over B ∈ A with respect to θ, then it is integrable over B with respect to T ˝ S and we have (∫ f d(T ˝ θ))(ν) = (∫ f dθ)(T ˚ (ν)) B

B

for all ν ∈ H ˚ . Now suppose in addition that G equals either ℝ or ℂ and that T is a linear functional μ ∈ F ˚ . The adjoint operator T ˚ then maps the scalars α ∈ G˚ into αμ ∈ F ˚ , and Theorem 2.3.28 returns that (∫ f d(μ ˝ θ))(α) = (∫ f dθ)(αμ) B

B

holds for all α in ℝ or ℂ in this case. We shall formulate this observation for α = 1 as a corollary. 2.3.29 Corollary. If f ∈ ℱ(θ,B) (X, Conv(E)) for B ∈ A, then f ∈ ℱ(μ˝θ,B) (X, Conv(E)) for every μ ∈ F ˚ and (∫ f dθ)(μ) = (∫ f d(μ ˝ θ))(+1). B

B

Now suppose that H = F with the identity operator for T. Theorem 2.3.28 states that if both functions f ∈ ℱ (X, Conv(G)) and S ˝ f ∈ ℱ (X, Conv(E)) are integrable over B with respect to θ ˝ S and θ, respectively, then

2.4 The convergence theorems

� 123

∫ f d(θ ˝ S) = ∫(S ˝ f ) dθ B

B

In particular, if G is a subspace of E and S is the imbedding operator, then S ˝ f = f for a Conv(G)-valued function and θ ˝ S arises from the restriction of the values of θ to the subspace G of E. Hence if f is integrable both with respect to θ and to θ ˝ S, then the values of these integrals coincide. Let us consider the case that E = F and that θ is real- or complex-valued. (We shall formulate the complex-valued case.) Given a linear functional μ ∈ E ˚ , we choose E in place of G and ℂ for E, F and H in Theorem 2.3.28. The functional μ stands for the operator S and T is the identity on ℂ. The measure θ is L(ℂ, ℂ)-valued. For f ∈ ℱ (X, Conv(E)), then μ ˝ f is the Conv(ℂ)-valued function x 󳨃→ μ(f (x)). The measure θ ˝ μ is L(E, ℂ)-valued, that is, E ˚ -valued. Theorem 2.3.28 states that ∫ f d(θ ˝ μ) = ∫(μ ˝ f ) dθ B

B

holds for all functions f ∈ ℱ (X, Conv(E)) such that f is integrable with respect to θ ˝ μ and μ ˝ f is integrable with respect to θ. The integrals in the preceding equation are 𝒬ℂ -valued. Alternatively, in the same setting, where E = F, we may consider the complexvalued measure θ as being L(E, E)-valued (see Example 2.3.26(a)). Corollary 2.3.29 then states that (∫B f dθ)(μ) = (∫B f d(μ ˝ θ))(+1) holds for every f ∈ ℱ(θ,B) (X, Conv(E)). We realize that the measures θ ˝ μ and μ ˝ θ are both L(E, ℂ)-valued and coincide. Indeed, we have (θ ˝ μ)A (a) = (μ ˝ θ)A (a) = μ(a) θA for all A ∈ R and a ∈ E. Moreover, integrability over B for an Conv(E)-valued function with respect to θ implies integrability with respect to θ ˝ μ = μ ˝ θ by the above. Hence if f ∈ ℱ (X, Conv(E)) and if both f and μ ˝ f are integrable over B with respect to θ (see 2.3.26(a)), then (∫ f dθ)(μ) = (∫ f d(μ ˝ θ))(+1) = (∫(μ ˝ f ) dθ)(+1). B

B

B

2.4 The convergence theorems We continue to use the previous assumptions and go forward to establish a range of general convergence results for sequences of functions and their respective integrals. These results are modeled after the dominated convergence theorem from classical measure theory. However, the presence of unbounded elements in the range of Conv(E)-valued

124 � 2 Integration functions and the general absence of negatives will considerably complicate some technical aspects of the approach, which is modeled after the corresponding method for locally convex cones in Chapter II.5 of [50]. As in the preceding section, we shall assume that θ is a bounded L(E, F ˚‚ )-valued measure. Subsequently, we shall set up suitable notions for convergence of sequences and series of Conv(E)-valued functions. Convergence for sequences of integrals will generally refer to the topology of pointwise convergence in 𝒬F . Theorem 2.3.18(b) states that a function in ℱ (X, Conv(E)) that is integrable over each of finitely many disjoint sets in A is also integrable over their union. If the sets are indeed in R, then this holds true as well if they are not necessarily disjoint (see the remark following 2.3.18). Simple examples show that these statements do however generally not extent to countable unions. For the following, we recall from Section 1.4 that a function f ∈ ℱ (X, Conv(E)) is bounded above (or below) if for every function space neighborhood v there is λ ⩾ 0 such that f ⩽ λv (or 0 ⩽ f + λv). It is bounded, if it is both bounded above and below. A subset of ℱ (X, Conv(E)) is bounded above (or below) if these conditions hold with the same λ ⩾ 0 for all functions in this subset. 2.4.1 Lemma. Suppose that |θ| is strongly additive at all neighborhoods V ∈ 𝒱 . Let An ∈ R such that A = ⋃n∈ℕ An ∈ R. If f ∈ ℱ (X, Conv(E)) is bounded and integrable over all sets An , then it is integrable over A. Proof. Set Bn = ⋃ni=1 Ai ∈ R. Then Bn ⊂ Bn+1 and ⋃n∈ℕ Bn = ⋃n∈ℕ An = A. If the function f ∈ ℱ (X, Conv(E)) is integrable over all sets An , then it is integrable over all sets Bn . If in addition |θ| is strongly additive and f is bounded, we shall verify its integrability over A. For this, let W ∈ 𝒲 . There is V ∈ 𝒱 such that |θ|A (V ) ⩽ (1/2)σW , and there is λ ⩾ 0 such that both f ⩽ λχX ⋅V and 0 ⩽ f +λχX ⋅V . As |θ| is supposed to be strongly additive at V , there is n ∈ ℕ such that |θ|A\Bn ⩽ (1/(6λ))σW . By the definition of integrability of f over Bn , we find measurable functions g and n satisfying f ⩽ g ⩽ f +n a.e. Bn

a.e. Bn

and

∫ n dθ ⩽ (1/2)σW . Bn

There is a subset B ∈ R of Bn such that Bn \ B is of measure zero, and the above inequalities hold everywhere on B. We have |θ|A\B = |θ|A\Bn + |θ|Bn \B = |θ|A\Bn ⩽ (1/(6λ))σW . Since χA ⋅ f = χB ⋅ f + χA\B ⋅ f ⩽ χB ⋅ f + λχA\B ⋅ V and χB ⋅ f ⩽ χB ⋅ g ⩽ γ χB ⋅ f + n, we conclude that χA ⋅ f ⩽ (χB ⋅ g + λχA\B ⋅ V )

⩽ γχB ⋅ f + λχA\B ⋅ V + n

⩽ γχB ⋅ f + λχA\B ⋅ V + n + γχA\B ⋅ (f + λχX ⋅ V ) ⩽ γχA ⋅ f + (n + λ(1 + γ)χA\B ⋅ V ).

2.4 The convergence theorems

� 125

The function g̃ = χB ⋅ g + λχA\B ⋅ V is measurable, likewise the neighborhood function ñ = n + λ(1 + γ)χA\B ⋅ V , and we calculate that ∫ ñ dθ ⩽ ∫ n dθ ⩽ σW + (1 + γ)λ|θ|A\B (V ). A

A

Here, we use that both ∫A n dθ ⩽ (1/2)σW and (1 + γ)λ|θ|A\B (V ) ⩽ 3λ|θ|A\B (V ) ⩽ (1/2)σW . Our definition of integrability for f over A is therefore satisfied with the measurable function g̃ ∈ ℱ (X, Conv(E)) and the measurable neighborhood function ñ. 2.4.2 Proposition. Let f ∈ ℱ(θ,A) (X, Conv(E)) for A ∈ R. (a) If An ∈ R are disjoint sets and A = ⋃n∈ℕ An , then ∞

∫ f dθ = ∑ ∫ f dθ. n=1 A n

A

(b) If An ∈ R such that An ⊂ An+1 for all n ∈ ℕ, and A = ⋃n∈ℕ An , then ∫ f dθ = lim ∫ f dθ. n→∞

A

An

(c) If An ∈ R such that A ⊃ An ⊃ An+1 for all n ∈ ℕ, and ⋂n∈ℕ An = H, then lim (∫ f dθ)(μ) = 0

n→∞

An

for all μ ∈ F ˚ such that (∫A f dθ)(μ) < +∞. Proof. We recall that a function that is integrable over a set in A is integrable over all its intersections with sets in R. Therefore, integrability over A = ⋃n∈ℕ An ∈ R implies integrability over each of the sets An . Parts (a) and (b) of the proposition are clearly equivalent, and we shall verify the second. Let An ∈ R such that An ⊂ An+1 for all n ∈ ℕ, and A = ⋃n∈ℕ An ∈ R. We shall first assume that the function f ∈ ℱ(θ,A) (X, Conv(E)) is measurable. For W ∈ 𝒲 , there is a neighborhood V ∈ 𝒱 such that θA (V ) ⩽ σW and by Lemma 2.2.2 there is λ ⩾ 0 such that 0 ⩽ f + λχX ⋅ V . This implies χAn ⋅ f ⩽ χAn ⋅ f + χA\An ⋅ (f + λχX ⋅ V ) = χA ⋅ f + λχA\An ⋅ V .

126 � 2 Integration Thus ∫ f dθ ⩽ ∫(f + λχA\An ⋅ V ) dθ = ∫ f dθ + λ|θ|A\An (V ).

An

A

A

Following Lemma 2.1.2(b), this yields lim sup(∫ f dθ)(μ) ⩽ (∫ f dθ)(μ) n→∞

An

A

for all μ ∈ W ˝ . For any U ∈ 𝒲 let h ∈ ℱ𝒮 (X, Conv(E)) be a step function such that h ⩽ f + vU , that is, h ⩽ f + n for some neighborhood-valued step function n such that ∫X n dθ ⩽ σU . Then ∫A h dθ ⩽ ∫A f dθ + σU by Theorem 2.3.8(b) and (c). Hence using n n the countable additivity of |θ| for the step function h, we conclude that ∫ h dθ = lim ∫ h dθ ⩽ lim inf ∫ f dθ + σU , n→∞

A

n→∞

An

An

holds pointwise on F ˚ . Taking the supremum over all such step functions h ⩽ f + vU renders (U)

∫ f dθ ⩽ lim inf ∫ f dθ + σU . n→∞

A

An

Taking the infimum over all U ∈ 𝒲 in the preceding inequality yields ∫ f dθ ⩽ lim inf ∫ f dθ, A

n→∞

An

and combining all of the above, we infer that (∫ f dθ)(μ) ⩽ lim inf(∫ f dθ)(μ) A

n→∞

An

⩽ lim sup(∫ f dθ)(μ) n→∞

An

⩽ (∫ f dθ)(μ) A

holds for all μ ∈ W ˝ . Since every μ ∈ F ˚ is contained in the polar of some neighborhood W ∈ 𝒲 , this delivers our claim for a measurable function f ∈ ℱ (X, Conv(E)). Now for the general case, let f ∈ ℱ(θ,A) (X, Conv(E)). Given W ∈ 𝒲 , we choose the

2.4 The convergence theorems

� 127

measurable functions fW and nW as in our definition of integrability over a set in A ∈ R. That is, f ⩽ fW ⩽ f + nW a.e. A

and

a.e. A

∫ nW dθ ⩽ σW . A

Using the previously established claim for measurable functions for each of the functions fW , we obtain ∫ f dθ ⩽ ∫ fW dθ = lim ∫ fW dθ ⩽ lim inf ∫ f dθ + σW , A

n→∞

A

n→∞

An

An

and lim sup ∫ f dθ ⩽ lim sup ∫ fW dθ = ∫ fW dθ ⩽ ∫ f dθ + σW , n→∞

An

n→∞

An

A

A

pointwise on F ˚ . Given μ ∈ F ˚ , there is W ∈ 𝒲 such that μ ∈ εW ˝ . Then a combination of the above inequalities returns lim sup(∫ f dθ)(μ) ⩽ (∫ f dθ)(μ) + ε ⩽ lim inf(∫ f dθ)(μ) + 2ε. n→∞

An

n→∞

A

An

Letting ε > 0 tend to 0 renders our claim in Part (b). For Part (c), let An ∈ R such that A ⊃ An ⊃ An+1 for all n ∈ ℕ and ⋂n∈ℕ An = H. Using Part (a) for the sets A\An yields ∫ f dθ = lim ∫ f dθ. n→∞

A

A\An

We have ∫A f dθ = ∫A\A f dθ + ∫A f dθ by Theorem 2.3.15(b), and consequently, n

n

lim (∫ f dθ)(μ) = (∫ f dθ)(μ) − lim ( ∫ f dθ)(μ) = 0,

n→∞

An

A

n→∞

A\An

provided that (∫A f dθ)(μ) < +∞. The convergence statements in Proposition 2.4.2 do generally not apply to the stronger symmetric topology of 𝒬F . This consolidation can however be established under some additional assumptions. In this vein, for a function f ∈ ℱ (X, Conv(E)) we shall say that its integral over a set A ∈ R is strongly additive if it is integrable over A, if ∫A f dθ is a bounded element of the locally convex cone 𝒬F and if the convergence claims in Proposition 2.4.2 hold for the symmetric topology of 𝒬F . We note that boundedness of the integral implies that (∫A f dθ)(μ) < +∞ for all μ ∈ F ˚ , and the statement

128 � 2 Integration in 2.4.2(c) implies those in 2.4.2(a) and (b). The integral of a function f over a set B ∈ R is strongly additive if for all A ∈ R its integrals over the sets A∩B are strongly additive and if the limit ∫B f dθ = limA∈R ∫A∩B f dθ refers to the symmetric topology of 𝒬F . If θ is L(E, F)-valued, then according to Proposition 2.1.1 it is strongly additive at all elements of E, hence the integral of any E-valued step function over a set in R is strongly additive. The same statement holds for integrable E-valued functions in general. 2.4.3 Corollary. Let f ∈ ℱ(θ,A) (X, E) for A ∈ R. If θ is L(E, F)-valued, then the integral of f over A is strongly additive. Proof. Suppose that θ is L(E, F)-valued and let f ∈ ℱ(θ,A) (X, E) for A ∈ R. Proposition 2.3.23(a) states that ∫A f dθ ∈ F ˚‚ , hence is a bounded element of 𝒬F . We shall verify the statement in 2.4.2(c) for the symmetric topology of 𝒬F , which in this case implies the remaining parts. Let An ∈ R such that A ⊃ An ⊃ An+1 for all n ∈ ℕ and ⋂n∈ℕ An = H. Given W ∈ 𝒲 , according to Corollary 2.3.22, there is a step function h ∈ ℱ𝒮 (X, E) and a measurable neighborhood function n such that h − f ⩽ n a.e. A

and ∫A n dθ ⩽ σW . Because ∫A h dθ is strongly additive, there is n0 such that both ∫ h dθ ⩽ σW

and

0 ⩽ ∫ h dθ + σW

An

An

for all n ⩾ n0 . We infer that ∫ f dθ ⩽ ∫ h dθ + σW ⩽ 2σW An

An

and 0 ⩽ ∫ h dθ + σW ⩽ ∫ f dθ + 2σW An

An

holds for all n ⩾ n0 . Our claim follows. 2.4.4 Corollary. Let f ∈ ℱ(θ,A) (X, Conv(E)) for A ∈ R. If f is bounded and if |θ| is strongly additive at all neighborhoods V ∈ 𝒱 , then the integral of f over A is strongly additive. Proof. Let A ∈ R and f ∈ ℱ(θ,A) (X, Conv(E)). Given W ∈ 𝒲 , there is V ∈ 𝒱 such that |θ|A (V ) ⩽ σW , and there is λ > 0 such that both 0 ⩽ f + λχX ⋅ V and f ⩽ λχX ⋅ V . Hence 0 ⩽ ∫A f dθ + λσW and ∫A f dθ ⩽ λσW . That is, ∫A f dθ ∈ λW s (0), and ∫A f dθ is seen to be a bounded element of the locally convex cone 𝒬F (see Section 1.4). Next, we shall verify the statement in 2.4.2(c) for the symmetric topology of 𝒬F , which implies the remaining parts. Let An ∈ R such that A ⊃ An ⊃ An+1 for all n ∈ ℕ and ⋂n∈ℕ An = H.

2.4 The convergence theorems

� 129

Given W ∈ 𝒲 , we choose V ∈ 𝒱 and λ ⩾ 0 from above. Because |θ| is supposed to be strongly additive at V , we find n0 such that |θ|An ⩽ (1/λ)σW for all n ⩾ n0 . Hence 0 ⩽ ∫A f dθ + σW and ∫A f dθ ⩽ σW . Our claim follows. n

n

Recall from Example 2.1.8(e) that every F ˚‚ -valued measure θ may be considered to be an operator-valued measure if we choose E = ℝ or E = ℂ. 2.4.5 Corollary. If f ∈ ℱ (X, Conv(E)) is integrable over all sets in R, then the mapping A 󳨃→ ∫ f dθ : R → F ˚‚ A

defines an F ˚‚ -valued measure. If all integrals are F-valued, then this measure is Fvalued and strongly additive. If f is bounded, then this measure is bounded. Proof. Let f ∈ ℱ (X, Conv(E)) be as stated and set ΘA = ∫A f dθ for all A ∈ R. Then Θ is F ˚‚ -valued, that is, L(ℝ, F ˚‚ )- or L(ℂ, F ˚‚ )-valued, and countably additive as stated in Proposition 2.4.2(a). If Θ is in fact F-valued, then according to Proposition 2.1.1 it is strongly additive. If f is bounded, then given A ∈ R and W ∈ 𝒲 , there is V ∈ 𝒱 such that |θ|A (V ) ⩽ σW , and there is λ > 0 such that both 0 ⩽ f + λχX ⋅ V and f ⩽ λχX ⋅ V , that is, 0 ⩽ ∫B f dθ + λ|θ|B (V ) and ∫B f dθ ⩽ λ|θ|B (V ) for every B ∈ R. For disjoint subsets Ai ∈ R of A and μ ∈ F ˚ this yields n 󵄨󵄨 n 󵄨󵄨󵄨 󵄨 ∑ 󵄨󵄨󵄨∫ f dθ (μ)󵄨󵄨󵄨 ⩽ λ ∑ |θ|Ai (V )(μ) ⩽ λ|θ|A (V )(μ). 󵄨 󵄨󵄨 i=1 󵄨 i=1 Ai

Hence for the unit ball 𝔹 of ℂ or ℝ we compute using 2.1.8(e) n 󵄨 󵄨 󵄨󵄨 |Θ|A (𝔹)(μ) = sup{∑ 󵄨󵄨󵄨ΘAi (μ)󵄨󵄨󵄨 󵄨󵄨󵄨 Ai ∈ R disjoint Ai ⊂ A} ⩽ λσW (μ). 󵄨 i=1

Thus |Θ|A ((1/λ)𝔹) ⩽ σW . The F ˚‚ -valued measure Θ is therefore bounded as claimed.

The relative topologies Chapter II.5 in [50] presents a variety of convergence theorems simultaneously involving sequences of measures and integrable functions. They involve the (generally coarser) relative rather that the given topologies of a locally convex cone which we will now establish for Conv(E). For a neighborhood V ∈ 𝒱 and ε > 0, we define the upper, lower and symmetric relative neighborhoods for an element C ∈ Conv(E) by Vε (C) = {D ∈ Conv(E) | B ⩽ γC + εV for some 1 ⩽ γ ⩽ 1 + ε}

130 � 2 Integration (C)Vε = {D ∈ Conv(E) | C ⩽ γD + εV for some 1 ⩽ γ ⩽ 1 + ε} and Vεs (C) = Vε (C) ∩ (C)Vε , respectively. These neighborhoods are convex subsets of Conv(E) and one easily checks the conditions for neighborhood bases defining a topology (see Theorem 4.5 in [63] and Lemma I.4.1 in [50]). They create the upper, lower and symmetric relative topologies on Conv(E). These are generally coarser than the given upper, lower and symmetric topologies, which are generated by the neighborhoods V (C) = {D ∈ Conv(E) | D ⊂ C + V }, (C)V = {D ∈ Conv(E) | C ⊂ D + V } and V s (C) = V (C) ∩ (C)V , respectively (see Section 1.4). The relative topologies are however locally equivalent to the given topologies on bounded sets C ∈ Conv(E). Indeed, for V ∈ 𝒱 and ε > 0 we have (εV )(C) ⊂ Vε (C) for all C ∈ Conv(E). If C is bounded, then C ⊂ λV for some λ ⩾ 0, and we have V1/(1+λ) (C) ⊂ V (C). On the subspace E of Conv(E), all of the above topologies coincide with the given one. We had implicitly employed the relative topologies in the earlier sections about measurability and integrability. The main justification for the use of the relative rather than the given topologies on a locally convex cone is that they render continuity not only for the addition but also the multiplication by positive scalars (see Proposition I.4.2 in [48]). This property is indispensable for the investigation of continuous cone-valued functions, as we shall illustrate in an example below.

Sequences of set-valued functions We shall use different patterns of pointwise convergence for sequences of functions in ℱ (X, Conv(E)). For sequences (Cn )n∈ℕ in Conv(E) and C ∈ Conv(E), we shall write Cn ↘ C,

Cn ↗ C,

or

Cn → C

if (Cn )n∈ℕ converges to C with respect to the upper, lower or symmetric relative topology of Conv(E), that is, if for every V ∈ 𝒱 and ε > 0 there is n0 ∈ ℕ such that Cn ∈ Vε (C), Cn ∈ (C)Vε or Cn ∈ Vεs (C) for all n ⩾ n0 , respectively. Similarly, for functions (fn )n∈ℕ and f in ℱ (X, Conv(E)), and a set B ∈ A we shall write fn B↘ f ,

fn ↗B f ,

or fn → B f,

if (fn )n∈ℕ converges to f pointwise on B with respect to the upper, lower or symmetric relative topology of Conv(E), respectively. We shall also use these notions for double sequences (fn )n∈ℕ and (gn )n∈ℕ in ℱ (X, Conv(E)). That is to say, we shall write fn B↘ gn

2.4 The convergence theorems

� 131

if for every x ∈ B, V ∈ 𝒱 and ε > 0 there is n0 ∈ ℕ such that fm (x) ∈ Vε (gn (x)) for all m, n ⩾ n0 . Correspondingly, we shall denote fn

↘ f,

a.e. B

fn ↗a.e. B f ,

fn 󳨀→ f a.e. B

or

fn

↘ gn ,

a.e. B

if this convergence holds θ-almost everywhere on B, that is, pointwise everywhere on a subset B \ Z with some set Z ∈ A of measure zero. All the above notions of convergence are compatible with the algebraic operations in Conv(E), that is, the limit rules apply with sums and multiples by positive scalars. 2.4.6 Lemma. Let α, αn ⩾ 0 and let C ∈ Conv(E). (a) If lim supn→∞ αn ⩽ α, if C ⩾ 0, and if either α > 0 or C is bounded, then αn C ↘ αC. (b) If lim infn→∞ αn ⩾ α and if C ⩾ 0, then αn C ↗ αC. (c) If limn→∞ αn = α and if either α > 0 or C is bounded, then αn C → αC. Proof. Let α, αn ⩾ 0 and C ∈ Conv(E) Let V ∈ 𝒱 and ε > 0. There is λ > 0 such that 0 ⩽ C + λV . For those statements in (a) and (c) which require C to be bounded, we shall also assume that C ⩽ λV . For (a), suppose that lim supn→∞ αn ⩽ α and that C ⩾ 0. In case that α > 0, we set δ = αε. There is n0 ∈ ℕ such that αn ⩽ α + δ for all n ⩾ n0 . Since C ⩾ 0, this implies αn C ⩽ (α + δ)C ⩽ (1 + ε)αC for all n ⩾ n0 . In case that α = 0 and C is bounded, we set δ = ε/λ. There is n0 ∈ ℕ such that 0 ⩽ αn ⩽ δ for all n ⩾ n0 . Then αn C ⩽ δC ⩽ δλV ⩽ 0 C + εV . Thus αn C ∈ Vε (αC) holds for all n ⩾ n0 in both cases. For (b), suppose that lim supn→∞ αn ⩾ α and that C ⩾ 0. In case that α > 0, we set δ = αε/(1 + ε). There is n0 ∈ ℕ such that α ⩽ αn + δ for all n ⩾ n0 . Then δ(1 + ε) ⩽ εα ⩽ ε(αn + δ), hence δ ⩽ εαn and αC ⩽ (αn + δ)C ⩽ (1 + ε)αn C for all n ⩾ n0 . In case that α = 0, we have 0 C ⩽ αn C for all n ∈ ℕ. Thus αC ∈ Vε (αn C), that is, αn C ∈ (αC)Vε , holds for all n ⩾ n0 in both cases. For (c), suppose that limn→∞ αn = α. In case that α > 0, we set δ = min{ε/λ, αε/(1 + ε)}. There is n0 ∈ ℕ such that |αn − α| ⩽ δ for all n ⩾ n0 . We claim that this implies that αn C ∈ Vεs (αC). Indeed, if αn ⩽ α, then αn C ⩽ αn C + (α − αn )(C + λV ) ⩽ αC + δλV ⩽ αC + εV

132 � 2 Integration and αC = γαn C with some 1 ⩽ γ ⩽ 1 + ε. The former follows since α − δ ⩽ αn ⩽ α, hence 0 ⩽ α − αn ⩽ δ ⩽ ε/λ by our choice for δ, and the latter since α − δ ⩾ α(1 − ε/(1 + ε)) = α/(1 + ε), hence 1 ⩽ γ = α/αn ⩽ α/(α − δ) ⩽ 1 + ε. If α < αn , then αn C = γαC and αC ⩽ αC + (αn − α)(C + λV ) ⩽ αn C + εV , with some 1 ⩽ γ ⩽ 1 + ε. The former follows since α < αn ⩽ α + δ ⩽ α(1 + ε/(1 + ε)) ⩽ α(1 + ε), hence 1 ⩽ γ = αn /α ⩽ 1 + ε, and the latter since 0 < αn − α ⩽ δ ⩽ ε/λ. We therefore have αn C ∈ Vεs (αC) for all n ⩾ n0 in both instances. Finally, in case that α = 0 and C is bounded, we set δ = ε/λ. There is n0 ∈ ℕ such that 0 ⩽ αn ⩽ δ for all n ⩾ n0 . Then αn C ⩽ αn λV ⩽ δλV ⩽ 0 C + εV and 0 C ⩽ αn (C + λV ) ⩽ αn C + δλV ⩽ αn C + εV . Thus αn C ∈ Vεs (0 C) for all n ⩾ n0 . For bounded sets C ∈ Conv(E), the statements of Lemma 2.4.6 hold indeed with the given rather than the relative topologies of Conv(E). But they fail with the given topologies for unbounded sets as simple examples can demonstrate. Indeed, let E = ℝ2 with the Euclidean unit ball 𝔹. Let C = {(x, y) | y ⩾ x 2 + 1} ∈ Conv(ℝ2 ). Then αC = {(x, y) | y ⩾ (1/α)x 2 + α} for α > 0, and we realize that αC ⊄ C + 𝔹, that is, αC ∉ 𝔹(C) for any choice of α > 1. However, given 0 < ε ⩽ 1, a brief calculation shows that αC ∈ 𝔹sε (C) holds whenever |α − 1| < ε. The following statement is an immediate consequence of the preceding lemma. Recall from Section 1.4 that a subscript to the symbol of a relation between functions signifies that this relation holds pointwise on the specified subset.

2.4 The convergence theorems

� 133

2.4.7 Proposition. Let B ∈ A. Let φ, φn ∈ ℱ (X) be nonnegative functions and let C ∈ Conv(E). (a) If lim supn→∞ φn ⩽B φ, if C ⩾ 0 and if either φ ⩾B 0 or C is bounded, then φn ⋅ C B↘ φ ⋅ C. (b) If lim infn→∞ φn ⩾B φ and if C ⩾ 0, then φn ⋅ C ↗B φ ⋅ C. (c) If limn→∞ φn =B φ and if either φ >B 0 or C is bounded, then φn ⋅ C → B φ ⋅ C. The next result is our main convergence theorem for sequences of integrable functions in ℱ (X, Conv(E)). Recall from Section 1.4 that we consider the upper, lower and symmetric locally convex cone topologies on 𝒬F . That is, φ ∈ W (ψ) for φ, ψ ∈ 𝒬F and W ∈ 𝒲 means that φ ⩽ ψ + σW . 2.4.8 Theorem. Let fn , gn , f ˚ , g˚ , g˚˚ ∈ ℱ(θ,B) (X, Conv(E)) for B ∈ A and suppose that fn ⩽ f ˚ and g˚˚ ⩽ gn + g˚ for all n ∈ ℕ. If fn a.e. B↘ gn , then for every μ ∈ F ˚ such a.e. B

a.e. B

that both (∫B f ˚ dθ)(μ) < +∞ and (∫B g˚ dθ)(μ) < +∞ and every ε > 0 there is n0 such that (∫ fm dθ)(μ) ⩽ (∫ gn dθ)(μ) + ε B

B

holds for all m, n ⩾ n0 . If the integrals of f ˚ , g˚ and g˚˚ over B are strongly additive, then the above holds uniformly on the polars of all neighborhoods in 𝒲 . That is, for every W ∈ 𝒲 there is n0 such that ∫B fm dθ ⩽ ∫B gn dθ + σW for all m, n ⩾ n0 . Proof. Without loss of generality, we may assume that B = X. Indeed, the respective integrals over B ∈ A equal the integrals over X for the products of the concerned functions with χB , and these products satisfy the conditions of the theorem with X in place of B. Also, we may assume that the required convergence and boundedness properties hold everywhere on X instead of θ-almost everywhere. Indeed, the convergence condition for fm and gn holds for all x ∈ X \ Z1 with some set Z1 ∈ A of measure zero. Using the fact that countable unions of sets of measure zero are again of measure zero (see Proposition 2.1.10(c)), we can find a set Z2 ∈ A of measure zero such that both χX\Z2 ⋅ fn ⩽ χX\Z2 ⋅ f ˚ and 0 ⩽ χX\Z2 ⋅ (gn + g˚ ) holds for all n ∈ ℕ everywhere on X. The set Z = Z1 ∪ Z2 ∈ A is also of measure zero, and the functions

′

fn′ = χX\Z ⋅ fn ,

f ˚ = χX\Z ⋅ f ˚ ,

gn′ = χX\Z ⋅ gn ,

g˚ ′ = χX\Z ⋅ g˚

and

g˚˚ ′ = χX\Z ⋅ g˚˚

fulfill all the assumptions of the theorem everywhere on X, and their respective integrals coincide with those of the given functions.

134 � 2 Integration Let W ∈ 𝒲 . In order to argue the general and the strongly additive instance of our statement simultaneously, we choose Ω = {μ}, where μ is an element of W ˝ such that both (∫B f ˚ dθ)(μ) < +∞ and (∫B g˚ dθ)(μ) < +∞, for the general case and Ω = W ˝ for the strongly additive case. For any μ ∈ Ω, there is nothing to prove in case that (∫X gn dθ)(μ) = +∞ for all n ∈ ℕ. Otherwise the relation g˚˚ ⩽ gn +g˚ guarantees that a.e. B

(∫B g˚˚ dθ)(μ) < +∞ and that the sequence ((∫X gn dθ)(μ))n∈ℕ is bounded below in ℝ. The same observation applies to the sequences ((∫A gn dθ)(μ))n∈ℕ for all A ∈ R since Theorem 2.3.18(b) guarantees that (∫A g˚ dθ)(μ) < +∞ also holds. In a first step of our argument, we consider a set A ∈ R. Let ε > 0. There is V ∈ 𝒱 such that |θ|A (V ) ⩽ σW . According to Proposition 2.3.13, there is a measurable neighborhood function n and λ ⩾ 0 such that 0 ⩽ f ˚ + n and ∫A n dθ ⩽ λσW . We choose δ > 0 such that both δ(1 + λ) ⩽ ε and δ (∫A f ˚ dθ)(μ) ⩽ ε for all μ ∈ Ω. For the latter, we recall that in the strongly additive case ∫A f ˚ dθ is bounded in 𝒬F . If fm (x) ∈ Vδ (gn (x)) for x ∈ A, then fm (x) ⩽ γgn (x) + δV ,

hence (1/γ)fm (x) ⩽ gn (x) + δV ,

with some 1 ⩽ γ ⩽ 1+δ. Using that fm ⩽ f ˚ and that both n(x) ⩾ 0 and n(x)+f ˚ (x) ⩾ 0, we continue to argue that fm (x)(x) = (1/γ)fm (x) + (1 − 1/γ)fm (x)

⩽ (gn (x) + δV ) + (1 − 1/γ)(f ˚ (x) + n(x)) ⩽ gn (x) + δ(f ˚ (x) + n(x) + V ).

The latter follows since (γ − 1)/γ ⩽ δ. According to Corollary 2.3.17, there are subsets n Bm ∈ R of A that contain all points x ∈ A such that fm (x) ⩽ gn (x) + δ(f ˚ (x) + n(x) + V ), therefore all points x ∈ A such that fm (x) ∈ Vδ (gn (x)), and we have ∫ fm dθ ⩽ ∫(gn + δ (f ˚ + n + χX ⋅ V )) dθ C

C

⩽ ∫ gn dθ + δ ∫ f ˚ dθ + δ(λ + 1)σW dθ C

C

n for every subset C ∈ R of Bm . In fact, Corollary 2.3.17 states that the above inequality ˚ holds at all μ ∈ F such that σW (μ) < +∞, but it obviously also holds if σW (μ) = +∞. n We set Ak = ⋂{Bm | m, n ⩾ k} ∈ R and observe that Ak ⊂ Ak+1 , and ⋃k∈ℕ Ak = A by our assumption. By our choice of δ, we have

∫ fm dθ ⩽ ∫ gn dθ + δ ∫ f ˚ dθ + εσW dθ Ak

Ak

Ak

2.4 The convergence theorems

� 135

for all m, n ⩾ k. From fm ⩽ f ˚ and g˚˚ ⩽ gn + g˚ for all m, n ∈ ℕ, we infer that ∫ fm dθ ⩽ ∫ f ˚ dθ A\Ak

and

A\Ak

∫ g˚˚ dθ ⩽ ∫ gm dθ + ∫ g˚ dθ. A\Ak

A\Ak

A\Ak

Adding these two inequalities to the prior one, we obtain ∫ fm dθ + ∫ g˚˚ dθ ⩽ ∫ gn dθ + δ ∫ f ˚ dθ + ∫ (f ˚ + g˚ ) dθ + εσW . A

A

A\Ak

A

A\Ak

This inequality remains true for all k ∈ ℕ and m, n ⩾ k. Following Proposition 2.4.2(c), there is k0 such that both ( ∫ g˚˚ dθ)(μ) ⩾ −ε

and ( ∫ (f ˚ + g˚ ) dθ)(μ) ⩽ ε

A\Ak

A\Ak

holds for all μ ∈ Ω and all k ⩾ k0 . Hence, using our choice for δ, we conclude that (∫ fm dθ)(μ) ⩽ (∫ gn dθ)(μ) + 4ε A

A

holds for all m, n ⩾ k0 and all μ ∈ Ω. Now in the second step of our argument, we shall extend the preceding inequality from integrals over sets A ∈ R to the corresponding integrals over X. Given ε > 0, there is A ∈ R such that (∫ f ˚ dθ)(μ) ⩽ (∫ f ˚ dθ)(μ) + ε, X

(∫ g˚ dθ)(μ) ⩽ (∫ g˚ dθ)(μ) + ε

A

X

A

and (∫ g˚˚ dθ)(μ) ⩽ (∫ g˚˚ dθ)(μ) + ε A

X

for all μ ∈ Ω. Recall that all these evaluations are finite. By our first step, there is n0 such that (∫A fn dθ)(μ) ⩽ (∫A gm dθ)(μ) + ε holds for all m, n ⩾ n0 and all μ ∈ Ω. We have fm ⩽ f ˚ for all m ∈ ℕ and, therefore, ∫ fm dθ + ∫ f ˚ dθ ⩽ ∫ fm dθ + ∫ f ˚ dθ, X

A

A

X

by Theorem 2.3.18(c). By our choice for A ∈ R, this yields

136 � 2 Integration

(∫ fm dθ)(μ) ⩽ (∫ fm dθ)(μ) + ε, X

A

for all m ∈ ℕ and μ ∈ Ω. Likewise, g˚˚ ⩽ gn + g˚ for all n ∈ ℕ, implies that ∫ g˚˚ dθ + ∫(gn + g˚ ) dθ ⩽ ∫ g˚˚ dθ + ∫(gn + g˚ ) dθ X

A

A

X

by 2.3.18(c) and (∫ gn dθ)(μ) ⩽ (∫ gn dθ)(μ) + 2ε A

X

for all n ∈ ℕ and μ ∈ Ω. Now combining all of the above, we calculate that (∫ fm dθ)(μ) ⩽ (∫ fm dθ)(μ) + ε X

A

⩽ (∫ gn dθ)(μ) + 2ε A

⩽ (∫ gn dθ)(μ) + 4ε X

holds for all m, n ⩾ n0 and all μ ∈ Ω. Hence our claim. We take notice that the claim in the general case of Theorem 2.4.8 can be reformulated, stating that lim sup(∫ fm dθ)(μ) ⩽ lim inf(∫ gn dθ)(μ) m→∞

n→∞

B

B

holds for all μ ∈ F ˚ such that both (∫B f ˚ dθ)(μ) < +∞ and (∫B g˚ dθ)(μ) < +∞. The result of this theorem becomes more recognizable if one of the sequences involved is constant. We formulate these special cases as our second convergence theorem. 2.4.9 Theorem. Let fn , f , f˚ , f ˚ ∈ ℱ(θ,B) (X, Conv(E)) for B ∈ A. (a) If fn ⩽ f ˚ for all n ∈ ℕ and fn a.e. B↘ f , then for every μ ∈ F ˚ such that a.e. B

(∫B f ˚ dθ)(μ) < +∞ and every ε > 0 there is n0 such that (∫ fn dθ)(μ) ⩽ (∫ f dθ)(μ) + ε B

for all n ⩾ n0 .

B

2.4 The convergence theorems

� 137

(b) If 0 ⩽ fn + f˚ for all n ∈ ℕ, and fn ↗a.e. B f , then for every μ ∈ F ˚ such that both a.e. B

(∫B f dθ)(μ) < +∞ and (∫B f˚ dθ)(μ) < +∞ and every ε > 0 there is n0 such that (∫ f dθ)(μ) ⩽ (∫ fn dθ)(μ) + ε B

B

for all n ⩾ n0 . (c) If 0 ⩽ fn + f˚ and fn ⩽ f ˚ for all n ∈ ℕ and fn 󳨀→ f , then a.e. B a.e. B

a.e. B

(∫ f dθ)(μ) = lim (∫ fn dθ)(μ) n→∞

B

B

for all μ ∈ F ˚ such that both (∫B f˚ dθ)(μ) < +∞ and (∫B f ˚ dθ)(μ) < +∞. If the integrals of f , f˚ and f ˚ over B are strongly additive, then the inequalities in (a) and (b) hold uniformly on the polars of all neighborhoods in 𝒲 , and the limit in (c) holds with respect to the symmetric topology of 𝒬F . Proof. We use Theorem 2.4.8 with the following insertions. For Part (a), we set gn = g˚˚ = f , g˚ = 0, and our claim follows from 2.4.8. For Part (b), we insert f for fn , rename fn for gn , f˚ for g˚ and choose g˚˚ = 0. Part (c) combines both Parts (a) and (b). Again, in the general case the claims of Theorem 2.4.9(a) and (b) can be reformulated, stating that lim sup(∫ fm dθ)(μ) ⩽ (∫ f dθ)(μ) m→∞

B

B

holds for all μ ∈ F ˚ such that (∫B f ˚ dθ)(μ) < +∞ in (a), and (∫ f dθ)(μ) ⩽ lim inf(∫ gn dθ)(μ) B

n→∞

B

holds for all μ ∈ Ω such that both (∫B f dθ)(μ) < +∞ and (∫B f˚ dθ)(μ) < +∞ in Part (b). 2.4.10 Examples. (a) In case that E = F is a Banach space over ℝ or ℂ and θ is a real- or complex-valued measure, then according to Corollary 2.4.3 the integrals for integrable E-valued functions over sets in R are strongly additive. For every absorbing set C ∈ Conv(E) and an ℝ+ -valued A-measurable function φ on X, the function φ ⋅ C ∈ ℱ (X, Conv(E)) is integrable over every set B ∈ A and its integral is evaluated as

138 � 2 Integration

(∫ φ ⋅ C dθ)(μ) = lim ( ∫ φ d|θ|) pC (μ), A∈R

B

A∩B

for μ ∈ F ˚ , where |θ| denotes the usual modulus of a real- or complex-valued measure and pC (μ) = sup{|μ(c)| | c ∈ C} ∈ ℝ (see Example 2.3.26(a)). If C is bounded, then pC is bounded in 𝒬F and if limA∈R (∫A∩B φ d|θ|) < +∞ for B ∈ A, then the integral of φ ⋅ C over B is strongly additive. Theorem 2.4.9(c) therefore implies the dominated convergence theorem (Theorem 3 in II.2 of [21]) for Bochner integrable functions. (b) For a concrete example, let X = ℝ, let R be the family of all relatively compact Borel subsets of X and let θ be the Lebesgue measure on X. For E = F, we choose the space of all continuous real-valued functions on [0, +∞) that vanish at infinity, endowed with the norm of uniform convergence. Its dual F ˚ then consists of all bounded regular Borel measures on [0, +∞). We consider the balanced set C ∈ Conv(E) consisting of all functions ρ ∈ E such that ρ(x) ⩽ x 2 for all x ∈ [0, +∞). According to Example 2.1.8(f), we have n 󵄨󵄨 |θ|A (C)(μ) = sup{∑ |θAi | 󵄨󵄨󵄨 Ai ∈ R disjoint, Ai ⊂ A}pC (μ) = θA pC (μ) 󵄨 i=1

for A ∈ R and μ ∈ F , where ˚

∞

󵄨 󵄨 pC (μ) = sup{󵄨󵄨󵄨μ(ρ)󵄨󵄨󵄨 | ρ ∈ C} = ∫ x 2 d|μ| ∈ ℝ+ , 0

according to Example 2.1.8(f). In this context, |μ| denotes the usual modulus of the bounded real-valued measure μ ∈ F ˚ . Following Example 2.3.26(a) for an ℝ+ -valued measurable function φ on X and a Borel subset B of X, we compute (∫ φ ⋅ C dθ)(μ) = lim ( ∫ φ d|θ|) pC (μ) A∈R

B

A∩B ∞

= ∫ φ dx ∫ x 2 d|μ| ∈ ℝ+ B

0

for all μ ∈ F ˚ . This integral is generally neither a bounded element of 𝒬F nor is it strongly additive. Now consider a sequence of θ-almost everywhere positive measurable functions φn ∈ ℱ (X) converging pointwise to φ ∈ ℱ (X). For every Borel subset B of X, we have φn ⋅ C 󳨀→ φ ⋅ C by Proposition 2.4.7(c). If there is a measurable function a.e. B φ˚ ∈ F such that 0 < φn ⩽ φ˚ for all n ∈ ℕ, then Theorem 2.4.9(c) applies with a.e. B

a.e. B

f˚ = 0 and f ˚ = φ˚ ⋅ C. If for example we choose φn (x) = |x|n , then φ(x) = 0 for |x| < 1, φ(±1) = 1 and φ(x) = +∞ for |x| > 1, and we choose φ˚ (x) = 1 for |x| < 1, φ˚ (±1) = 1 and φ˚ (x) = +∞ for |x| > 1. Thus Theorem 2.4.9(c) applies to all Borel subsets B of [−1, +1] and μ ∈ F ˚ that as measures on [0, +∞) have a compact support.

2.5 Measures as linear operators � 139

2.5 Measures as linear operators Using the general assumptions from the previous sections, we proceed to investigate the integral as a linear operator from the integration cones ℱ(θ,B) (X, Conv(E)) into 𝒬F . For an L(E, F ˚‚ )-valued measure θ and a set B ∈ A, we shall denote the operator f 󳨃→ ∫ f dθ : ℱ(θ,B) (X, Conv(E)) → 𝒬F B

by 𝒯(θ,B) . According to Theorem 2.3.15, this operator is linear. For every W ∈ 𝒲 , we denote by v(θ,W ) the function space neighborhood, which is generated by the family 𝒩v(θ,W ) of all measurable neighborhood functions n such that ∫X n dθ ⩽ σW . The requirements for a function space neighborhood are readily verified. By Vθ , we denote the function space neighborhood system in the sense of Section 1.2 comprised of the neighborhoods v(θ,W ) for all W ∈ 𝒲 . We elaborated earlier (see Section 2.2) that the concept of function space neighborhoods canonically extends to the set-valued functions in ℱ (X, Conv(E)). Locally convex cones were introduced in Section 1.4. 2.5.1 Proposition. For every A ∈ R, the cone ℱ(θ,A) (X, Conv(E)) endowed with ⩽ as a.e. A

its order and the neighborhood system Vθ forms a locally convex cone. The subcone of measurable functions is dense in its symmetric topology. 𝒯(θ,A) is a continuous linear operator from ℱ(θ,A) (X, Conv(E)) into 𝒬F .

Proof. Let A ∈ R. We shall establish that ℱ(θ,A) (X, Conv(E)) is a subcone of a full locally convex cone (𝒫 , Vθ ), which contains the neighborhoods in Vθ as its elements. Sums, positive multiples and the order for function space neighborhoods were defined ̃ θ the cone generated by the sums of elements of Vθ in Section 1.2. We denote by V adjoined by a formal zero element 0 and set ̃ θ }. 𝒫 = {(f , v) | f ∈ ℱ(θ,A) (X, Conv(E)), v ∈ V Endowed with the componentwise algebraic operations 𝒫 forms a cone. Its canonical order is given by (f , v) ⩽ (g, u) if for every n ∈ 𝒩v there is m ∈ 𝒩u such that f + n ⩽ g + m. If either v = 0 or u = 0, the latter means f ⩽ g + m or f + n ⩽ g, a.e. A

a.e. A

a.e. A

respectively. The embeddings f 󳨃→ (f , 0) of ℱ(θ,A) (X, Conv(E)) and v(θ,W ) 󳨃→ (0, v(θ,W ) ) of Vθ into 𝒫 preserve the respective algebraic and order structures. Furthermore, every element (f , u) ∈ 𝒫 is bounded below with respect to the neighborhood system Vθ . Indeed, let v(θ,W ) ∈ Vθ with W ∈ 𝒲 . According to Proposition 2.3.13, there is a measurable neighborhood function n and λ ⩾ 0 such that 0 ⩽ f +n and ∫A n dθ ⩽ λσW . Hence m = χA ⋅ n ∈ 𝒩λv(θ,W ) . We choose any neighborhood function u ∈ 𝒩u and set s = u + m ∈ 𝒩u+λv(θ,W ) . Then 0 ⩽ f + s and, therefore, a.e. A

(0, 0) ⩽ (f , u) + λ(0, v(θ,W ) ) = (f , u + λv(θ,W ) ).

140 � 2 Integration In this way, (𝒫 , 𝒱θ ) becomes a full locally convex cone, which contains the cone ℱ(θ,A) (X, Conv(E)) of integrable functions as a subcone. A brief inspection of our def-

inition of integrability shows that every symmetric neighborhood of a function in ℱ(θ,A) (X, Conv(E)) contains measurable functions, which therefore form a dense subcone. Finally, the operator 𝒯(θ,A) : ℱ(θ,A) (X, Conv(E)) → 𝒬F

is real-linear and continuous. For the latter, let W ∈ 𝒲 . If f ⩽ g + v(θ,W ) for f , g ∈ a.e. A

ℱ(θ,A) (X, Conv(E)), then there exists a neighborhood function n ∈ 𝒩v(θ,W ) such that

f ⩽ g + n. Following Theorem 2.3.15(b) and (c), this renders a.e. A

𝒯(θ,A) (f ) = ∫ f dθ ⩽ ∫ g dθ + ∫ n dθ ⩽ 𝒯(θ,A) (g) + σW . A

A

A

We note that the order ⩽ induces the usual equivalence relation for integration a.e. A

spaces. The locally convex cone ℱ(θ,B) (X, Conv(E)) is closed in ℱ (X, Conv(E)) in the following sense.

2.5.2 Proposition. Let f ∈ ℱ (X, Conv(E)) and let B ∈ A. If for every W ∈ 𝒲 , there is g ∈ ℱ(θ,B) (X, Conv(E)) such that f ⩽ g + v(θ,W ) and g ⩽ f + v(θ,W ) , then f ∈ a.e. B

ℱ(θ,B) (X, Conv(E)).

a.e. B

Proof. Suppose that f ∈ ℱ (X, Conv(E)) satisfies our assumptions. Using Theorem 2.3.18(a), we may assume that B = X. Given W ∈ 𝒲 , we set U = (1/3)W ∈ 𝒲 . There are g ∈ ℱ(θ,X) (X, Conv(E)) and neighborhood functions n1 , n2 ∈ 𝒩v(θ,U) such that f ⩽ g + n1 , a.e. X

g ⩽ f + n2 a.e. B

and

∫ n1 dθ, ∫ n2 dθ ⩽ σU . X

X

Given A ∈ R, there are a measurable functions gU and nU in ℱ (X, Conv(E)) as in the definition of the integrability of g over A. That is, g ⩽ gU ⩽ g +nU and ∫A nU dθ ⩽ σU . a.e. A

Hence

a.e. A

f ⩽ gU + n1 ⩽ g + nU + n1 ⩽ f + nU + n1 + n2 . a.e. A

a.e. A

a.e. A

Now setting fW = gU + n1 and nW = nU + n1 + n2 , we have f ⩽ fW ⩽ f + nW , a.e. A

a.e. A

where fW is measurable and nW is a measurable neighborhood function such that ∫ nW dθ ⩽ 3σU = σW . A

2.5 Measures as linear operators

� 141

In this way, f satisfies the definition of integrability over every set A ∈ R. Furthermore, the above yields that ∫ f dθ ⩽ ∫ g dθ + σU

and

∫ g dθ ⩽ ∫ f dθ + σU A

A

A

A

holds for all A ∈ R. Since g is integrable over X, the limit ∫X g dθ = limA∈R ∫A g dθ exist pointwise on F ˚ and is an element of 𝒬F . This yields lim sup(∫ f dθ)(μ) ⩽ (∫ g dθ)(μ) + σU (μ) A∈R

A

X

⩽ lim inf(∫ f dθ)(μ) + 2σU (μ) A∈R

A

for all μ ∈ F ˚ . Since this last inequality holds for any choice of W , we can infer that the limit limA∈R (∫A f dθ)(μ) ∈ ℝ exists for all μ ∈ F ˚ . Moreover, for every W ∈ 𝒲 the integral ∫X g dθ as an element of 𝒬F is bounded below on W ˝ . Hence the above inequality also implies that the pointwise limit limA∈R ∫A f dθ is similarly bounded below on W ˝ and, therefore, also an element of 𝒬F . We infer that f ∈ ℱ(θ,X) (X, Conv(E)), our claim. The approximation procedure described in Proposition 2.5.2 uses the symmetric neighborhoods created by Vθ . Together with the order ⩽ they define a topola.e. A

ogy on all of ℱ (X, Conv(E)), which does however not render a locally convex cone since the elements need not be bounded below. Propositions 2.5.1 and 2.5.2 yield that ℱ(θ,A) (X, Conv(E)) is the closure in this topology of the cone of measurable functions in ℱ (X, Conv(E)). The comparison of function space neighborhood systems was defined in Section 1.2. In accordance with this definition, V is finer than U if for every u ∈ U there is v ∈ V such that v ⩽ u. In this sense, the earlier (see Section 2.3) specified function space neighborhood system {vW | W ∈ 𝒲 } is finer than Vθ . 2.5.3 Proposition. Suppose that V is finer than Vθ and let A ∈ R. If for every v ∈ V, there is n ∈ 𝒩v such that n(x) = E for all x ∈ A, then A is of measure zero. Proof. Suppose that V is finer that Vθ . Let A ∈ R and let B ∈ R be any subset of A. For W ∈ 𝒲 , choose a function space neighborhood v ∈ V such that v ⩽ v(θ,W ) . Following our assumption, there is n ∈ 𝒩v , such that n(x) = E for all x ∈ A. Then for every a ∈ E the function χB ⋅ a is measurable (Proposition 2.2.9), hence integrable, and we have χB ⋅ a ⩽ n, that is, χB ⋅ a ⩽ v. Thus θB (a) = ∫X χB ⋅ a dθ ⩽ σW . As W ∈ 𝒲 was arbitrarily chosen, this shows θB (a) = 0. Hence θB = 0.

142 � 2 Integration Properties of operators defined by an integral We shall now return to the investigation of function spaces, that is, spaces of E-valued functions as introduced in Section 1.2 and to the topological case. Throughout the remainder of this section and indeed of this text, we shall assume that X is a locally compact Hausdorff space and that R consists of all relatively compact Borel subsets of X. Any measure θ on R is presumed to be L(E, F ˚‚ )-valued and bounded. Recall that integrals of E-valued functions are elements of F ˚‚ (Proposition 2.3.23(a)) and, therefore, finitely-valued on F ˚ . The function space neighborhood system V is assumed to be measurable. As a reference, we shall use the function space neighborhood systems Vθ and the system Vc of compact convergence from Example 1.2.7(b). Using the conventions from Section 1.3, the space of all continuous linear operators from 𝒞V (X, E) ̂ F ˚˚ or F ˚‚ is denoted by to F, F, L(𝒞V (X, E), F),

̂ L(𝒞V (X, E), F),

L(𝒞V (X, E), F ˚˚ ) or

L(𝒞V (X, E), F ˚‚ ),

respectively. Proposition 2.5.1 implies that the operator 𝒯(θ,B) is continuous on every function space contained in ℱ(θ,B) (X, Conv(E)) provided that its topology is finer than Vθ . In our first set of results, we shall derive properties of the operator 𝒯(θ,B) from those of the measure θ. 2.5.4 Proposition. Suppose that V is finer than Vθ , let f ∈ 𝒞V (X, E) and B ∈ A. (a) The function f is integrable over B with respect to θ and the net (∫A∩B f dθ)A∈R converges in the symmetric topology of 𝒬F to ∫B f dθ. Hence 𝒯(θ,B) ∈ L(𝒞V (X, E), F ˚‚ ). ̂ the (b) If θ is L(E, F)-valued, then ∫B f dθ is strongly additive and an element of F, ̂ completion of F. Hence 𝒯(θ,B) ∈ L(𝒞V (X, E), F).

(c) If θ is L(E, F)-valued and weakly compact, then ∫A f dθ ∈ F for every A ∈ R. Hence 𝒯(θ,A) ∈ L(𝒞V (X, E), F). Proof. Suppose that V is finer than Vθ and let f ∈ 𝒞V (X, E). According to Proposition 2.3.24, every continuous E-valued function is integrable over all sets A ∈ R and the integral is an element of F ˚‚ by 2.3.23(a). Let B ∈ A. Given W ∈ 𝒲 , since V is finer than Vθ , there is A ∈ R such that χX\D ⋅ f ⩽ v(θ,W ) Then for any choice of C, D ∈ R, both containing A we have C ∩ B = (C ∩ D ∩ B) ∪ (C \ (C ∩ D ∩ B)) and D ∩ B = (C ∩ D ∩ B) ∪ (D \ (C ∩ D ∩ B)).

2.5 Measures as linear operators � 143

Since both sets C \ (C ∩ D ∩ B) and D \ (C ∩ D ∩ B) are contained in X \ A, the integral of f over each of these sets is smaller than σW . Consequently, with 2.3.18(b), we realize that both ∫ f dθ ⩽ ∫ f dθ + σW C∩B

and

∫ f dθ ⩽ ∫ f dθ + σW . D∩B

D∩B

C∩B

Hence (∫A∩B f dθ)A∈R is a Cauchy net in the symmetric topology of 𝒬F and, therefore, convergent in this topology to ∫B f dθ. For the latter, we recall from Section 2.2 that 𝒬F is complete in this topology. The integral defines a real- or complex-linear operator from ℱ(θ,A) (X, E) into F ˚‚ by 2.3.23(b). Part (a) follows. Part (b) is a consequence of Corollary 2.4.3 and Proposition 2.3.23(c), and Part (c) follows from Proposition 2.3.23(d). Part (c) of Proposition 2.5.4 can be further strengthened in case that the neighborhood system V is also finer than Vc . 2.5.5 Proposition. If V is finer than both Vθ and Vc and if θ is L(E, F)-valued and compact (or weakly compact), then for every A ∈ R the operator 𝒯(θ,A) ∈ L(𝒞V (X, E), F) is compact (or weakly compact). Proof. Suppose that V is finer than both Vθ and Vc and let A ∈ R. According to Propositions 2.5.1 and 2.5.4, 𝒯(θ,A) is a continuous linear operator from 𝒞V (X, E) to F. Let K be a compact subset of X that contains the closure of A in its interior. There is φ ∈ 𝒞𝒦 (X) such that χA ⩽ φ ⩽ χK . Let ℬ be a bounded subset of 𝒞V (X, E). Following Lemma 1.2.1(a), the set ℬ̃ = {φ ⋅ f | f ∈ ℬ} is also bounded and since 𝒯(θ,A) (f ) = ∫ f dθ = ∫ φ ⋅ f dθ = 𝒯(θ,A) (φ ⋅ f ), A

A

we have 𝒯(θ,A) (ℬ) = 𝒯(θ,A) (ℬ̃). By our assumption, for every V ∈ 𝒱 there is v ∈ 𝒱 such that v ⩽ v(K,V ) (see Example 1.2.7(b)). There is λ ⩾ 0 such that ℬ̃ ⩽ λv, that is, f ⩽ λv, hence f (x) ∈ λV for all x ∈ K and all f ∈ ℬ̃. The set C = ⋃{f (K) | f ∈ ℬ̃} ̃ By the definition of is therefore seen to be bounded in E, and so is its convex hull C. (weak) compactness of a measure, the convex set n

n

i=1

i=1

󵄨󵄨 ̃ Ai ∈ R disjoint, ⋃ Ai = K} D = {∑ θAi (ci ) 󵄨󵄨󵄨 ci ∈ C, 󵄨 ̃ is relatively compact (or relatively weakly compact) in F. Every function f ∈ ℬ̃ is C̃ valued and can be approximated in the topology of 𝒞V (X, E) by C-valued step functions that are supported by K. (Proposition 1.2.5(ii)). The integrals of these step functions are therefore elements of D, and as the pointwise limit of these integrals, the value 𝒯(θ,A) (f ) = ∫A f dθ is contained in the weak closure of D. As D is convex, its closure in

144 � 2 Integration F coincides with its weak closure. We infer that 𝒯(θ,A) (ℬ) = 𝒯(θ,A) (ℬ̃) is indeed relatively compact (or relatively weakly compact) in F. We shall supply a reverse of the assertion of Proposition 2.5.5 below, but continue with some technical, but nevertheless highly practicable considerations. For the following statements, recall the definition of boundedness for subsets of ℱ (X, Conv(E)) and ℱ (X, E) from Section 1.4. 2.5.6 Lemma. For every set A ∈ R, there is a net (φi )i∈ℐ in 𝒞𝒦 (X) such that for every compact subset K of A and every open set O ∈ O containing A there is i0 ∈ ℐ such that χK ⩽ φi ⩽ χO for all i ⩾ i0 . Every net (φi )i∈ℐ in 𝒞𝒦 (X) with this property converges pointwise on X to χA , and for every choice of E there is i0 ∈ ℐ such that for every f ∈ 𝒞 (X, E) the set {φi ⋅ f | i ∈ ℐ , i ⩾ i0 } is bounded in ℱ (X, E). For every choice of F and every regular L(E, F ˚‚ )-valued measure θ, we have lim 𝒯(θ,X) (φi ⋅ f ) = ∫ f dθ. i∈ℐ

A

Convergence is meant pointwise on F ˚ . Proof. Given a set A ∈ R, we use the index set ℐ consisting of all pairs (O, K), where O ∈ O is an open set containing A and K is a compact subset of A, ordered by (O, K) ⩽ (O′ , K ′ ) if O ⊃ O′ and K ⊂ K ′ . We choose functions φ(O,K) ∈ 𝒞𝒦 (X) such that χK ⩽ φ(O,K) ⩽ χO . The net (φi )i∈ℐ then satisfies our first statement. If (φi )i∈ℐ is any such net in 𝒞𝒦 (X), it converges pointwise to χA since for every x ∈ A, K = {x} is compact subset of A, and for x ∉ A there is O ∈ O containing A but not x. For the latter, consider any set U ∈ O containing A and choose O = U \ {x}. Next, for any choice of O and K choose i0 ∈ ℐ such that χK ⩽ φi ⩽ χO for all i ⩾ i0 . Let E be a locally convex topological vector space and let f ∈ 𝒞 (X, E). We claim that the set ℱ = {φi ⋅ f | i ∈ ℐ , i ⩾ i0 }

is bounded in ℱ (X, E). Indeed, given a function space neighborhood v by Lemma 1.2.2 there is V ∈ 𝒱 and λ ⩾ 0 such that χO ⋅ f ⩽ λv. Thus ℱ ⊂ λv by 1.2.1(a) and our claim follows. For our final statement, let F be a Hausdorff locally convex topological vector space and let θ be a regular L(E, F ˚‚ )-valued measure. According to Proposition 2.3.24, every function f ∈ 𝒞 (X, E) is integrable over A with respect to θ. Let μ ∈ F ˚ . Since θ is regular, there is V ∈ 𝒱 such that lim |θ|O\A (V )(μ) = lim |θ|A\K (V )(μ) = 0.

O⊃A

K⊂A

We choose U ∈ O such that A ⊂ U. Since U is relatively compact in X and f is continuous, the set f (U) is bounded in E. Hence there is λ > 0 such that f (U) ⊂ λV . Given ε > 0, we find a subset O ∈ O of U containing A and a compact subset K

2.5 Measures as linear operators � 145

of A such that |θ|O\K (V )(μ) ⩽ ε/(2λ). There is i0 ∈ ℐ such that χK ⩽ φi ⩽ χO for all i ⩾ i0 . Since (φi ⋅ f )(U) ⊂ λV holds as well, we infer that 󵄨󵄨 󵄨󵄨 ε 󵄨󵄨 󵄨 󵄨󵄨( ∫ f dθ)(μ)󵄨󵄨󵄨 ⩽ 󵄨󵄨 󵄨󵄨 2 A\K

󵄨󵄨 ε 󵄨󵄨 󵄨 󵄨 and 󵄨󵄨󵄨( ∫ φi ⋅ f dθ)(μ)󵄨󵄨󵄨 ⩽ 󵄨󵄨 2 󵄨󵄨 O\K

holds for all i ⩾ i0 . Now combining the latter and using that 𝒯(θ,X) (φi ⋅ f ) = ∫X φi ⋅ f dθ = ∫O φi ⋅ f dθ and ∫K f dθ = ∫K φi ⋅ f dθ, we conclude that 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 󵄨 󵄨 󵄨󵄨(∫ f dθ)(μ) − 𝒯(θ,X) (φi ⋅ f )(μ)󵄨󵄨󵄨 = 󵄨󵄨󵄨( ∫ f dθ)(μ) − ( ∫ φi ⋅ f dθ)(μ)󵄨󵄨󵄨 ⩽ ε. 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 A

A\K

O\K

Our claim follows. A similar but slightly weaker statement is available for all sets in the σ-field A associated with R and functions in 𝒞V (X, E). Following Proposition 2.5.4(a), every function f ∈ 𝒞V (X, E) is integrable over every set B ∈ A with respect to a measure θ, provided that V is finer than Vθ , and its integral is an element of F ˚‚ . We shall construct a diagonal net for the proof of the following lemma. 2.5.7 Lemma. For every set B ∈ A, there is a net (φi )i∈ℐ in 𝒞𝒦 (X) such that (φi )i∈ℐ converges pointwise to χB and for every compact subset K of B and every open set U containing B there is i0 ∈ ℐ such that χK ⩽ φi ⩽ χU for all i ⩾ i0 . For every choice of E, of 𝒞V (X, E) and F, for every regular L(E, F ˚‚ )-valued measure θ such that V is finer than Vθ and every f ∈ 𝒞V (X, E) we have lim 𝒯(θ,X) (φi ⋅ f ) = ∫ f dθ i∈ℐ

B

Convergence is meant pointwise on F ˚ . Proof. Let B ∈ A. For every A ∈ R, let (φAi )i∈ℐA be a net in 𝒞𝒦 (X) converging to B ∩ A ∈ R as in Lemma 2.5.6. We define the index set for a diagonal net as follows. Let ℐ = {(A, ξ) | A ∈ R and ξ ∈ ΠA∈R ℐA }.

We notice that the elements of ΠA∈R ℐA are functions ξ on R such that ξ(A) ∈ ℐA for all A ∈ R. The order on ℐ is defined by (A1 , ξ1 ) ⩽ (A2 , ξ2 ) if A1 ⊂ A2 and ξ1 (A) ⩽ ξ2 (A) for all A ∈ R. We shall verify that the net (φAξ(A) )(A,ξ)∈ℐ satisfies our requirements. Indeed, let K ⊂ B ⊂ U for a compact set K and an open set U. We choose ξ0 ∈ ΠA∈R ℐA such that for all sets A ∈ R that contain K we have χK ⩽ φAi ⩽ χU for all i ∈ ℐA such that i ⩾ ξ0 (A). Then χK ⩽ φAξ(A) ⩽ χU for all (A, ξ) ⩾ (K, ξ0 ). This implies pointwise convergence to χB since for every x ∈ X we have either K = {x} ⊂ B or x ∉ B and U = X \{x} is an open set containing B, but not x. Hence our first claim. For our second

146 � 2 Integration claim, let E, 𝒞V (X, E), F and θ be as stated and let f ∈ 𝒞V (X, E). Then φi ⋅ f ∈ 𝒞V (X, E) for all i ∈ ℐ , hence φi ⋅ f is integrable over X with respect to θ by 2.5.4. Let μ ∈ F ˚ and ε > 0. Choose A0 ∈ R such that |(∫B f dθ)(μ) − (∫B∩A f dθ)(μ)| ⩽ ε/2 for all A ∈ R such that A0 ⊂ A. Then according to Lemma 2.5.6, we choose ξ0 ∈ ΠA∈R ℐA such that for every A ∈ R we have |(∫B∩A f dθ)(μ) − (𝒯(θ,X) (φAi ⋅ f ))(μ)| ⩽ ε/2 for all i ∈ ℐA such that i ⩾ ξ0 (A). Thus 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 A 󵄨󵄨(∫ f dθ)(μ) − (𝒯(θ,X) (φξ(A) ⋅ f ))(μ)󵄨󵄨󵄨 ⩽ ε 󵄨󵄨 󵄨󵄨 B

holds for all (A, ξ) ⩾ (A0 , ξ0 ), and our claim follows. Properties of measures representing an operator Lemmas 2.5.6 and 2.5.7 permit to conveniently retrieve properties of a regular measure from those of the operator 𝒯(θ,X) that it induces on function spaces. Using Lemma 2.5.6, we can directly compute the values of the measure θ in terms 𝒯(θ,X) . Indeed, let A ∈ R and a ∈ E. We apply 2.5.6 with a net (φi )i∈ℐ in 𝒞𝒦 (X) as stated and the constant function x 󳨃→ a for f . This renders θA (a) = lim 𝒯θ,X (φi ⋅ a) ∈ F ˚‚ . i∈ℐ

As an immediate consequence, we obtain the following statements. 2.5.8 Proposition. Let θ and ϑ be regular measures. If 𝒯(θ,X) (φ ⋅ a) = 𝒯(ϑ,X) (φ ⋅ a) for all φ ∈ 𝒞𝒦 (X) and a ∈ E, then θ = ϑ. 2.5.9 Proposition. Let θ be a regular measure and let B ∈ A be a closed set. If 𝒯(θ,B) (φ ⋅ a) = 0 for all φ ∈ 𝒞𝒦 (X) and a ∈ E such that φ is supported by X \ B, then θ is supported by B. Proof. We shall verify that X \B is of measure zero. Let A ∈ R be a subset of X \B and let a ∈ E. We choose the net (φi )i∈ℐ in 𝒞𝒦 (X) from Lemma 2.5.6 for the set A with K = H and U = X \ B. For f ∈ 𝒞 (X, E), we choose the constant function x 󳨃→ a. Then all the functions φi ⋅ a vanish on B, hence 𝒯 (φi ⋅ a) = 0. Thus θA (a) = ∫A f dθ = 0 by Lemma 2.5.6. Our claim follows. 2.5.10 Proposition. Suppose that V is finer than Vθ , that θ is a regular measure and that 𝒯(θ,X) ∈ L(𝒞V (X, E), F). (a) Then θ is L(E, F ˚˚ )-valued and 𝒯(θ,A) ∈ L(𝒞V (X, E), F ˚˚ ) for all A ∈ R. (b) If for every A ∈ R and W ∈ 𝒲 , there is a function space neighborhood v such ˝ ) ̂ ̂ for that ‖θ‖(v,W < +∞, then θ is L(E, F)-valued and 𝒯(θ,A) ∈ L(𝒞V (X, E), F) A all A ∈ R.

2.5 Measures as linear operators

� 147

(c) If 𝒯(θ,X) is compact (or weakly compact), then θ is L(E, F)-valued and compact (or weakly compact) and 𝒯(θ,A) ∈ L(𝒞V (X, E), F) for all A ∈ R. Proof. Suppose that V is finer than Vθ , that θ is regular and that 𝒯(θ,X) (f ) ∈ F for all f ∈ 𝒞V (X, E). Let A ∈ R and f ∈ 𝒞 (X, E). Choose a net (φi )i∈ℐ in 𝒞𝒦 (X) as in Lemma 2.5.6. Then limi∈ℐ 𝒯(θ,X) (φi ⋅ f ) = ∫A f dθ pointwise on F ˚ , and there is i0 ∈ ℐ such that the set ℱ = {φi ⋅ f | i ⩾ i0 }

is bounded in ℱ (X, E). For Part (a) we continue as follows. Since every continuous operator is bounded, the set 𝒯(θ,X) (ℱ ) is bounded in F and its polar is a neighborhood in the strong topology of F ˚ . For every μ in this neighborhood, we have Re μ(c) ⩽ 1 for all c ∈ T(θ,X) (ℱ ). Hence Re(∫A f dθ)(μ) ⩽ 1 holds as well. But this shows that the linear functional ∫A f dθ is bounded on a neighborhood of F ˚ , hence is an element of its dual F ˚˚ . If for a ∈ E, we use the constant function x 󳨃→ a for f , this yields θA (a) ∈ F ˚˚ . Moreover, for every α in the common scalar field of E and F, we have θA (αa)(μ) = lim 𝒯(θ,X) (φi ⋅ (αa))(μ) i∈ℐ

= lim 𝒯(θ,X) (φi ⋅ a)(αμ) i∈ℐ

= θA (a)(αμ) Thus θ is indeed L(E, F ˚˚ )-valued, as claimed. Furthermore, if V is finer than Vθ , then the operator 𝒯(θ,A) is continuous on 𝒞V (X, E) and F ˚˚ -valued by the above. This yields Part (a). Under the assumptions of (b), we shall verify that (𝒯(θ,X) (φi ⋅ f ))i∈ℐ is a Cauchy net in F. For this, let W ∈ 𝒲 and choose an open set O0 ∈ O containing ˝ ) A and a function space neighborhood v such that ‖θ‖(v,W < +∞. There is V ∈ 𝒱 O0 such that χA ⋅ V ⩽ v and there is ρ > 0 such that φi ⋅ f ⩽ ρχX ⋅ V for all i ⩾ i0 . Now according to Proposition 2.1.16, we find a compact set K and an open set O ∈ O ˝ ) such that K ⊂ A ⊂ O and ‖θ‖(v,W ⩽ 1/(2ρ). According to 2.5.6, there is i1 ⩾ i0 such O\K that χK ⩽ φi ⩽ χO for all i ⩾ i1 . Thus |φi − φj | ⩽ χO\K for all i, j ⩾ i1 . We have ((φi − φj ) ⋅ f )(x) ∈ 2ρV for all x ∈ O \ K, therefore ∫ (φi − φj ) ⋅ f dθ ⩽ 2ρ|θ|O\K (V ) O\K

pointwise on F ˚ . Using Lemma 2.1.5(a), for all μ ∈ W ˝ we calculate that |θ|O\K (V )(μ) ⩽ ‖θ‖(v,W O\K Hence

˝

)

⩽ 1/(2ρ).

148 � 2 Integration

(𝒯(θ,X) (φi ⋅ f ) − 𝒯(θ,X) (φj ⋅ f ))(μ) = ( ∫ (φi − φj ) ⋅ f dθ)(μ) ⩽ 1. O\K

This implies that 𝒯(θ,X) (φi ⋅ a) − 𝒯(θ,X) (φj ⋅ a) ∈ W for all i, j ⩾ i1 , our claim. The net (𝒯(θ,X) (φi ⋅ a))i∈ℐ therefore converges to an element of F̂ ⊂ F ˚‚ . As in the argument ̂ and consequently, θA (a) ∈ F̂ for all a ∈ E. For (c), for (a), we infer that 𝒯(θ,A) (f ) ∈ F, if the operator 𝒯(θ,X) is compact (or weakly compact), then the set 𝒯(θ,X) (ℱ ) is relatively weakly compact in F and its polar is a neighborhood consistent with the duality (F ˚ , F). The argument from (a) then shows that ∫A f dθ ∈ F, hence θA (a) ∈ F for all a ∈ E and 𝒯(θ,A) ∈ L(𝒞V (X, E), F). All left to show for Part (c) is that the measure θ is compact (or weakly compact). For this, let A ∈ R, let O ∈ O be an open set containing A and let C ∈ Conv(E) be bounded. We may also assume that C is balanced. We shall establish that the set n n 󵄨󵄨 D = {∑ θAi (ci ) 󵄨󵄨󵄨 ci ∈ C, Ai ∈ R disjoint, ⋃ Ai = A} 󵄨 i=1

i=1

is relatively compact (or relatively weakly compact) in F. Let ℋ be the subset of all functions f ∈ 𝒞V (X, E) that are supported by O and C-valued. For every v ∈ V, there is V ∈ 𝒱 such that χA ⋅ V ⩽ v and there is λ ⩾ 0 such that C ⊂ λV . Thus f ⩽ λv for all f ∈ ℋ, and the set ℋ is understood to be bounded in 𝒞V (X, E). Its image 𝒯(θ,X) (ℋ) is therefore relatively compact (or relatively weakly compact) in F. Since 𝒯(θ,X) (ℋ) is convex, its closure 𝒯(θ,X) (ℋ) in the given and the weak topologies of F coincide and is compact (or weakly compact). We shall verify that D ⊂ 𝒯(θ,X) (ℋ). For this, let ∑nk=1 θAk (ck ) ∈ D and choose compact subsets Kk of Ak . There are disjoint open sets Ok ⊂ O such that Kk ⊂ Ok . Now for each of the sets Kk , we choose a net (φkik )ik ∈ℐk in 𝒞𝒦 (X) with Kk for K and Ok for O in Lemma 2.5.6. We consider the index set ℐ = Πnk=1 ℐk with the componentwise order and for i = (i1 , . . . , in ) ∈ ℐ set fi = ∑nk=1 φkik ∈ ℋ. Then according to Lemma 2.5.6 the net (T(θ,X) (fi ))i∈ℐ in 𝒯(θ,X) (ℋ) converges pointwise on F ˚ , that is, in the weak topology of F to ∑nk=1 θKk (ck ). Hence n

∑ θKk (ck ) ∈ 𝒯(θ,X) (ℋ).

k=1

Next, we construct an index set 𝒦 = {(K1 , . . . , Kn )}, where Kk are compact subsets of Ak , ordered componentwise by set inclusion. Then finally, using the regularity of θ and Lemma 2.1.12, we conclude that n

n

k=1

k=1

∑ θAk (ck ) = ∑ lim θKk (ck ) =

Our claim in (c) follows.

Kk ⊂Ak

lim

(K1 ,...,Kn )∈𝒦

n

∑ θKk (ck ) ∈ 𝒯(θ,X) (ℋ).

k=1

2.6 Integral representation

� 149

2.5.11 Example. The following simple example will demonstrate that for a regular L(E, F)-valued measure θ with bounded variation the operator 𝒯(θ,X) is not necessarily weakly compact (compare VI.3 in [21]). Let X = [0, 1] and let E = F be any normed space with unit ball 𝔹. Suppose that 𝒞V (X, E) is endowed with the topology of uniform convergence (Example 1.2.7(a)). We consider the point evaluation measure δ0I , where I is the identity operator on E. That is, (δ0I )A = I if 0 ∈ A and (δ0I )A = 0 if 0 ∉ A. Then 𝒯(δI ,X) (f ) = f (0) for f ∈ 𝒞 (X, E). We choose the unit ball 𝔹𝒞 of 𝒞 (X, E) 0

(𝔹𝒞 ,𝔹˝ )

for v and the dual unit ball 𝔹˝ of E = F for Π and calculate that ‖θ‖A 𝒯 (𝔹𝒞 ) = 𝔹, which in general is not relatively weakly compact.

= 1. But

2.6 Integral representation The Riesz representation theorem for linear functionals on a space of real-valued continuous functions on a compact space is the starting point for a variety of subsequent results. In the preceding section, we investigated linear operators defined by measures on function spaces and the interplay of their properties. In this section, we shall formulate and prove an integral representation theorem, which establishes that every continuous linear operator on a function space is of this type. It is an adaptation (and in some aspects, strengthening) of a more general result for locally convex cones, that is, Theorem III.5.1 together with Corollary III.6.10 in [48]. There are numerous earlier versions of this variety. For the special case of a compact space X, for E = ℝ, a Banach space F and weakly compact operators, for example, a similar result is due to Bartle, Dunford and Schwartz [7]. Our approach to integral representations will use the assumptions and notation from the previous sections while considering only the topological case. Accordingly, throughout the remainder of this text, X represents a locally compact Hausdorff space and the σ-ring R consists of all relatively compact Borel subsets of X. All measures are supposed to be bounded. A function space neighborhood system is said to be measurable if all neighborhood functions involved in the formation of its neighborhoods are measurable. It is weakly lower continuous (see Section 1.2) if these neighborhood functions are weakly lower continuous. All function space neighborhood systems in our Examples 1.2.7(a) to (d) in Section 1.2 are both measurable (see Corollary 2.2.21) and weakly lower continuous. The space L(𝒞V (X, E), F) consists of all continuous linear operators from the function space 𝒞V (X, E) into the locally convex space F. We recall from Section 1.3 that in case that F is semireflexive, every operator 𝒯 ∈ L(𝒞V (X, E), F) is weakly compact. 2.6.1 Theorem. If the function space neighborhood system V is measurable and weakly lower continuous, then every operator 𝒯 ∈ L(𝒞V (X, E), F) can be represented as an integral on X. More precisely, there exists a unique regular L(E, F ˚˚ )-valued measure θ

150 � 2 Integration on R such that V is finer than Vθ , all functions in 𝒞V (X, E) are integrable over every set in A with respect to θ, their integrals are F ˚˚ -valued and ∫ f dθ = 𝒯 (f )

for all

f ∈ 𝒞V (X, E).

X

If 𝒯 is compact (or weakly compact), then the measure θ is L(E, F)-valued and compact (or weakly compact), countably additive with respect to the strong operator topology of L(E, F), and the integrals of functions in 𝒞V (X, E) over sets in R are F-valued. Proof. Let 𝒯 ∈ L(𝒞V (X, E), F). In our construction of a representation measure θ for the operator 𝒯 on the σ-ring R, we shall use a version of the classical Riesz representation theorem, which states that every real-valued continuous linear functional Φ on the space 𝒞𝒦 (X) of continuous real-valued functions with compact support on a locally compact Hausdorff space X, endowed with the topology of uniform convergence, can be uniquely represented by a regular real-valued Borel measure ϑ. That is, Φ(φ) = ∫X φ dϑ holds for all φ ∈ 𝒞𝒦 (X) and ‖Φ‖ = |ϑ|X . This version can for instance be found in Rudin’s book [55] (Theorem 6.19). We shall go forward with a step-by-step construction of our representation measure for the operator 𝒯 . Every open subset O of X is itself a locally compact Hausdorff space, and we denote by BO the σ-field of all Borel subsets of O. We claim that BO = {B ∈ BX | B ⊂ O}. Indeed, the family {B ∈ BX | B ⊂ O} forms a σ-field in O that contains all open subsets of O. Hence this family contains BO . For the reverse inclusion, consider the family C = {B ⊂ X | B ∩ O ∈ BO }. This is a σ-field in X that contains all open subsets of X. Hence BX ⊂ C, and for every B ∈ BX such that B ⊂ O we have B ∈ C and, therefore, B = B ∩ O ∈ BO . Moreover, if a Borel subset of O is relatively compact in O, then its closure in O, which is contained in its closure in X, is compact, hence also closed in X. Thus this set is also relatively compact in X. In summary, the family RO of all relatively compact Borel subsets of O is a subfamily of R. We shall use this observation in our construction. In a first step, we fix elements a ∈ E and μ ∈ F ˚ and choose a neighborhood W ∈ 𝒲 such that μ ∈ W ˝ . There is v ∈ V such that 𝒯 (v) ⊂ W . For every O ∈ O, there is V ∈ 𝒱 such that χO ⋅ V ⩽ v, and there is λW ⩾ 0 such that a ∈ λW V . Since O ⊂ X is itself locally compact, we can apply the above version of the Riesz representation theorem to every continuous linear functional on 𝒞𝒦 (O), and because O is open, every function φ ∈ 𝒞𝒦 (O) if extended by zero on X \O is continuous on X. In this way, (a,μ) on 𝒞𝒦 (O) setting 𝒞𝒦 (O) is a subspace of 𝒞𝒦 (X). We define a linear functional ΦO (a,μ)

ΦO

(φ) = (𝒯 (φ ⋅ a))(μ)

2.6 Integral representation

� 151

for φ ∈ 𝒞𝒦 (O). As |φ| ⩽ 1 implies that φ⋅a ⩽ λW χO ⋅V ⩽ λW v, we have 𝒯 (φ⋅a) ∈ λW W (a,μ) (a,μ) and, therefore, ΦO (φ) ⩽ λW . The functional ΦO is therefore continuous on 𝒞𝒦 (O) (a,μ)

and can be represented by a unique regular Borel measure ϑO (a,μ) |ϑO |O

=

(a,μ) ‖ΦO ‖

on O such that

≤ λW . If O, U ∈ O, then BO∩U ⊂ BO ∩ BU by the above, and 𝒞𝒦 (O ∩ U) ⊂ 𝒞𝒦 (O) ∩ 𝒞𝒦 (U).

For every function φ ∈ 𝒞𝒦 (O ∩ U), we have (a,μ)

(a,μ)

ΦO∩U (φ) = ΦO

(a,μ)

(φ) = ΦU

(φ),

and consequently, (a,μ)

(a,μ)

∫ φ dϑO∩U = ∫ φ dϑO

O∩U

O

= ∫φ U (a,μ)

(a,μ)

(a,μ) 󵄨󵄨 󵄨󵄨O∩U

= ∫ φ dϑO O∩U

(a,μ) dϑU

(a,μ) 󵄨󵄨 󵄨󵄨O∩U .

= ∫ φ dϑU O∩U

(a,μ)

The measures ϑO∩U , ϑO |O∩U and ϑU |O∩U all are regular on O ∩ U (Proposition 2.1.18) and their integrals coincide on 𝒞𝒦 (O ∩ U). By the uniqueness statement of the Riesz representation theorem (see also Proposition 2.5.8) they are therefore equal. Thus for any set A ∈ BO∩U we have (a,μ)

(ϑO

(a,μ) 󵄨󵄨 󵄨󵄨O∩U )A

)A = (ϑO

(a,μ) 󵄨󵄨

= (ϑU

(a,μ)

󵄨󵄨O∩U )A = (ϑU

)A .

For every A ∈ R and a ∈ E, we can therefore consistently define a real-valued function θA (a) on F ˚ by (a,μ)

θA (a)(μ) = (ϑO

)A ,

where μ ∈ F ˚ and O is any set in O containing A. We shall proceed to demonstrate that for every fixed element a ∈ E the mapping μ 󳨃→ (θA (a))(μ) on F ˚ is an element of 𝒬F . For A ∈ R and a ∈ E, choose O ∈ O such that A ⊂ O. Then (a,λμ)

ΦO

(a,μ)

= λΦO

and

(a,μ+ν)

ΦO

(a,μ)

= ΦO

+ Φ(a,ν) O

for μ, ν ∈ F ˚ and λ ∈ ℝ. We agree that sums and scalar multiples of real-valued measures are defined setwise and using the uniqueness of the corresponding real-valued representation measures conclude that (a,λμ)

ϑO

(a,μ)

= λϑO

and

(a,μ+ν)

ϑO

(a,μ)

= ϑO

+ ϑO(a,ν) .

152 � 2 Integration Hence θA (a) is seen to be a linear functional on F ˚ . We still need to verify that it is bounded on the polars of the neighborhoods in 𝒲 . For this, let W ∈ 𝒲 . For every μ ∈ W ˝ , we have (a,μ)

(θA (a))(μ) = (ϑO

󵄨 (a,μ) 󵄨 󵄨 (a,μ) 󵄨 )A ⩽ 󵄨󵄨󵄨ϑO 󵄨󵄨󵄨A ⩽ 󵄨󵄨󵄨ϑO 󵄨󵄨󵄨O ⩽ λW .

Thus indeed θA (a) ∈ 𝒬F . Next, we shall establish that θA ∈ L(E, 𝒬F ). Again, we choose a set O ∈ O containing A. For every μ ∈ F ˚ , we have (λa,μ)

ΦO

(a,μ)

and

ΦO

(a,μ)

and

ϑO

= λΦO

(a+b,μ)

= ΦO

(a,μ)

(a+b,μ)

= ϑO

(b,μ)

+ ΦO

for a, b ∈ E and λ ∈ ℝ. Hence (λa,μ)

ϑO

= λϑO

(a,μ)

(b,μ)

+ ϑO

.

This yields θA (λa) = λθA

and

θA (a + b) = θA (a) + θA (b).

Furthermore, given W ∈ 𝒲 , there is v ∈ V such that 𝒯 (v) ⊂ W and V ∈ 𝒱 such that χO ⋅ V ⩽ v. Then for all a ∈ V we have λW = 1 and, therefore, (θA (a))(μ) ⩽ 1 for all μ ∈ W ˝ by the above. Hence θA (a) ⩽ σW (see Section 1.4). We infer that θA is indeed a continuous linear operator from E into 𝒬F . Next, we shall reason that θ is countably additive, hence an L(E, 𝒬F )-valued measure on R. For this, suppose that Ai ∈ R are disjoint sets and that A = ⋃i∈ℕ Ai ∈ R. Again, we choose O ∈ O containing A, an element a ∈ E and μ ∈ F ˚ . Using that the (a,μ) real-valued measure ϑO is countably additive, we compute that (a,μ)

(θA (a))(μ) = (ϑO

∞

(a,μ)

)A = ∑(ϑO i=1

∞

)A = ∑(θAi (a))(μ). i

i=1

The arguments for boundedness and for regularity of the L(E, 𝒬F )-valued measure θ are somewhat more elaborate. Some complications arise from the feature that (a,μ) the expression defining its modulus involves different real-valued measures ϑO . In preparation, for a set A ∈ R and a neighborhood V ∈ 𝒱 , we choose an open set O ∈ O containing A and calculate that n 󵄨󵄨 |θ|A (V )(μ) = sup{∑ θAi (ai )(μ) 󵄨󵄨󵄨 ai ∈ V , Ai ∈ R disjoint, Ai ⊂ A} 󵄨 i=1 n

󵄨󵄨 (a ,μ) = sup{∑(ϑO i )A 󵄨󵄨󵄨 ai ∈ V , Ai ∈ R disjoint, Ai ⊂ A} i 󵄨 i=1

2.6 Integral representation

� 153

n 󵄨󵄨 (a ,μ) = sup{∑(ϑO i )K 󵄨󵄨󵄨 ai ∈ V , Ki disjoint compact, Ki ⊂ A} i 󵄨 i=1

The last line in this equality, that is, the fact that we only need to consider compact (a ,μ) subsets Ki of A, results from the regularity of the measures ϑO i . Indeed, let Ai ∈ R be disjoint subsets of A, and let ai ∈ V for i = 1, . . . , n and let μ ∈ F ˚ . Because the (a ,μ) real-valued measures ϑO i are regular, and we have (a ,μ)

(a ,μ)

(ϑO i )A = lim (ϑO i )K , i

Ki ⊂Ai

i

where Ki are compact subsets of Ai , for i = 1, . . . , n, we infer that n

n

(a ,μ)

(a ,μ)

∑(ϑO i )A = lim ∑(ϑO i )K . i

i=1

Ki ⊂Ai

i

i=1

This demonstrates that the supremum in the last line of the precedent equation using only compact subsets of A equals the supremum using all sets in R in the second line. Now for the boundedness of θ let A ∈ R and choose an open set O ∈ O containing A. Let W ∈ 𝒲 . There is v ∈ V such that 𝒯 (v) ⊂ W and there is V ∈ 𝒱 such that χO ⋅ V ⩽ v. We shall verify that |θ|A (V ) ⩽ σW . For this, let μ ∈ W ˝ . Let Kk be disjoint compact subsets of A, let ak ∈ V for k = 1, . . . , n and let μ ∈ F ˚ . There are disjoint open sets Ok ∈ R such that Kk ⊂ Ok ⊂ O. For each of the sets Kk , there is a net (φkik )i∈ℐk in 𝒞𝒦 (X) as in Lemma 2.5.6 with Ok in place of O. Hence lim ∫(φkik ⋅ ak ) dϑO k

(a ,μ)

ik ∈ℐk

X

(a ,μ)

= (ϑO k )K . k

For different choices of k, the functions φkik are supported by disjoint sets and, therefore, n

∑ φkik ⋅ ak ⩽ χO ⋅ V ⩽ v

k=1

for any choice of the indexes ik ∈ ℐk . Since ∫X (φkik ⋅ ak ) dϑO k yields

(a ,μ)

n

∑ ∫(φkik ⋅ ak ) dϑO k

k=1 X

(a ,μ)

= 𝒯 (φkik ⋅ ak )(μ), this

n

= 𝒯 ( ∑ φkik ⋅ ak )(μ) ⩽ 1. k=1

Thus taking the limits over all index sets ℐk yields ∑nk=1 (ϑO k )Kk ⩽ 1. Our claim now follows from our preparatory remarks. The argument for the regularity of θ is more straightforward. Following Proposition 2.1.13, it suffices to verify inner regularity for all sets in R. Let A ∈ R and choose (a ,μ)

154 � 2 Integration O ∈ O containing A. The already proven boundedness of θ implies that for every μ ∈ F ˚ there is V ∈ 𝒱 such that |θ|O (V )(μ) < +∞. The inequality |θ|A (V )(μ) ⩾ lim |θ|K (V )(μ) = sup |θ|K (V )(μ) K⊂A

K⊂A

is obvious since |θ|B (V )(μ) ⩽ |θ|C (V )(μ) whenever B ⊂ C for B, C ∈ R. For the reverse inequality, let Ki be disjoint compact subsets of A, let ai ∈ V for i = 1, . . . , n and let μ ∈ F ˚ . Set K = ⋃ni=1 Ki ⊂ A. Then n

(a ,μ)

∑(ϑO i )K ⩽ |θ|K (V ) ⩽ lim |θ|K (V )(μ). K⊂A

i

i=1

This renders |θ|A (V )(μ) ⩽ limK⊂A |θ|K (V )(μ), our claim. Next, we shall argue that V is finer than Vθ and proceed in several steps. Let W ∈ 𝒲 . There is v ∈ V such that 𝒯 (v) ⊂ W . Let h = ∑k=1 χAk ⋅ ak be an E-valued step function such that h ⩽ v. We may assume that the sets Ak are disjoint and choose compact subsets Kk of Ak . Given ε > 0, according to Lemma 1.2.4(a) there are disjoint open sets Ok ∈ O such that Kk ⊂ Ok and ∑k=1 χOk ⋅ ak ⩽ (1 + ε)v. We choose nets (φkik )i∈ℐk in 𝒞𝒦 (X) as in the prior argument for boundedness. Hence n

∑ φkik ⋅ ak ⩽ (1 + ε)v,

k=1

that is

n

k

𝒯 ( ∑ φik ⋅ ak ) ⩽ (1 + ε)σW k=1

for any choice of the indexes ik ∈ ℐk . Taking the limits first over the indexes ik ∈ ℐk as in Lemma 2.5.6 and over all ε > 0 then returns n

∫( ∑ χAk ⋅ ak ) dθ = ∑ θKk (ak ) ⩽ σW .

X

k=1

k=1

Finally, now using the previously established regularity of θ and passing to the limits over the compact subsets Kk of Ak in the last inequality renders n

∫ h dθ = ∑ θAk (ak ) ⩽ σW .

X

k=1

In the light of Lemma 2.3.3, this observation extends to Conv(E)-valued step functions h ⩽ v since the integral of a set-valued step function h was seen to be the supremum of the integrals of all vector-valued step functions smaller than h. Now let n ∈ 𝒩v be any of the measurable weakly lower continuous neighborhood functions that define v. We shall proceed to demonstrate that ∫X n dθ ⩽ σW . For this, we recall the definition of the integral of a measurable function from Section 2.3. Let

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ε > 0 and let h ∈ ℱ𝒮 (X, Conv(E)) be a step function such that h ⩽ n + εv ⩽ (1 + ε)v. Then ∫X h dθ ⩽ (1 + ε)σW by the above. Hence (εv)

∫ n dθ ⩽ ∫ n dθ ⩽ (1 + ε)σW X

X

The latter holds for all ε > 0 and proves that ∫X n dθ ⩽ σW , our claim. From this, we infer that n ∈ 𝒩v(θ,W ) . In summary, we have established that, given W ∈ 𝒲 , there is v ∈ V such that 𝒩v ⊂ 𝒩v(θ,W ) . Accordingly, the neighborhood system V is finer that Vθ . We are now in a position to reason that θ represents the operator 𝒯 . First, we shall prove that 𝒯 (φ ⋅ a) = ∫X φ ⋅ a dθ holds for all φ ∈ 𝒞𝒦 (X) and a ∈ E. Let μ ∈ F ˚ . We shall use Theorem 2.3.28 with G = H = ℝ and the operators S ∈ L(ℝ, E) and T ∈ L(F, ℝ) such that S(α) = αa and T(c) = Re μ(c) for α ∈ ℝ and c ∈ F. The real-valued composition measure T ˝ θ ˝ S then is given by (a,μ)

(T ˝ θ ˝ S)A = ((T ˝ θ ˝ S)A (1))(1) = θA (a)(μ) = (ϑO

)A ,

where O ∈ O is any open set containing A. The 𝒬F -valued measure θ ˝ S, on the other hand is defined by (a,μ)

(θ ˝ S)A (μ) = (θA (a))(μ) = (ϑO

)A

for all μ ∈ F ˚ . We choose f = φ in 2.3.28 and observe that φ is integrable over X with respect to θ ˝ S (Proposition 2.3.24). The function S ˝ φ, that is, φ ⋅ a, is integrable over X with respect to θ (also see 2.3.24). Theorem 2.3.28 then yields that (a,μ)

𝒯 (φ ⋅ a)(μ) = ∫ φ dϑO

= (∫ φ d(T ˝ θ ˝ S))(1) = (∫(φ ⋅ a) dθ)(μ)

X

X

X

holds for all μ ∈ F ˚ . Hence our claim. The preceding implies that 𝒯(X,θ) , that is, the linear operator defined by the integral over X with respect to θ with values in 𝒬F , is continuous on the subspace 𝒞V (X, E) of ℱ(θ,X) (X, Conv(E)). By the prior argument, it coincides with the given operator 𝒯 ∈ L(𝒞V (X, E), F) on the dense subspace of 𝒞V (X, E) spanned by the functions φ ⋅ a for all φ ∈ 𝒞𝒦 (X) and a ∈ E (Proposition 1.2.5(i)). Hence ∫ f dθ = 𝒯 (f ) X

holds for all functions f ∈ 𝒞V (X, E) since both operators 𝒯 and 𝒯(X,θ) are continuous. There are a few more loose ends and claims that we need to justify. Proposition 2.5.10(a) yields that the measure θ is indeed L(E, F ˚˚ )-valued and the integrals of

156 � 2 Integration functions in 𝒞V (X, E) over sets in A exist and are F ˚˚ -valued. The uniqueness of θ as a representation measure for the operator 𝒯 ∈ L(𝒞V (X, E), F) follows from Proposition 2.5.8. If 𝒯 is compact (or weakly compact), the measure θ is L(E, F)-valued and compact (or weakly compact), and the integrals of functions in 𝒞V (X, E) over sets in R are F-valued by Proposition 2.5.10(c). Moreover, θ is countably additive with respect to the strong operator topology of L(E, F) in this case by Proposition 2.1.1. Theorem 2.6.1 permits us to consider the elements of L(𝒞V (X, E), F) both as linear operators on a function space and as operator-valued measures, thus permitting to employ techniques from either perspective. This interchangeability will be used to a great extend in the following chapter. Several special cases are worth mentioning. Positive operators If both E and F are ordered topological vector spaces, then an operator T ∈ L(E, F) is called monotone or positive if it maps positive elements of E into positive elements of F. L(E, F)+ denotes the subcone of all positive operators in L(E, F). Recall that the order relations transfer naturally to the function space 𝒞V (X, E) (see Proposition 1.2.8(a)) and to the respective dual spaces of E and F. In this context, F+˚ is the positive cone of F ˚ , which consists of all μ ∈ F ˚ such that Re μ(c) ⩾ 0 for all c ∈ F+ , that is, F+˚ is the negative of the polar of F+ . Similarly, F+˚˚ is the negative of the polar of F+˚ . That is, an element of F ˚˚ is positive if its embedding into 𝒬F takes nonnegative values on the elements of F+˚ ⊂ F ˚ . 2.6.2 Corollary. If in addition to the assumptions of Theorem 2.6.1, both (E, 𝒱 ) and (F, 𝒲 ) are ordered topological vector spaces, then the operator 𝒯 is positive if and only if all values of its representation measure θ are positive operators in L(E, F ˚˚ ). Proof. First, suppose that the operator 𝒯 ∈ L(𝒞V (X, E), F) is positive, that is, 𝒯 maps E+ -valued functions into F+ . Let θ be the representation measure for 𝒯 . Let a ∈ E+ and A ∈ R. Choose a net (φi )i∈ℐ in 𝒞𝒦 (X) as in Lemma 2.5.6. Then for every μ ∈ F+˚ we have θA (a)(μ) = lim 𝒯 (φi ⋅ a)(μ) ⩾ 0. i∈ℐ

Hence θA (a) ∈ F+˚˚ . Conversely, if all values of θ are positive operators in L(E, F ˚˚ ) and if f ∈ 𝒞V (X, E) is E+ -valued, then f can be approximated with respect to the neighborhood system 𝒱 by E+ -valued step functions (Proposition 1.2.5(ii)). The integrals of these step functions with respect to θ are elements of F+˚˚ , and as F+˚˚ is closed in 𝒬F with respect to its topology of pointwise convergence, the integral of f is also contained in F+˚˚ . Finally, 𝒯 (𝒞V (X, E)+ ) ⊂ F+˚˚ follows since 𝒞V (X, E)+ is the closure of the subcone of all E+ -valued functions in 𝒞V (X, E) (Proposition 1.2.8(a)).

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� 157

Lattice homomorphisms If both E and F are indeed topological vector lattices over ℝ, and if 𝒞V (X, E) satisfies the conditions of Proposition 1.2.8(b), then we may inquire about additional properties of the representation measure for a lattice homomorphism in L(𝒞V (X, E), F). As usual, we denote the respective lattice operations by ∨ and ∧. A lattice homomorphism is an operator in L(𝒞V (X, E), F) that is compatible with these operations. We recall that the dual and second dual of a topological vector lattice are again a topological vector lattices if endowed with their respective strong topologies (Theorem V.7.4 in [56]). 2.6.3 Corollary. Suppose that in addition to the assumptions of Theorem 2.6.1 both (E, 𝒱 ) and (F, 𝒲 ) are topological vector lattices over ℝ and that 𝒞V (X, E) fulfills the conditions of Proposition 1.2.8(b). If 𝒯 ∈ L(𝒞V (X, E), F) is a lattice homomorphism, then its representation measure θ satisfies θA (a ∨ b)(δ) = θA (a)(δ) ∨ θA (b)(δ) and θA (a)(δ) ∨ θB (b)(δ) = θA (a)(δ) + θB (b)(δ) for all real-valued lattice homomorphisms δ ∈ F ˚ , all a, b ∈ E+ and disjoint sets A, B ∈ R. Proof. Let A ∈ R and a, b ∈ E. We choose the net (φi )i∈ℐ in 𝒞𝒦 (X) as in Proposition 2.5.6 and understand that θA (a) = lim T(φi ⋅ a), i∈ℐ

θA (b) = lim T(φi ⋅ b) i∈ℐ

and θA (a ∨ b) = lim T(φi ⋅ (a ∨ b)). i∈ℐ

We have φi ⋅ (a ∨ b) = (φi ⋅ a) ∨ (φi ⋅ b), hence T(φi ⋅ (a ∨ b)) = T(φi ⋅ a) ∨ T(φi ⋅ b) ∈ F, since T is a lattice homomorphism in L(𝒞V (X, E), F). Therefore, T(φi ⋅ (a ∨ b))(δ) = T(φi ⋅ a)(δ) ∨ T(φi ⋅ b)(δ) holds for every real-valued lattice homomorphism δ ∈ F ˚ . Now passing to the limits over i ∈ ℐ returns θA (a ∨ b)(δ) = θA (a)(δ) ∨ θA (b)(δ),

158 � 2 Integration our first claim. For the second claim, let A, B ∈ R be disjoint sets and let a, b ∈ E+ . For any choice of compact subsets K and L of A and B, respectively, there are disjoint open sets O, U ∈ O such that K ⊂ O and L ⊂ U. We choose nets (φi )i∈ℐ and (ψj )j∈𝒥 in 𝒞𝒦 (X) for K and L, respectively. That is, φi ⩽ χO and ψj ⩽ χU and, therefore, (φi ⋅ a) ∨ (ψj ⋅ b) = φi ⋅ a + ψj ⋅ b since a, b ⩾ 0. Hence, again passing to the limits over i ∈ ℐ and j ∈ 𝒥 yields θK (a)(δ) ∨ θL (b)(δ) = θK (a)(δ) + θL (b)(δ) for every lattice homomorphism δ ∈ F ˚ . Now using the inner regularity of θ and passing to the limits over the compact sets K ⊂ A and L ⊂ B yields our claim. An element of F ˚ is known to be a real-valued lattice homomorphism if and only if it generates an extreme ray in F+˚ (see V.1.7 in [56]). If the closure of the span of all lattice homomorphisms with respect to the strong topology of F ˚ is all of F ˚ , then any two elements of F ˚˚ that coincide on all lattice homomorphisms are equal. That is, the statements of Corollary 2.6.3 read as θA (a ∨ b) = θA (a) ∨ θA (b) and θA (a) ∨ θB (b) = θA (a) + θB (b). In case that the operator 𝒯 is weakly compact, its representation measure θ is L(E, F)valued, and it suffices to consider the closure of the span of all lattice homomorphisms in the weak* topology of F ˚ for the preceding observation. Algebra homomorphisms If both E and F are topological algebras and if 𝒞V (X, E) satisfies the condition of Proposition 1.2.8(c), then we may ask how multiplicativity of an operator 𝒯 ∈ L(𝒞V (X, E), F) transfers to its representation measure θ. We also consider the case that both E and F contain an involutions a 󳨃→ a˚ and c 󳨃→ c˚ , respectively. Other than in the case of topological vector lattices the continuity of the operations with respect to the given topology of a Hausdorff topological algebra implies weak continuity (see Section 1.3) and allows a more efficient use of Lemma 2.5.6. We shall formulate our result for the case that the representation measure of an algebra homomorphism is L(E, F)valued. A more general and more technical result can be found in Corollary III.6.13 from [48]. 2.6.4 Corollary. Suppose that in addition to the assumptions of Theorem 2.6.1 both (E, 𝒱 ) and (F, 𝒲 ) are topological algebras and that 𝒞V (X, E) fulfills the condition of Proposition 1.2.8(c). If 𝒯 ∈ L(𝒞V (X, E), F) is an algebra homomorphisms and its representation measure θ is L(E, F)-valued, then it satisfies θA (ab) = θA (a) θA (b) and

θA (a) θB (b) = 0

2.6 Integral representation

� 159

for all a, b ∈ E and disjoint sets A, B ∈ R. If both E and F contain involutions satisfying the condition in 1.2.8(c) and if 𝒯 (f ˚ ) = (𝒯 (f ))˚ for all f ∈ 𝒞V (X, E), then θA (a˚ ) = θA (a)˚ holds in addition. Proof. Let A ∈ R and a, b ∈ E. Let K be compact subset of A and choose the net (φi )i∈ℐ in 𝒞𝒦 (X) for the set K as in Lemma 2.5.6. We may assume that χK ⩽ φi for all i ∈ ℐ . For every fixed k ∈ ℐ , the net (φi φk )i∈ℐ also satisfies the assumption of Lemma 2.5.6 and with f = φk ⋅ ab we infer that lim 𝒯 (φi ⋅ (φk ⋅ ab)) = ∫ φk ⋅ ab dθ = θK (ab). i∈ℐ

K

On the other hand, we have 𝒯 (φi ⋅ (φk ⋅ ab)) = 𝒯 ((φi ⋅ a)(φk ⋅ b)) = 𝒯 (φi ⋅ a)𝒯 (φk ⋅ b)

since 𝒯 is an algebra homomorphism. For every d ∈ F, the linear operators c 󳨃→ cd and c 󳨃→ dc are supposed to be continuous with the given and, therefore, also with respect to the weak topology of F (see IV.7.4 in [56]). For this, we recall that the space F is generally supposed to be Hausdorff. We therefore conclude that lim(𝒯 (φi ⋅ a)𝒯 (φk ⋅ b)) = (lim 𝒯 (φi ⋅ a))𝒯 (φk ⋅ b) = θK (a)𝒯 (φk ⋅ b) i∈ℐ

i∈ℐ

pointwise on F ˚ and, therefore, θK (ab) = θK (a)𝒯 (φk ⋅ b). Now passing to the limit over k ∈ ℐ on the right-hand side yields θK (ab) = θK (a) θK (b). Our first claim follows with the inner regularity of θ. For the second claim, let A, B ∈ R be disjoint sets and let a, b ∈ E. For any choice of compact subsets K and L of A and B, respectively, there are disjoint open sets O, U ∈ O such that K ⊂ O and L ⊂ U. We choose nets (φi )i∈ℐ and (ψj )j∈𝒥 in 𝒞𝒦 (X) for K and L, respectively. That is, φi ⩽ χO and ψj ⩽ χU and, therefore, (φi ⋅ a)(ψj ⋅ b) = 0. Hence passing to the limits pointwise on F ˚ over i ∈ ℐ and j ∈ 𝒥 renders θK (a) θL (b) = 0. Now again using the inner regularity of θ, passing to the limits over K ⊂ A and L ⊂ B yields our second claim. Finally, suppose that E contains an involution satisfying the condition in Proposition 1.2.8(c) and that 𝒯 (f ˚ ) = (𝒯 (f ))˚ holds for all f ∈ 𝒞V (X, E). Let A ∈ R and a ∈ E and choose the net (φi )i∈ℐ in 𝒞𝒦 (X) for A as in Lemma 2.5.6. Then ˚

θA (a˚ ) = lim 𝒯 (φi ⋅ a˚ ) = lim(𝒯 (φi ⋅ a)˚ ) = (lim 𝒯 (φi ⋅ a)) = θA (a)˚ . i∈ℐ

i∈ℐ

i∈ℐ

This last equality uses once more the fact that the involution in F is continuous, hence also weakly continuous. If F itself is an operator space Lb (H, K) endowed with the strong operator topology, if K is reflexive and if an operator 𝒯 ∈ L(𝒞V (X, E), Lb (H, K)) is indeed bounded with respect to the uniform operator topology of Lb (H, K), then we may employ Propo-

160 � 2 Integration sition 1.3.6 and Proposition 1.3.3 with the uniform operator topology for τ. We infer that 𝒯 is weakly compact and summarize the following. 2.6.5 Corollary. Suppose that Lb (H, K) is endowed with the strong operator topology and that K is reflexive. If an operator in L(𝒞V (X, E), Lb (H, K)) is bounded with respect to the uniform operator topology of Lb (H, K), then it is weakly compact. Hence its representation measure is L(E, Lb (H, K))-valued and weakly compact, countably additive with respect to the strong operator topology of L(E, Lb (H, K)) and the integrals of functions in 𝒞V (X, E) over sets in R are Lb (H, K)-valued. The case that E = ℝ or E = ℂ is of particular interest. In the case of Corollary 2.6.5, then 𝒯 is an operator from 𝒞 (X) or 𝒞 (X, ℂ) into Lb (H, K), its representation measure is Lb (H, K)-valued, countably additive and regular with respect to the strong operator topology of Lb (H, K). The spectral theorem Corollaries 2.6.4 and 2.6.5 yield the classical Spectral theorem (see II.44 in [30] or X.2 in [26]) as an immediate application. Indeed, let B be a normal operator on a Hilbert space H, let X ⊂ ℂ be its compact spectrum, and consider 𝒞 (X, ℂ) with the topology of uniform convergence. Let F = L(H, H) be endowed with the strong operator topology. According to the Gelfand–Naimark theorem (Theorem IX.3.7 in [26]), there exists a multiplicative linear operator 𝒯 : 𝒞 (X, ℂ) → L(H, H),

compatible with the involution and continuous with respect to the uniform operator topology of L(H, H). It maps the identity function x 󳨃→ x in 𝒞 (X, ℂ) to the operator B. According to Corollary 2.6.5, the operator 𝒯 ∈ L(𝒞V (X, E), Lb (H, H))

is weakly compact, its representation measure θ is L(H, H)-valued, weakly compact and countably additive with respect to the strong operator topology of L(H, H). Furthermore, according to Corollary 2.6.4 with E = ℂ and F = L(H, H), we have θA = θA (1) = θA (1 ⋅ 1) = θA ˝ θA

and

θA = θA˚

for all A ∈ R. Hence the values of θ are indeed projections on H, and θ is understood to be the spectral measure of the operator B. From now on, we shall always assume that 𝒞V (X, E) satisfies the assumptions of Theorem 2.6.1 and, therefore, every operator in L(𝒞V (X, E), F) admits a unique integral representation with the stated properties.

2.6 Integral representation

� 161

Compositions of operators with real- or complex-valued functions We continue with some observations about compositions of operators with scalarvalued functions. Let 𝒯 ∈ L(𝒞V (X, E), F) and let φ be a bounded real- or complexvalued (depending on the scalar field of E) continuous function on X. For every v ∈ V and f ⩽ v, we have φ ⋅ f ∈ ρv, where ρ ⩾ 0 is a bound for |φ| (see Lemma 1.2.1(a)). In particular, we have φ ⋅ f ∈ 𝒞V (X, E) for all f ∈ 𝒞V (X, E). Hence the formula 𝒯φ (f ) = 𝒯 (φ ⋅ f ) defines an operator 𝒯φ ∈ L(𝒞V (X, E), F). 2.6.6 Proposition. If θ is the representation measure for 𝒯 ∈ L(𝒞V (X, E), F) and φ and ψ are bounded real- or complex-valued continuous functions on X, then Tφ = Tψ if and only φ(x) = ψ(x) for all x in the support of θ. Proof. Our claim is equivalent to the statement that Tφ = 0 if and only φ(x) = 0 for all x in the support of θ. We shall prove the latter. Let A be the support of θ. Then 𝒯 (f ) = ∫A f dθ for all f ∈ 𝒞V (X, E), hence 𝒯φ (f ) = 0, provided that φ(x) = 0 for all x ∈ A. Conversely, if φ(x) ≠ 0 for some x ∈ A, then there are ε > 0 and an open neighborhood U of x such that |φ(y)| ⩾ ε for all y ∈ U. Since A \ U is no longer the support of θ, using inner regularity we find a compact subset K of U and an element a ∈ E such that θK (a) ≠ 0. We choose μ ∈ F ˚ such that μ(θK (a)) = 1. By outer regularity, there is V ∈ 𝒱 such that |θ|A (V )(μ) < +∞ and an open set K ⊂ O ⊂ U such that |θ|O\K (V )(μ) ⩽ 1/(2λ), where λ > 0 is such that a ∈ λV . Moreover, according to Urysohn’s lemma there is ψ ∈ 𝒞 (X) such that 0 ⩽ ψ ⩽ 1, ψ(x) = 1 for all x ∈ K and supp(ψ) ⊂ O. Set f (x) = (ψ(x)/φ(x))a for x ∈ O and f (x) = 0, else. Then f ∈ 𝒞V (X, E) and 𝒯φ (f ) = 𝒯 (φf ) = ∫ ψ ⋅ a dθ = ∫ ψ ⋅ a dθ + θK (a). O

O\K

Finally, since ψ ⋅ a ≤ λχX ⋅ V , hence 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 󵄨󵄨( ∫ ψ ⋅ a)(μ)󵄨󵄨󵄨 ⩽ λ|θ|O\K (V )(μ) ⩽ 1/2, 󵄨󵄨 󵄨󵄨 O\K

we infer that 𝒯φ (f )(μ) ≠ 0 and, therefore, that Tφ ≠ 0. Lemma 2.5.6 facilitates the identification of the representation measure θφ of the operator 𝒯φ associated with 𝒯 ∈ L(𝒞V (X, E), F), its representation measure θ and a bounded continuous real- or complex-valued function φ. For a set A ∈ R, we use a net (φi )i∈ℐ in 𝒞𝒦 (X) as in 2.5.6. Given a ∈ E, we choose the constant function x 󳨃→ a for f and calculate (θφ )A (a) = ∫ f dθφ = lim(𝒯φ )φi (f ) = lim 𝒯φi (φ ⋅ f ) = ∫ φ ⋅ a dθ. A

i∈ℐ

i∈ℐ

A

162 � 2 Integration Thus, according to Theorem 2.6.1 (θφ )A (a) is indeed an element of F ˚˚ , as required. If the operator 𝒯 ∈ L(𝒞V (X, E), F) is compact (or weakly compact), we may use the characteristic function χA of a set A ∈ R in place of φ and obtain the restriction 𝒯χA of 𝒯 to A. For f ∈ 𝒞V (X, E), we set 𝒯χA (f ) = ∫ f dθ = ∫ f dθ|A . A

X

According to Theorem 2.6.1, 𝒯χA (f ) is an element of F. Moreover, given W ∈ 𝒲 there is v ∈ V such that | ∫X f dθ(μ)| ⩽ 1 for all integrable functions f ∈ ℱ (X, E) and all μ ∈ W ˝ . Since χA ⋅ f ⩽ v, whenever f ⩽ v (Lemma 1.2.1(a)) we have 𝒯χA (f ) ∈ W for all f ⩽ v. Hence 𝒯χA is continuous and an element of L(𝒞V (X, E), F). As observed in Proposition 2.1.9, the restriction of a regular compact (or weakly compact) measure is again regular and compact (or weakly compact). Thus according to Proposition 2.5.5, the operator 𝒯χA is also compact (or weakly compact). Finally, because 𝒯χ(X\A) (f ) = 𝒯 (f ) − 𝒯χA (f ) ∈ F, the same conclusions hold for the operator 𝒯χX\A . We summarize the following. 2.6.7 Proposition. If θ is the representation measure for 𝒯 ∈ L(𝒞V (X, E), F) and φ is a bounded real-or complex-valued continuous function on X, then the representation measure θφ for 𝒯φ ∈ L(𝒞V (X, E), F) is defined by (θφ )A (a) = ∫ φ ⋅ a dθ A

for all A ∈ R and a ∈ E. If 𝒯 ∈ L(𝒞V (X, E), F) is compact (or weakly compact), if either A ∈ R or X \A ∈ R, then the operator 𝒯χA is also in L(𝒞V (X, E), F), compact (or weakly compact) and its representation measure θχA is the restriction θ|A of θ to A. 2.6.8 Proposition. If θ is the representation measure for 𝒯 ∈ L(𝒞V (X, E), F) and G is a subspace of E, then the L(G, F ˚˚ )-valued representation measure θ‖G for the restriction of 𝒯 to 𝒞V (X, G) is the restriction of θ to G, that is, it coincides with θ on all sets A ∈ R and a ∈ G. Proof. Let 𝒯 , θ and θ‖G be as stated. For A ∈ R and the net (φi )i∈ℐ in 𝒞𝒦 (X) from Lemma 2.5.6, we observe that all the functions φi ⋅ a are contained in 𝒞V (X, G). Thus, again using Lemma 2.5.6 for both θ and 𝒯 as well as θ‖G and the restriction of 𝒯 to 𝒞V (X, G), we infer that θA (a) = lim 𝒯 (φi ⋅ a) = (θ‖G )A (a) i∈ℐ

as claimed.

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Point evaluation operators For a point x ∈ X and a linear operator T from E into F, the formula f 󳨃→ T(f (x)) defines a linear operator 𝒟xT from 𝒞V (X, E) to F. Since {f (x) | f ∈ 𝒞V (X, E)} = E, this operator is not zero if and only if T ≠ 0. For 𝒟xT to be continuous, that is, an element of L(𝒞V (X, E), F), it is necessary that T ∈ L(E, F). Indeed, if 𝒟xT ∈ L(𝒞V (X, E), F), given W ∈ 𝒲 there is v ∈ V such that 𝒟xT (v) ⊂ W . According to the definition of a function space neighborhood, there is V ∈ 𝒱 and φ ∈ 𝒞𝒦 (X) such that φ(x) = 1 and φ ⋅ a ⩽ v for all a ∈ V . Thus T(a) = 𝒟xT (φ ⋅ a) ∈ W . Hence our claim. However, depending on the neighborhood systems V for 𝒞V (X, E) and 𝒲 for F not all choices for x ∈ X and T ∈ L(E, F) render operators 𝒟xT ∈ L(𝒞V (X, E), F). The point evaluation measure δxT does nevertheless represent this operator. 2.6.9 Lemma. If x ∈ X and T ∈ L(E, F), then 𝒟xT ∈ L(𝒞V (X, E), F) if and only if V is finer than VδxT , where δxT is the representation measure for 𝒟xT . This condition is satisfied if there is v ∈ V such that the set {f (x) | f ⩽ v} is bounded in E. Proof. Let x ∈ X and T ∈ L(E, F). If 𝒟xT ∈ L(𝒞V (X, E), F), then according to Theorem 2.6.1 the neighborhood system VδxT defined by its representation measure δxT is coarser than V. Conversely, if V is finer than VδxT , then for W ∈ 𝒲 the neighborhood v(δxT ,W ) is defined by the family 𝒩(δxT ,W ) consisting of all neighborhood functions n such that ∫ n dδxT = sup{T(a) | a ∈ n(x)} ⩽ σW ,

X

that is, T(n(x)) ⊂ W . There is v ∈ V such that v ⩽ v(δxT ,W ) . Hence f ⩽ v for f ∈

𝒞V (X, E) implies that f ⩽ n for some n ∈ 𝒩(δT ,W ) and, therefore, 𝒟xT (f ) = T(f (x)) ∈ x

W . Thus 𝒟xT ∈ L(𝒞V (X, E), F), as claimed. Now suppose that there is v ∈ V such that the set A = {f (x) | f ⩽ v} is bounded in E. Given W ∈ 𝒲 , there is V ∈ 𝒱 such that T(V ) ⊂ W . There is λ > 0 such that A ⊂ λV . Thus 𝒟xT ((1/λ)v) ⊂ W .

A nonzero operator, 𝒟xT ∈ L(𝒞V (X, E), F), is called a point evaluation operator. It is represented by the point evaluation measure δxT . Point evaluations will play an important part in the development of Choquet theory for linear operators in the subsequent sections. 2.6.10 Lemma. If 𝒟xT and 𝒟yS are point evaluation operators in L(𝒞V (X, E), F), then 𝒟xT = 𝒟yS if and only if x = y and T = S.

Proof. Let 𝒟xT , 𝒟yS ∈ L(𝒞V (X, E), F). If x ≠ y, there is φ ∈ 𝒞𝒦 (X) such that φ(x) = 1 and φ(y) = 0. Since T ≠ 0, there is a ∈ E such that T(a) ≠ 0 and since φ ⋅ a ∈ 𝒞V (X, E), we infer that 𝒟xT (φ ⋅ a) = T(a), whereas 𝒟yS (φ ⋅ a) = 0. Thus 𝒟xT ≠ 𝒟yS . If

164 � 2 Integration x = y but T ≠ S, there is a ∈ E such that T(a) ≠ S(a). Choosing φ ∈ 𝒞𝒦 (X) such that φ(x) = 1, this yields 𝒟xT (φ ⋅ a) ≠ 𝒟xS (φ ⋅ a). Hence 𝒟xT ≠ 𝒟xS . For the following, we recall the function space topology of pointwise convergence from Example 1.2.7(b). 2.6.11 Lemma. Let 𝒟xT ∈ L(𝒞V (X, E), F) be a point evaluation operator. If 𝒟xT is compact (or weakly compact), then the operator T ∈ L(E, F) is compact (or weakly compact). Conversely, if T ∈ L(E, F) is compact (or weakly compact) and if V is finer than the topology of pointwise convergence, then 𝒟xT is compact (or weakly compact). Proof. Suppose that the operator 𝒟xT ∈ L(𝒞V (X, E), F) is compact (or weakly compact) and let C be a bounded subset of E. We choose a function φ ∈ 𝒞𝒦 (X) such that 0 ⩽ φ ⩽ 1 and φ(x) = 1. The set 𝒢 = {φ ⋅ c | c ∈ C} is bounded in 𝒞V (X, E). Indeed, given v ∈ V there is V ∈ 𝒱 such that χK ⋅ V ⩽ v, where K denotes the compact support of φ. We have C ⊂ λV with some λ ⩾ 0, and thus 𝒢 ⩽ λv. Hence the set 𝒟xT (𝒢 ) = T(C) is relatively compact (or relatively weakly compact) in F. Conversely, suppose that V is finer than the topology of pointwise convergence and that the operator T ∈ L(E, F) is compact (or weakly compact). Let 𝒢 be a bounded subset of 𝒞V (X, E). Then the set C = {f (x) | f ∈ 𝒢 } is bounded in E. Indeed, given V ∈ 𝒱 , by our assumption there is v ∈ V such that v ⩽ v({x},V ) = {f ∈ ℱ (X, E) | f (x) ∈ V }. We have 𝒢 ⊂ λv ⩽ λv({x},V ) with some λ ⩾ 0, and therefore C ⊂ λV . Hence the set 𝒟xT (𝒢 ) = T(C) is relatively compact (or relatively weakly compact) in F, our claim. 2.6.12 Lemma. Let 𝒟xT ∈ L(𝒞V (X, E), F) be a weakly compact point evaluation operator. If 𝒟xT = 𝒮 + 𝒰 for weakly compact operators 𝒮 , 𝒰 ∈ L(𝒞V (X, E), F), then there is a weakly compact operator ℛ ∈ L(𝒞V (X, E), F) whose representation measure is supported by X \{x} and such that both 𝒮 − ℛ and 𝒰 + ℛ are either zero or weakly compact point evaluation operators at x. Proof. Let 𝒟xT , 𝒮 and 𝒰 be as stated. According to Proposition 2.6.7, the restriction of 𝒮 to {x} is a weakly compact operator in L(𝒞V (X, E), F), hence equals 𝒟xS for some S ∈ L(E, F). The operator ℛ = 𝒮 − 𝒟xS is also weakly compact, and we have T

T

S

(T−S)

𝒰 = 𝒟x − 𝒮 = 𝒟x − (𝒟x + ℛ) = 𝒟x

− ℛ.

2.6.13 Examples and Remarks. (a) If E is indeed a function space 𝒞U (Y , G), where G is a normed space, Y is compact and U renders the topology of uniform convergence on Y , then according to Proposition 1.2.9 the function space 𝒞V (X, 𝒞U (Y , G)) is ̃ isomorphic to a function space 𝒞V ̃ (X × Y , G) via the bijection f 󳨃→ f (see 1.2.9), that is, ̃f (x, y) = f (x)(y) for f ∈ 𝒞 (X, 𝒞 (Y , G)). Thus an operator 𝒯 ∈ L(𝒞 (X, 𝒞 (Y , G)), F) V U V U corresponds to an operator 𝒯̃ ∈ L(𝒞V ̃ (X × Y , G), F)

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defined as 𝒯̃ (̃f ) = 𝒯 (f ). The representation measure θ for 𝒯 is defined on subsets of X and L(𝒞U (Y , G), F ˚˚ )-valued, whereas the representation measure θ̃ for 𝒯̃ is defined on subsets of X × Y and L(G, F ˚˚ )-valued. We shall indicate how to establish a ̃ For compact subsets A of X and B of Y , let relation between the measures θ and θ. (φi )i∈ℐ and (ψj )j∈𝒥 be nets in 𝒞𝒦 (X) and 𝒞𝒦 (Y ) = 𝒞 (Y ) converging to χA and χB as in Lemma 2.5.6, respectively. A straightforward compactness argument shows that for every open set O in X × Y containing A × B there are open sets U in X and V in Y such that A × B ⊂ U × V ⊂ O. Thus the net (φi ⋅ ψj )(i,j)∈ℐ×𝒥 in 𝒞𝒦 (X × Y ) (the index set ℐ × 𝒥 is ordered componentwise) satisfies the assumptions of Lemma 2.5.6 for the set A × B and we have θ̃A×B (c) =

lim

(i,j)∈ℐ×𝒥

𝒯̃ (φi ψj ⋅ c),

for every c ∈ G. On the other hand, for every j ∈ 𝒥 we have θA (ψj ⋅ c) = lim 𝒯 (φi ⋅ ψj ⋅ c), i∈ℐ

also by Lemma 2.5.6. This renders lim θA (ψj ⋅ c) = lim lim 𝒯 (φi ⋅ ψj ⋅ c) = j∈𝒥

j∈𝒥 i∈ℐ

lim

(i,j)∈ℐ×𝒥

̃ (c) 𝒯̃ (φi ψj ⋅ c) = θ A×B

for all c ∈ G. All limits are taken pointwise on F ˚ , that is, in the weak* topology of F ˚˚ . If in particular 𝒯 = 𝒟xT is a point evaluation and A = {x}, then θ̃{x}×B (c) = lim T(ψj ⋅ c), j∈𝒥

that is, the L(G, F ˚˚ )-valued set function B 󳨃→ θ̃{x}×B on Y determines the representation measure on Y for the operator T ∈ L(𝒞U (Y , G), F). The case that G = ℝ is of particular interest. The measures θ and θ̃ then are L(𝒞U (Y ), F ˚˚ )- and F ˚˚ -valued, respectively. The above reads as θ̃A×B = lim θA (ψj ) j∈𝒥

for compact subsets A of X and B of Y in this case. The case that G = ℂ is similar. (b) An L(E, F)-valued measure θ can also be considered to be L(ℝ, L(E, F))or L(ℂ, L(E, F))-valued, depending on the scalar field for both E and F (see Example 2.1.8(e)), if we interpret the element θA ∈ L(E, F) as the operator z 󳨃→ z θA , where z ∈ ℝ or z ∈ ℂ. For a closer look at this perspective, we consider the complex case (the real one is similar) and endow L(E, F) with its strong operator topology. According to Proposition 1.3.1, the dual L(E, F)˚ of L(E, F) is the tensor product Ẽ ⊗ F ˚ , where Ẽ ̃ denotes the quotient space E/E0 and E0 = ⋂V ∈𝒱 V . If E is Hausdorff, then E = E. ̃ The L(E, F)-valued measure θ transfers to an L(ℂ, 𝒬L(E,F) )-valued measure θ via

166 � 2 Integration θ̃A (z)(ω) = Re ω(zθA ) = θA (za1 )(μ1 ) + ⋅ ⋅ ⋅ + θA (zan )(μn ) for A ∈ R, z ∈ ℂ, ω = ã1 ⊗ μ1 + ⋅ ⋅ ⋅ + ãn ⊗ μn ∈ L(E, F)˚ and any ai ∈ ãi . Then θ̃A (z) ∈ L(E, F) ⊂ 𝒬L(E,F) , and the mapping A 󳨃→ θ̃A defines an L(E, F)-valued measure. Its countable additivity follows directly from the countable additivity of θ. For the unit ball 𝔹 of ℂ, we evaluate the modulus of θ̃ as n ̃ (𝔹)(ω) = sup{∑ θ̃ (z )(ω) 󵄨󵄨󵄨󵄨 |z | ⩽ 1, A ∈ R disjoint, A ⊂ A} |θ| i i A A i 󵄨󵄨 i i=1

n 󵄨 󵄨 󵄨󵄨 = sup{∑󵄨󵄨󵄨ω(θAi )󵄨󵄨󵄨 󵄨󵄨󵄨 Ai ∈ R disjoint, Ai ⊂ A} 󵄨 i=1

̃ let A ∈ R and let for ω ∈ L(E, F)˚ . For boundedness of the measure θ, W

𝒰B = {T ∈ L(E, F) | T(B) ⊂ W }

be a neighborhood in the strong operator topology of L(E, F), whereby is W ∈ 𝒲 and B is a finite subset of E. Since 𝒰BW is the polar in L(E, F) of the set Ω = {ã ⊗ μ | a ∈ B, μ ∈ W ˝ } ⊂ L(E, F)˚ , the polar of 𝒰BW in L(E, F)˚ is the bipolar, that is, the weak* closed convex hull of Ω. There is V ∈ 𝒱 such that |θ|A (V ) ⩽ σW and there is λ ⩾ 0 such that B ⊂ λV . Then for every ã ⊗ μ ∈ Ω we have n 󵄨 ̃ (𝔹)(ã ⊗ μ) = sup {∑ θ (z a)(μ) 󵄨󵄨󵄨󵄨 |zi | ⩽ 1, ai ∈ ãi , |θ| A Ai i 󵄨󵄨 A ∈ R disjoint, A ⊂ A} . 󵄨 i i i=1

̃ (𝔹)(ω) ⩽ |θ| (λV )(μ) ⩽ λ holds for all ω ∈ Ω and, therefore, for all ω in Thus |θ| A A the weak* closed convex hull of Ω, that is, the polar of 𝒰BW . The latter follows since ̃ (𝔹) is a weak* lower semicontinuous sublinear functional on L(E, F)˚ . We infer |θ| A that the L(ℂ, 𝒬L(E,F) )-valued measure θ̃ is indeed bounded. If V is a measurable weakly lower continuous function space neighborhood system and θ is the representation measure for an operator 𝒯 ∈ L(𝒞V (X, E), F), then ̃ as can immediately be understood from its prior definition. θ is regular, and so is θ, The L(ℂ, 𝒬L(E,F) )-valued measure θ̃ induces a function space neighborhood system Vθ̃ for complex-valued functions as elaborated in Section 2.5 and according to Proposition 2.5.4(b) defines the continuous linear operator ? 𝒯(θ,X) : 𝒞Ṽ (X, ℂ) → L(E, F) ̃ θ

̃ by 𝒯(θ,X) ̃ (φ) = ∫X φ d θ for φ ∈ 𝒞Vθ̃ (X, ℂ). We shall apply Theorem 2.3.28 in order to directly relate the operators 𝒯 and 𝒯(θ,B) ̃ . We proceed to establish that

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� 167

̃ ã ⊗ μ) = (∫ φ ⋅ a dθ)(μ) (∫ φ d θ)( X

X

holds for all φ ∈ 𝒞𝒦 (X, ℂ), all ã ⊗ μ ∈ L(E, F)˚ and a ∈ ã. Integrability of these functions with respect the measures θ̃ and θ is of course guaranteed due to the fact that φ has a compact support. First, we use Corollary 2.3.29 for ̃ ã ⊗ μ) = (∫ f d((ã ⊗ μ) ˝ θ))(+1). ̃ (∫ φ d θ)( X

X

Next, we use Theorem 2.3.28 with ℂ for both G and H and the operators S ∈ L(ℂ, E) such that z 󳨃→ za and T ∈ L(F, ℂ) such that c 󳨃→ μ(c). Then S ˝ φ = φ ⋅ a and Theorem 2.3.28 states that (∫ φ ⋅ a dθ)(μ) = (∫ f d(μ ˝ θ ˝ S))(+1). X

X

Both measures (ã ⊗ μ) ˝ θ̃ and μ ˝ θ ˝ S are complex-valued, that is, L(ℂ, Qℂ )-valued, and indeed equal, as for every A ∈ R and z ∈ ℂ we calculate that ̃ (z)(+1) = θ (za)(μ) = (μ ˝ θ ˝ S)(z)(+1). ((ã ⊗ μ) ˝ θ) A A We infer that ̃ ⊗ μ) = 𝒯 (φ ⋅ a)(μ) 𝒯(θ,X) ̃ (φ)(a holds for all φ ∈ 𝒞𝒦 (X, ℂ) and ã ⊗ μ ∈ L(E, F)˚ . Moreover, this argument also implies that 𝒯(θ,X) ̃ (φ) is an element of the space L(E, F) rather than its completion. Indeed, let φ ∈ 𝒞𝒦 (X, ℂ) and let A ∈ R be the support of φ. We examine the operator L : E → F such that L(a) = 𝒯 (φ⋅a) for all a ∈ E. The linearity of L is obvious, and for continuity let W ∈ 𝒲 . There is v ∈ V such 𝒯 (v) ⊂ W , and in turn there is V ∈ 𝒱 such that χA ⋅ V ⩽ v. Then φ ⋅ a ⩽ v for all a ∈ V and, therefore, L(V ) ⊂ W . Hence L ∈ L(E, F). But we have ̃ ⊗ μ) L(ã ⊗ μ) = 𝒯 (φ ⋅ a)(μ) = 𝒯(θ,X) ̃ (φ)(a for all ã ⊗ μ ∈ L(E, F)˚ and, therefore, 𝒯(θ,X) ̃ (φ) = L ∈ L(E, F),

our claim.

3 Choquet theory In this final chapter, we shall develop an extension of classical Choquet theory to spaces of linear operators on function spaces utilizing the results of Chapters 1 and 2. The generality of our setting requires substantial adaptations to the technology that is used in the original situation (see [1] or [41]), which deals with continuous linear functionals on spaces 𝒞 (X) of continuous real-valued functions on a compact set X, endowed with the topology of uniform convergence. There, given a subspace G of 𝒞 (X), an order relation is established in the dual of 𝒞 (X), that is, the space of all regular Borel measures on X. This order allows for maximal elements that deliver integral representations for continuous linear functionals on the subspace G with particular properties. The support of any such maximal representation measure is localized on the Choquet boundary, a subset of X characterized by G. The transfer of this technique to our situation is far from trivial. We deal with locally compact rather than compact domains X, with spaces 𝒞V (X, E) of vector-valued rather than real-valued functions, and with linear operators into a second-vector space F rather than linear functionals. In the classical situation, the order on its dual space 𝒞 (X)˚ is established using a suitable subcone of 𝒞 (X), and the choice of this subcone guarantees that the set of functionals that dominate a given one in this order is weak* compact. This assures the existence of maximal elements. All these features turn out to be problematic in our more general environment. We shall establish an order on the space L(𝒞V (X, E), F) of continuous linear operators from 𝒞V (X, E) into F, but the use of a subcone of 𝒞V (X, E) is no longer suitable for this purpose. We shall use sub and superlinear functionals on L(𝒞V (X, E), F) generated by certain subsets of the tensor product 𝒞V (X, E) ⊗ F ˚ instead. Section 3.1 is concerned with the study of this type of functionals and their properties. We shall however strive for minimal rather than maximal elements in this order, since this formulation fits more naturally into our general settings. This will be elaborated on in Section 3.2, which contains our main results. Some of the concepts and techniques in therein lean on ideas developed by the author in [47] and [52] and Batty in [9] for linear functionals on complexand vector-valued function spaces. Section 3.3 is concerned with some special cases and examples. The classical theory is based on the fundamental work by Choquet [14, 15, 18], Choquet and Deny [16], Choquet and Meyer [17], Mokobotzki [37, 38] and Bishop and de Leeuw [11]. Later material can be found in [9, 13, 28, 32, 34, 47, 52] and [59].

3.1 C(X)-convex sets and functionals We shall use the previously established notation and results and universally assume that the function space 𝒞V (X, E) satisfies the assumptions of Theorem 2.6.1. That is, the space X is locally compact and Hausdorff, and the σ-ring R consists of all relatively compact Borel subsets of X. The function space neighborhood system V is https://doi.org/10.1515/9783111315478-003

3.1 C(X)-convex sets and functionals

� 169

measurable and weakly lower continuous. The measures under consideration are supposed to be representation measures for some operator in L(𝒞V (X, E), F), hence are L(E, F ˚˚ )-valued and regular. We generally employ the weak operator topology on the operator space L(𝒞V (X, E), F), which was seen to be Hausdorff (see Section 1.3). Then according to Proposition 1.3.1 ? ? its dual is the tensor product 𝒞V (X, E) ⊗ F ˚ , where 𝒞V (X, E) is the quotient space 𝒞V (X, E)/𝒞V (X, E)0 and 𝒞V (X, E)0 = { f ∈ 𝒞V (X, E) | f ≤ v for all v ∈ V}

? (see Section 1.3). If 𝒞V (X, E) is Hausdorff, then 𝒞V (X, E) = 𝒞V (X, E). This is a requirement, which would limit our choice of function spaces (see Examples 1.2.7 and Proposition 1.2.6) and which we therefore prefer not to impose. Some of the subsequent deliberations would however be simplified in the Hausdorff case. If 𝒞V (X, E) is not Hausdorff, then the bilinear form n

n

i=1

i=1

(𝒯 , ∑ fi ⊗ μi ) 󳨃→ ∑ μi (𝒯 (fi )) on L(𝒞V (X, E), F) × 𝒞V (X, E) ⊗ F ˚ is degenerate. Nevertheless, every linear functional in L(𝒞V (X, E), F)˚ is represented by at least one element of 𝒞V (X, E) ⊗ F ˚ (Corollary 1.3.2), and every element of 𝒞V (X, E) ⊗ F ˚ defines a linear functional in L(𝒞V (X, E), F)˚ . Polars of sets and polar topologies are defined in the same way as in the nondegenerate case, and most results for the latter can be transferred using suitable quotient spaces. The weak operator topology of L(𝒞V (X, E), F) coincides with the weak topology induced by 𝒞V (X, E) ⊗ F ˚ , that is, the Hausdorff topology generated by the neighborhoods 󵄨󵄨 Re⟨𝒯 , ω ⟩ ≤ 1 󵄨 i V{ω1 ,...,ωn } = {𝒯 ∈ L(𝒞V (X, E), F) 󵄨󵄨󵄨 } 󵄨󵄨 for all i = 1, . . . , n corresponding to finitely many elements ω1 , . . . , ωn of 𝒞V (X, E) ⊗ F ˚ . It concurs with ? the weak topology of the duality of L(𝒞V (X, E), F) with its dual space 𝒞V (X, E) ⊗ F ˚ ˚ (Proposition 1.3.1). Similarly, we endow 𝒞V (X, E) ⊗ F with the weak topology induced by L(𝒞V (X, E), F) and generated by the neighborhoods 󵄨󵄨 Re⟨𝒯 , ω⟩ ≤ 1 󵄨 i U{𝒯1 ,...,𝒯n } = {ω ∈ 𝒞V (X, E) ⊗ F ˚ 󵄨󵄨󵄨 } 󵄨󵄨 for all i = 1, . . . , n corresponding to finitely many operators 𝒯1 , . . . , 𝒯n in L(𝒞V (X, E), F). This topology is Hausdorff if and only if 𝒞V (X, E) is Hausdorff. The dual of 𝒞V (X, E) ⊗ F ˚ under this topology is L(𝒞V (X, E), F).

170 � 3 Choquet theory We consider the pointwise order for ℝ- or ℝ-valued functions on the operator space L(𝒞V (X, E), F). The cone of all lower semicontinuous (with respect to the weak operator topology) ℝ-valued sublinear functionals on L(𝒞V (X, E), F) is denoted by P, and the cone of all upper semicontinuous ℝ-valued superlinear functionals by Q, that is, Q = −P. Their intersection P ∩ Q is the space of all real-valued, real-linear continuous functionals on L(𝒞V (X, E), F). Corresponding to a nonempty convex subset Ω of 𝒞V (X, E) ⊗ F ˚ we define sub and superlinear functionals pΩ ∈ P and qΩ ∈ Q by pΩ (𝒯 ) = sup{Re⟨𝒯 , ω⟩ | ω ∈ Ω} and qΩ (𝒯 ) = inf{Re⟨𝒯 , ω⟩ | ω ∈ Ω} for 𝒯 ∈ L(𝒞V (X, E), F). We observe that −pΩ = q(−Ω) . Following the above remarks, P ∩ Q consists of the functionals p{ω} = q{ω} for elements ω ∈ 𝒞V (X, E) ⊗ F ˚ , that is, the mappings 𝒯 󳨃→ Re⟨𝒯 , ω⟩,

for all 𝒯 ∈ L(𝒞V (X, E), F). Conversely, with a sublinear functional p ∈ 𝒫 we associate the subset of 𝒞V (X, E) ⊗ F ˚ , 󵄨󵄨 Re⟨𝒯 , ω⟩ ≤ p(𝒯 ) 󵄨 Ωp = {ω ∈ 𝒞V (X, E) ⊗ F ˚ 󵄨󵄨󵄨 }. 󵄨󵄨 for all 𝒯 ∈ L(𝒞V (X, E), F) Proposition 1.4.3(a) implies that there are sufficiently many continuous linear functionals on L(𝒞V (X, E), F) such that every p ∈ 𝒫 is the pointwise supremum of the real parts of the functionals that it dominates. Since every continuous linear functional on L(𝒞V (X, E), F) is represented by an element of 𝒞V (X, E) ⊗ F ˚ , we infer that p = pΩp . That is, every element of P is generated by a convex subset of 𝒞V (X, E) ⊗ F ˚ . We have p(Ω+Φ) = pΩ + pΦ and p(αΩ) = αpΩ for convex subsets Ω, Φ of 𝒞V (X, E) ⊗ F ˚ and α ≥ 0. Moreover, pΩ ≤ pΦ if and only if Ω ⊂ Φ, whereby the closure is taken in the weak topology of 𝒞V (X, E)⊗F ˚ (see Proposition 1.4.3(a) and the correspondence between the ? closures of sets in 𝒞V (X, E)⊗F ˚ and in 𝒞V (X, E)⊗F ˚ as elaborated in Section 1.3). Thus, pΩ = pΦ if and only if Ω = Φ. Dual statements apply to the superlinear functionals qΩ . Both pΩ and qΩ are real-valued if and only if Ω is bounded (Proposition 1.4.3(b)). We also notice that if Ω = conv(⋃i∈ℐ Ωi ), that is, the convex hull of the union of the sets Ωi , then pΩ = ∨i∈ℐ pΩi is the pointwise supremum of the functionals pΩi . If a functional p ∈ P (or q ∈ Q) is upper (or lower) semicontinuous at 0 ∈ L(𝒞V (X, E), F), then it is continuous and real-valued on all of L(𝒞V (X, E), F) (see Section 1.4). For a nonempty convex subset Ω of 𝒞V (X, E) ⊗ F ˚ , we denote HΩ = {𝒯 ∈ L(𝒞V (X, E), F) | pΩ (𝒯 ) ≤ 1},

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that is, the polar of Ω in L(𝒞V (X, E), F). This set is convex and closed in the weak operator topology of L(𝒞V (X, E), F). It is compact in the weak operator topology if and ? only if its polar in the dual 𝒞V (X, E) ⊗ F ˚ of L(𝒞V (X, E), F) is a neighborhood in a topology consistent with this duality (see Theorem IV.3.2 in [56]). 3.1.1 Lemma. Let Ω be a convex subset of 𝒞V (X, E) ⊗ F ˚ such that HΩ is compact in the weak operator topology, and let Φ be the closed convex hull of Ω ∪ {0}. (a) If pΩ (𝒯 ) ≤ 0 for 𝒯 ∈ L(𝒞V (X, E), F), then 𝒯 = 0. (b) Φ absorbs every bounded subset of 𝒞V (X, E) ⊗ F ˚ . (c) Λ ⊂ Λ + Φ for every convex subset Λ of 𝒞V (X, E) ⊗ F ˚ . (d) Let ψ be a linear functional on 𝒞V (X, E) ⊗ F ˚ that vanishes at all elements f ⊗ μ ∈ 𝒞V (X, E) ⊗ F ˚ such that f ∈ 𝒞V (X, E)0 . If Re ψ is bounded above on Φ, then there is T ∈ L(𝒞V (X, E), F) such that ψ(ω) = ⟨𝒯 , ω⟩ for all ω ∈ 𝒞V (X, E) ⊗ F ˚ . Proof. For Part (a), let 𝒯 ∈ L(𝒞V (X, E), F). If pΩ (𝒯 ) ≤ 0, then λ𝒯 ∈ HΩ for all λ ≥ 0. Given f ∈ 𝒞V (X, E) and μ ∈ F ˚ , the mapping 𝒮 󳨃→ μ(𝒮 (f )) is continuous, hence bounded on the compact set HΩ . Thus μ(𝒯 (f )) = 0 holds for all f ∈ 𝒞V (X, E) and μ ∈ F ˚ . This shows 𝒯 = 0. The arguments for Parts (b), (c) and (d) demand a closer look at the involved quotient spaces and their projections. In keeping with the notation from Section 1.3, for a function f ∈ 𝒞V (X, E), we denote by ̃f = f + 𝒞V (X, E)0 its projection into 𝒞V (X, E)/𝒞V (X, E)0 , and for n

ω = ∑ fi ⊗ μi ∈ 𝒞V (X, E) ⊗ F ˚ i=1

we set n

? ̃ = ∑ ̃fi ⊗ μi ∈ 𝒞V ω (X, E) ⊗ F ˚ . i=1

This notation will also be used for subsets of 𝒞V (X, E) and of 𝒞V (X, E) ⊗ F ˚ , respectively. Recall that every operator 𝒯 ∈ L(𝒞V (X, E), F) vanishes on 𝒞V (X, E)0 , that is, we ̃⟩ = ⟨𝒯 , ω⟩ for all ω ∈ 𝒞V (X, E) ⊗ F ˚ . If Φ is the polar of HΩ in 𝒞V (X, E) ⊗ have ⟨𝒯 , ω ˚ ̃ which ? ̃ is the polar of HΩ in 𝒞V F , then Φ (X, E) ⊗ F ˚ , that is, the second polar of Ω, ̃ is the closed convex hull of Ω ∪ {0} in its weak topology. Thus Φ is the closed convex ̃. hull of Ω ∪ {0} in 𝒞V (X, E) ⊗ F ˚ . We observe that η ∈ Φ whenever ω ∈ Φ and η̃ = ω ̃ is If HΩ is compact in the weak operator topology of L(𝒞V (X, E), F), then its polar Φ ? a neighborhood in 𝒞V (X, E) ⊗ F ˚ that is consistent in its duality with L(𝒞V (X, E), F). ̃ of a bounded subset Θ of 𝒞V (X, E)⊗F ˚ is bounded in 𝒞V ? The projection Θ (X, E)⊗F ˚ ̃ Hence Θ is absorbed by Φ, that is, our claim in Part (b). and, therefore, absorbed by Φ. ̃ If ω is an element of the closure of a convex subset Λ of 𝒞V (X, E) ⊗ F ˚ , then ω ̃ ̃ ̃ ? ̃ ∈ Λ + Φ = Λ + Φ. The latter follows, because the closure is in the closure of Λ. Hence ω ? ̃ coincides in all topologies of 𝒞V of Λ (X, E) ⊗ F ˚ that are consistent in its duality with

172 � 3 Choquet theory ̃ is a neighborhood in the Mackey topology. Thus we find L(𝒞V (X, E), F), and because Φ ? ̃. That is, η−λ ∈ Φ with some λ ∈ Λ, and since ω η ∈ Λ+Φ such that η̃ = ω − λ = η? − λ, also ω − λ ∈ Φ. We conclude that ω ∈ Λ + Φ, our claim in Part (c). A linear functional ψ on 𝒞V (X, E) ⊗ F ˚ that vanishes at all elements f ⊗ μ ∈ 𝒞V (X, E) ⊗ F ˚ such that f ∈ 𝒞V (X, E)0 , canonically corresponds to a linear functional ˚ ̃ on 𝒞 ? ψ setting V (X, E) ⊗ F n

n

i=1

i=1

̃ ∑ ̃f ⊗ μ ) = ψ(∑ f ⊗ μ ) ψ( i i i i ̃ is bounded above for any choice of fi ∈ ̃fi . If Re ψ is bounded above on Φ, then Re ψ ̃ on Φ, which was seen to be a neighborhood consistent with the duality. We infer that ? the linear functional ψ is continuous in the Mackey topology of 𝒞V (X, E) ⊗ F ˚ , hence ? the evaluation of an operator 𝒯 ∈ L(𝒞V (X, E), F) on 𝒞V (X, E) ⊗ F ˚ . That is, we have ̃ ω ̃) = ⟨𝒯 , ω ̃⟩ = ⟨𝒯 , ω⟩ ψ(ω) = ψ( for all ω ∈ 𝒞V (X, E) ⊗ F ˚ . Hence our claim in Part (d). Fortunately, we shall not be concerned with similar constructions that arise from the non-Hausdorff case in the sequel. We shall look at the elements of 𝒞V (X, E) ⊗ F ˚ as continuous linear functionals on L(𝒞V (X, E), F) and write ω(𝒯 )

for ⟨𝒯 , ω⟩,

whenever ω ∈ 𝒞V (X, E) ⊗ F ˚ and 𝒯 ∈ L(𝒞V (X, E), F). 𝒞 (X)-convex sets

In Section 3.2 below, we shall use a suitable family of functionals p ∈ P to define an order on L(𝒞V (X, E), F), which will yield minimal elements. Our purposes require these functionals to be additive for operators with mutually singular representation measures. We proceed as follows. For an element ω = ∑ni=1 fi ⊗ μi of 𝒞V (X, E) ⊗ F ˚ and a bounded function φ ∈ 𝒞 (X), we recall from Lemma 1.2.1(a) that φ ⋅ fi ∈ 𝒞V (X, E) and define n

φ ⋅ ω = ∑(φ ⋅ fi ) ⊗ μi ∈ 𝒞V (X, E) ⊗ F ˚ . i=1

We have (φ ⋅ ω)(𝒯 ) = ω(𝒯φ ) for all 𝒯 ∈ L(𝒞V (X, E), F). Similarly, for a subset Ω of 𝒞V (X, E) ⊗ F ˚ , we write φ ⋅ Ω = {φ ⋅ ω | ω ∈ Ω}. Clearly, φ ⋅ Ω is convex whenever Ω

is convex. The usual associative and distributive properties hold for this operation. We have φ ⋅ (ψ ⋅ Ω) = (φψ) ⋅ Ω

and

(φ + ψ) ⋅ Ω ⊂ φ ⋅ Ω + ψ ⋅ Ω

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for all bounded functions φ, ψ ∈ 𝒞 (X). A nonempty subset Ω of 𝒞V (X, E) ⊗ F ˚ is said to be 𝒞 (X)-convex if φ ⋅ ω1 + (1 − φ) ⋅ ω2 ∈ Ω whenever ω1 , ω2 ∈ Ω and 0 ≤ φ ≤ 1 for φ ∈ 𝒞 (X). This requirement implies of course convexity in the usual sense. Clearly, sums, scalar multiples and intersections of families of 𝒞 (X)-convex sets are again 𝒞 (X)-convex. If Ω is 𝒞 (X)-convex, so is φ ⋅ Ω for all bounded functions φ ∈ 𝒞 (X). Every subset of 𝒞V (X, E) ⊗ F ˚ possesses a 𝒞 (X)-convex hull. Correspondingly, a sublinear functional p ∈ P is said to be 𝒞 (X)-convex if there is a 𝒞 (X)-convex subset Ω of 𝒞V (X, E) ⊗ F ˚ such that p = pΩ . These functionals form a subcone of P. A dual notion applies to the superlinear functionals in Q. For the sake of simplicity, we shall formulate the following results only in terms of the sublinear functionals pΩ , but since −Ω is 𝒞 (X)-convex if and only Ω is 𝒞 (X)-convex, similar results apply to the corresponding superlinear functionals qΩ . We recall that 𝒞V (X, E)⊗F ˚ is endowed with the weak topology induced by L(𝒞V (X, E), F). The closure of a subset of 𝒞V (X, E) ⊗ F ˚ refers to this topology. 3.1.2 Proposition. If Ω is a convex subset of 𝒞V (X, E) ⊗ F ˚ such that pΩ (𝒯 ) = pΩ (𝒯φ ) + pΩ (𝒯1−φ ) for all 𝒯 ∈ L(𝒞V (X, E), F) and 0 ≤ φ ≤ 1 for φ ∈ 𝒞 (X), then Ω is 𝒞 (X)-convex. Proof. Let ω = φ ⋅ ω1 + (1 − φ) ⋅ ω2 for ω1 , ω2 ∈ Ω and 0 ≤ φ ≤ 1 for φ ∈ 𝒞 (X). Then both Re ω1 (𝒯 ) ≤ pΩ (𝒯 ) and Re ω2 (𝒯 ) ≤ pΩ (𝒯 ) holds for all 𝒯 ∈ L(𝒞V (X, E), F). Thus Re ω(𝒯 ) = Re ω1 (𝒯φ ) + Re ω2 (𝒯1−φ ) ≤ pΩ (𝒯φ ) + pΩ (𝒯1−φ ) = p(𝒯 ). Hence ω ∈ Ω by the prior remarks. 3.1.3 Corollary. The closure of a 𝒞 (X)-convex subset of 𝒞V (X, E)⊗F ˚ is also 𝒞 (X)-convex. Proof. Suppose that Ω ⊂ 𝒞V (X, E) ⊗ F ˚ is 𝒞 (X)-convex, let 𝒯 ∈ L(𝒞V (X, E), F) and 0 ≤ φ ≤ 1 for φ ∈ 𝒞 (X). Then pΩ (𝒯φ ) + pΩ (𝒯1−φ ) = sup {(φ ⋅ ω1 )(𝒯 ) + (1 − φ) ⋅ ω2 (𝒯 )} ω1 ,ω2 ∈Ω

= sup ω(𝒯 ) = pΩ (𝒯 ) ω∈Ω

since Ω is 𝒞 (X)-convex. Thus Proposition 3.1.2 applies. Consequently, a sublinear functional p ∈ P is 𝒞 (X)-convex if and only if the set Ωp ⊂ 𝒞V (X, E)⊗F ˚ is 𝒞 (X)-convex. The statement of Corollary 3.1.3 can be strengthened further as follows. 3.1.4 Proposition. If Ω is a 𝒞 (X)-convex subset of 𝒞V (X, E) ⊗ F ˚ and if φ1 , . . . , φn are bounded nonnegative functions in 𝒞 (X) such that ∑ni=1 φi is strictly positive, then (∑ni=1 φi ) ⋅ Ω = ∑ni=1 φi ⋅ Ω.

174 � 3 Choquet theory Proof. Clearly, (∑ni=1 φi ) ⋅ Ω ⊂ ∑ni=1 φi ⋅ Ω holds in any case. We shall verify the reverse inclusion first for the case that ∑ni=1 φi = 1. For n = 2, our claim follows from Corollary 3.1.3 in this case. Suppose it holds for some n ≥ 2 and let ω = ∑n+1 i=1 φi ⋅ ωi for n+1 ωi ∈ Ω and 0 ≤ φi ∈ 𝒞 (X) such that ∑i=1 φi = 1. Given any neighborhood Δ in the weak* topology of 𝒞V (X, E) ⊗ F ˚ , there is 0 < ε < 1 such that εφn+1 ⋅ (ωn+1 − ωn ) ∈ Δ. Set ψi = φi for i = 1, . . . , n − 1, ψn = φn + εφn+1 and ψn+1 = (1 − ε)φn+1 . Then 0 ≤ ψi n for all i, ∑n+1 i=1 ψi = 1 and setting ψ = ∑i=1 ψi we have ψ(x) > 0 for all x ∈ X. The latter follows since φi (x) > 0 for at least one i ∈ {1, . . . , n + 1}. The functions ψi /ψ are therefore nonnegative, continuous and ∑ni=1 ψi /ψ = 1. Thus according to our assumption η = ∑ni=1 (ψi /ψ) ⋅ ωi is an element of Ω. Moreover, since ψ + ψn+1 = 1, according to the first step of our argument, that is, the case n = 2, the element ζ = ψ ⋅ η + ψn+1 ⋅ ωn+1 is also contained in Ω. We have n−1

ζ = ∑ φi ⋅ ωi + (φn + εφn+1 ) ⋅ ωn + ((1 − ε)φn+1 ) ⋅ ωn+1 , i=1

hence ω − ζ = εφn+1 ⋅ (ωn+1 − ωn ) ∈ Δ by the above. Since the neighborhood Δ in 𝒞V (X, E) ⊗ F ˚ was arbitrarily chosen, this demonstrates that ω ∈ Ω, as claimed. Now for the general case suppose that ω = ∑ni=1 φi ⋅ωi for ωi ∈ Ω and 0 ≤ φi ∈ 𝒞 (X) and that the function φ = ∑ni=1 φi is strictly positive. The functions φi /φ are nonnegative, continuous and ∑ni=1 (φi /φ) = 1. Thus η = ∑ni=1 (φi /φ) ⋅ ωi is an element of Ω by our previous argument for the first case. Hence ω = φ ⋅ η ≤ φ ⋅ Ω, our claim. 3.1.5 Proposition. If Ω is a 𝒞 (X)-convex subset of 𝒞V (X, E) ⊗ F ˚ , then (a) pΩ (𝒯∑ni=1 φi ) = ∑ni=1 pΩ (𝒯φi ) for all 𝒯 ∈ L(𝒞V (X, E), F) and bounded nonnegative functions φi ∈ 𝒞 (X) such that ∑ni=1 φi is strictly positive. (b) pΩ (𝒯 + 𝒮 ) = pΩ (𝒯 ) + pΩ (𝒮 ) for all operators 𝒯 , 𝒮 ∈ L(𝒞V (X, E), F) with mutually singular representation measures. Proof. For (a), let 𝒯 ∈ L(𝒞V (X, E), F) and 0 ≤ φi ∈ 𝒞 (X) such that φ = ∑ni=1 φi is strictly positive. Using Proposition 3.1.4, we calculate n

∑ pΩ (𝒯φi ) = i=1

n

∑(φi ⋅ ωi )(𝒯 ) = sup ω(𝒯 ) = pφ⋅Ω (𝒯 ) = pΩ (𝒯φ ) = pΩ (𝒯φ ).

sup

ω1 ,...ωn ∈Ω i=1

ω∈φ⋅Ω

For Part (b), let 𝒯 , 𝒮 ∈ L(𝒞V (X, E), F) with mutually singular representation measures θ and ϑ, that is, there is B ∈ A supporting θ and such that X \ B supports ϑ. Then η = θ + ϑ is the measure representing the operator 𝒯 + 𝒮 . We have T(f ) = ∫ f dθ = ∫ f dθ X

B

and

S(f ) = ∫ f dϑ = ∫ f dϑ X

X\B

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for all f ∈ 𝒞V (X, E). According to Lemma 2.5.7, there is a net (φ)i∈ℐ of functions φi ∈ 𝒞 (X) such that 0 ≤ φi ≤ 1 and lim (T + S)φi (f ) = ∫ f dη = T(f ) i∈ℐ

B

for all f ∈ 𝒞V (X, E). Convergence is meant pointwise on F ˚ . This yields both lim (T + S)φi = T i∈ℐ

and

lim (T + S)1−φi = S, i∈ℐ

in the weak operator topology of L(𝒞V (X, E), F). The functional pΩ on L(𝒞V (X, E), F) is sublinear and lower semicontinuous. Sublinearity yields pΩ (𝒯 + 𝒮 ) ≤ pΩ (𝒯 ) + pΩ (𝒮 ). Lower semicontinuity yields pΩ (𝒯 ) ≤ lim inf pΩ ((𝒯 + 𝒮 )φi ) i∈ℐ

and pΩ (𝒮 ) ≤ lim inf pΩ ((𝒯 + 𝒮 )1−φi ). i∈ℐ

Hence using the usual limit rules together with Part (a), we obtain the reverse inequality, that is, pΩ (𝒯 ) + pΩ (𝒮 ) ≤ lim inf(pΩ ((𝒯 + 𝒮 )φi ) + pΩ ((𝒯 + 𝒮 )1−φi )) i∈ℐ

= lim inf(pΩ ((𝒯 + 𝒮 )φi + (𝒯 + 𝒮 )1−φi )) i∈ℐ

= pΩ (𝒯 + 𝒮 ). 3.1.6 Corollary. For every nonzero operator in L(𝒞V (X, E), F) that is not a point evaluation there are noncollinear operators 𝒯1 , 𝒯2 ∈ L(𝒞V (X, E), F) such that 𝒯 = 𝒯1 + 𝒯2 and pΩ (𝒯 ) = pΩ (𝒯1 ) + pΩ (𝒯2 ) for all 𝒞 (X)-convex subsets Ω of 𝒞V (X, E) ⊗ F ˚ . Proof. If 𝒯 ∈ L(𝒞V (X, E), F) is neither zero nor a point evaluation, then the support of its representing measure θ contains two distinct points x and y. There is φ ∈ 𝒞 (X) such that 0 ≤ φ ≤ 1, φ(x) = 0 and φ(y) = 1. Then 𝒯 = 𝒯φ + 𝒯1−φ and pΩ (𝒯 ) = pΩ (𝒯φ )+pΩ (𝒯1−φ ) for all 𝒞 (X)-convex subsets Ω of 𝒞V (X, E)⊗F ˚ by Proposition 3.1.5(a). Because the functions φ and 1 − φ are noncollinear, likewise are the operators 𝒯φ and 𝒯1−φ according to to Proposition 2.6.6. Ω-extremal point evaluations Given a 𝒞 (X)-convex subset Ω of 𝒞V (X, E) ⊗ F ˚ such that 0 ∈ Ω, we shall say that a point evaluation 𝒟xT ∈ L(𝒞V (X, E), F) is Ω-extremal if pΩ (𝒟xT ) < +∞ and if T = T1 +T2 such that pΩ (𝒟xT ) = pΩ (𝒟xT1 ) + pΩ (𝒟xT2 ) for T1 , T2 ∈ L(E, F) implies that pΩ (𝒟xT ) T1 = pΩ (𝒟xT1 ) T

and

pΩ (𝒟xT ) T2 = pΩ (𝒟xT2 ) T.

176 � 3 Choquet theory 3.1.7 Corollary. If Ω is a 𝒞 (X)-convex subset of 𝒞V (X, E) ⊗ F ˚ such that 0 ∈ Ω, then all nonzero extreme points of HΩ are Ω-extremal point evaluations 𝒟xT such that pΩ (𝒟xT ) = 1. If HΩ is compact in the weak operator topology and if all operators in HΩ are weakly compact, then all such point evaluations are extreme points of HΩ . Proof. We have pΩ (𝒯 ) ≥ 0 for all 𝒯 ∈ L(𝒞V (X, E), F) since 0 ∈ Ω. Let 0 ≠ 𝒯 ∈ HΩ . That is, pΩ (𝒯 ) ≤ 1. We go forward to list conditions that exclude 𝒯 from being an extreme point of HΩ . (i) If pΩ (𝒯 ) < 1, we choose pΩ (𝒯 ) < λ < 1 and set 𝒯1 = (1/λ)𝒯 and 𝒯2 = 0. Then 𝒯1 , 𝒯2 ∈ HΩ and 𝒯 = λ𝒯1 + (1 − λ)𝒯2 , hence 𝒯 is not an extreme point of HΩ in this case. (ii) If pΩ (𝒯 ) = 1 and 𝒯 = 𝒯1 + 𝒯2 for noncollinear operators 𝒯1 , 𝒯2 ∈ L(𝒞V (X, E), F) such that pΦ (𝒯 ) = pΦ (𝒯1 ) + pΦ (𝒯2 ) for all 𝒞 (X)-convex subsets Φ of 𝒞V (X, E) ⊗ F ˚ , we argue as follows: We abbreviate α1 = pΩ (𝒯1 ) and α2 = pΩ (𝒯2 ), that is, α1 + α2 = 1. If both α1 , α2 > 0, we set 𝒮1 = (1/α1 )𝒯1 and 𝒮2 = (1/α2 )𝒯2 . Then 𝒮1 , 𝒮2 ∈ HΩ since pΩ (𝒮1 ) = pΩ (𝒮2 ) = 1 and 𝒯 = α1 𝒮1 + α2 𝒮2 . Because 𝒯1 and 𝒯2 are noncollinear, the operators 𝒮1 and 𝒮2 are distinct. If α1 = 1 and α2 = 0, we set 𝒮1 = 𝒯1 and 𝒮2 = 𝒯1 + 2𝒯2 . Again we realize that 𝒮1 , 𝒮2 ∈ HΩ since pΩ (𝒮2 ) ≤ pΩ (𝒯1 ) + 2pΩ (𝒯2 ) = 1

and 𝒯 = (1/2)𝒮1 + (1/2)𝒯2 .

The case that α1 = 0 and α2 = 1 is similar. Thus in any case 𝒯 is a convex combination of distinct elements of HΩ and, therefore, not an extreme point of HΩ . Using this, we infer the following: (iii) If pΩ (𝒯 ) = 1 and 𝒯 is not a point evaluation, we choose 𝒯 = 𝒯1 + 𝒯2 for noncollinear operators T1 , 𝒯2 ∈ L(𝒞V (X, E), F) as in 3.1.6 and (ii) yields that 𝒯 is not an extreme point of HΩ in this case. (iv) If pΩ (𝒯 ) = 1, where 𝒯 = 𝒟xT is a point evaluation that is not Ω-extremal, then there are T1 , T2 ∈ L(E, F) such that T T T = T1 + T2 and α1 + α2 = 1, where α1 = pΩ (𝒟x 1 ) and α2 = pΩ (𝒟x 2 ), but α1 T ≠ T1 . If both α1 , α2 > 0, then both T1 , T2 ≠ 0 and we set S1 = (1/α1 )T1 and 𝒮2 = (1/α2 )T2 . Then S S S S S 𝒟x1 , 𝒟x2 ∈ HΩ and 𝒟xT = α1 𝒟x1 + α2 𝒟x2 . Since S1 ≠ T by the above, we have 𝒟xT ≠ 𝒟x1 T by Lemma 2.6.10. Hence 𝒟x is not an extreme point of HΩ in this case. If α1 = 1 and S S α2 = 0, we set S1 = T1 and S2 = T1 + 2T2 . Again we realize that 𝒟x1 , 𝒟x2 ∈ HΩ since pΩ (𝒟xS2 ) ≤ pΩ (𝒟xS1 ) + 2pΩ (𝒟xS2 ) = 1, S

S

and 𝒟xT = (1/2)𝒟x1 + (1/2)𝒟x2 . Moreover, since S1 ≠ T by the above, we infer 𝒟xT is not an extreme point of HΩ in this case. The case that α1 = 0 and α2 = 1 is similar. This completes our first line of arguments. Now suppose that HΩ is compact in the weak operator topology and that all its elements are weakly compact operators. Let 𝒟xT ∈ HΩ be an Ω-extremal point evaluation such that pΩ (𝒟xT ) = 1. We shall verify that 𝒟xT is an extreme point of HΩ . For this, suppose that 𝒟xT = α1 𝒯1 + α2 𝒯2 for 𝒯1 , 𝒯2 ∈ HΩ and α1 , α2 > 0 such that α1 + α2 = 1. Then according to Lemma 2.6.12 there are S, U ∈ L(E, F) and a weakly compact operator ℛ ∈ L(𝒞V (X, E), F) whose representation measure is supported by X \ {x} and such that α1 𝒯1 = 𝒟xS + ℛ and α2 𝒯2 = 𝒟xU − ℛ. We have

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pΩ (𝒟xS + ℛ) = pΩ (𝒟xS ) + pΩ (ℛ) by Proposition 3.1.5(b), so pΩ (𝒟xS ) ≤ α1 and likewise, pΩ (𝒟xU ) ≤ α2 . We have 𝒟xT = 𝒟xS + 𝒟xU , hence pΩ (𝒟xT ) ≤ p(𝒟xS ) + pΩ (𝒟xU ) and, therefore, pΩ (𝒟xS ) = α1 and pΩ (𝒟xU ) = α2 . This yields S = α1 T and U = α2 T since 𝒟xT is supposed to be Ω-extremal. The above also implies that pΩ (ℛ) = 0, hence ℛ = 0 by Lemma 3.1.1(a). Summarizing, we realize that α1 𝒯1 = α1 𝒟xT and α2 𝒯2 = α2 𝒟xT , that is, 𝒯1 = 𝒯2 = 𝒟xT . Our final claim follows. 3.1.8 Corollary. If Ω ⊂ 𝒞V (X, E) ⊗ F ˚ is 𝒞 (X)-convex, if 0 ∈ Ω and if HΩ is compact in the weak operator topology of L(𝒞V (X, E), F), then for every x ∈ X the set x

S

ℋΩ = {S ∈ L(E, F) | pΩ (𝒟x ) ≤ 1}

is convex and compact in the weak operator topology of L(E, F). A point evaluation 𝒟xT ∈ L(𝒞V (X, E), F) such that pΩ (𝒟xT ) < +∞ is Ω-extremal if and only if T is an extreme point of pΩ (𝒟xT ) ℋΩx . Proof. Let x ∈ X. Obviously, ℋΩx is a convex subset of L(E, F). Let (Ti )i∈ℐ be an T ultranet in ℋΩx . Then (𝒟x i )i∈ℐ is an ultranet in HΩ , hence convergent towards an element 𝒯 ∈ HΩ in the weak operator topology of L(𝒞V (X, E), F). We use Proposition 2.5.9 to argue that the representation measure for 𝒯 is supported by {x}. Indeed, T for all φ ∈ 𝒞𝒦 (X) and a ∈ E such that φ is supported by X \{x}, we have 𝒟x i (φ⋅a) = 0 for all i ∈ ℐ , hence 𝒯 (φ ⋅ a) = 0. This shows that 𝒯 = 𝒟xT for some T ∈ L(E, F). For T every a ∈ E, there is f ∈ 𝒞V (X, E) such that f (x) = a, hence 𝒟x i (f ) = Ti (a) conT verges weakly to 𝒟x (f ) = T(a). The ultranet (Ti )i∈ℐ therefore converges to T in ℋΩx , which is seen to be compact in the weak operator topology of L(E, F). For our second claim, let 𝒟xT ∈ L(𝒞V (X, E), F) be a point evaluation such that pΩ (𝒟xT ) < +∞. We have pΩ (𝒟xT ) > 0 by Lemma 3.1.1(a). If 𝒟xT is Ω-extremal and if T = λR + (1 − λ)S for R, S ∈ pΩ (𝒟xT ) ℋΩx and 0 < λ < 1, then 𝒟xT = 𝒟xλR + 𝒟x(1−λ)S and pΩ (𝒟xT ) ≤ pΩ (𝒟xλR ) + pΩ (𝒟x(1−λ)S )

= λpΩ (𝒟xR ) + (1 − λ)pΩ (𝒟xS )

≤ pΩ (𝒟xT ), that is, pΩ (𝒟xT ) = pΩ (𝒟xR ) = pΩ (𝒟xS ) and

pΩ (𝒟xT ) = pΩ (𝒟xλR ) + pΩ (𝒟x(1−λ)S ).

Hence pΩ (𝒟xT )(λR) = pΩ (𝒟xλR ) T = pΩ (𝒟xR )(λT)

178 � 3 Choquet theory by the above definition of 𝒟xT being Ω-extremal and, therefore, R = T and also S = T. Thus T is indeed an extreme point of pΩ (𝒟xT ) ℋΩx . Conversely, suppose that T is an extreme point of pΩ (𝒟xT ) ℋΩx and that T = T1 + T2 such that pΩ (𝒟xT ) = pΩ (𝒟xT1 ) + pΩ (𝒟xT2 ) for T1 , T2 ∈ L(E, F). There is nothing to prove if either T1 = 0 or T2 = 0. Otherwise, T T T we have both pΩ (𝒟x 1 ) > 0 and pΩ (𝒟x 2 ) > 0. Setting α1 = pΩ (𝒟x 1 )/pΩ (𝒟xT ) and α2 = T pΩ (𝒟x 2 )/pΩ (𝒟xT ), with S1 = (1/α1 )T1 and S2 = (1/α2 )T2 we have S1 , S2 ∈ pΩ (𝒟xT ) ℋΩx and T = α1 S1 + α2 S2 . Thus S1 = S2 = T since T is supposed to be an extreme point of pΩ (𝒟xT )ℋΩx . We conclude that both pΩ (𝒟xT )T1 = pΩ (𝒟xT1 )T

and

pΩ (𝒟xT )T2 = pΩ (𝒟xT2 )T

as is required for 𝒟xT to be Ω-extremal. Corollary 3.1.7 establishes the main motivation for the use of 𝒞 (X)-convex subsets of 𝒞V (X, E) ⊗ F ˚ and their associated functionals to define a Choquet-type order structure on L(𝒞V (X, E), F). To further our preparations, we shall introduce some concepts and examples how to build up suitable collections of these sets. For subsets u of 𝒞V (X, E)

and Π of F ˚ , we denote

n 󵄨󵄨 f ∈ u, μ ∈ Π, 0 ≤ φ ∈ 𝒞 (X) 󵄨 i i ΩΠu = {∑(φi ⋅ fi ) ⊗ μi 󵄨󵄨󵄨 i }. 󵄨󵄨 such that ∑ni=1 φi = 1 i=1

It is straightforward to verify that ΩΠu is a 𝒞 (X)-convex subset of 𝒞V (X, E) ⊗ F ˚ . We Π Π+Γ note that λΩΠu = ΩλΠ ⊂ ΩΠu + ΩΓu and u = Ωλu for λ in ℝ or in ℂ and that Ωu Π Π Π ˚ Ωu+w ⊂ Ωu + Ωw for subsets u, w of 𝒞V (X, E) and Π, Γ of F . The 𝒞 (X)-convex set ΩΠu corresponds to the functional in pΩΠu ∈ P. In order to compute the values of this functional, we return to the notation of the preceding sections and consider the embedding of F into 𝒬F . That is, for an element ω = f ⊗ μ ∈ 𝒞V (X, E) ⊗ F ˚ and an operator 𝒯 ∈ L(𝒞V (X, E), F) we write 𝒯 (f )(μ) for Re ω(𝒯 ) = Re μ(𝒯 (f )). We also use the ℝ-valued sublinear functional pΠ on F defined as pΠ (c) = sup Re μ(c) μ∈Π

for all c ∈ F. Utilizing this, we calculate that n 󵄨󵄨 f ∈ u, μ ∈ Π, 0 ≤ φ ∈ 𝒞 (X) 󵄨 i i pΩΠu (𝒯 ) = sup {∑(𝒯 (φi ⋅ fi )(μi )) 󵄨󵄨󵄨 i } 󵄨󵄨 such that ∑ni=1 φi = 1 i=1 n 󵄨󵄨󵄨 f ∈ u, 0 ≤ φi ∈ 𝒞 (X) = sup {∑ pΠ (𝒯 (φi ⋅ fi )) 󵄨󵄨󵄨 i } n 󵄨󵄨 such that ∑i=1 φi = 1 i=1

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� 179

for all 𝒯 ∈ L(𝒞V (X, E), F). We note that pΩΠu (𝒯 ) = pΩΠ (𝒯 ), where u and Π denote u

the respective closures of u in 𝒞V (X, E) in its given topology, and of Π in F ˚ in its weak* topology. The superlinear functional qΩΠu is similarly defined. 3.1.9 Proposition. If 𝒟xT ∈ L(𝒞V (X, E), F) is a point evaluation, then pΩΠu (𝒟xT ) = sup{pΠ (T(f (x))) | f ∈ u}.

Proof. If 𝒟xT ∈ L(𝒞V (X, E), F) is a point evaluation at x ∈ X, then 𝒟xT (f ) = T(f (x)) for all f ∈ 𝒞V (X, E). For any choice of fi ∈ u and 0 ≤ φi ∈ 𝒞 (X) such that ∑ni=1 φi = 1, there is i0 such that pΠ (T(fi0 (x))) = maxni=1 pΠ (T(fi (x))). Then n

n

i=1

i=1

∑ pΠ (T(φi (x)fi (x))) = ∑ φi (x) pΠ (T(fi (x))) ≤ pΠ (T(fi0 (x))) ∑ni=1 φi (x)

since is evident.

= 1. Thus pΩΠu (𝒟xT ) ≤ sup{pΠ (T(f (x))) | f ∈ u}. The reverse inequality

In keeping with the notation from Section 1.3, the symbol LΠu (𝒞V (X, E), F) will represent the set of all operators 𝒯 ∈ L(𝒞V (X, E), F) such that 𝒯 (u) ⊂ Π˝ . If pΩΠu (𝒯 ) ≤ 1, then Re μ(𝒯 (f )) ≤ 1 for all μ ∈ Π. Hence 𝒯 (f ) ∈ Π˝ for all f ∈ u, that is, 𝒯 ∈ ˝ LΠu (𝒞V (X, E), F). This renders ˝

HΩΠu ⊂ LΠu (𝒞V (X, E), F). ˝

Both sets HΩΠu and LΠu (𝒞V (X, E), F) were seen to be closed and convex in the weak ˝

operator topology of L(𝒞V (X, E), F). If the operator 𝒟xT ∈ L(𝒞V (X, E), F) is a point evaluation, then pΩΠu (𝒟xT ) = sup{pΠ (T(f (x))) | f ∈ u}

by Proposition 3.1.9. Thus 𝒟xT ∈ HΩΠu , that is, pΩΠu (𝒟xT ) ≤ 1, if and only if 𝒟xT ∈

LΠu (𝒞V (X, E), F). ˝ Corollary 1.3.9 identifies cases when the set LuΠ (𝒞V (X, E), F), hence also HΩΠu is compact in the weak operator topology. We summarize using Corollary 3.1.7 and the Krein–Milman theorem (Theorem VII.3.1.in [45]). ˝

3.1.10 Proposition. Let u be a subset of 𝒞V (X, E) and let Π ⊂ F ˚ such that ˝ LΠu (𝒞V (X, E), F) is compact in the weak operator topology. Then HΩΠu is also compact. If 0 ∈ ΩΠu , then HΩΠu is the closed convex hull of the set of all ΩΠu -extremal point evaluations together with 0.

We shall at times require that a subset Π of F ˚ satisfies one of the following conditions: (Π1) pΠ (c + d) = pΠ (c) + pΠ (d) for all c, d ∈ F such that pΠ (c), pΠ (d) < +∞. (Π2) pΠ (−c) = −pΠ (c) for all c ∈ F such that pΠ (c) < +∞.

180 � 3 Choquet theory These conditions are certainly stringent and fall just short of requiring that Π is a singleton set. We realize that (Π2) implies (Π1). Indeed, if (Π2) holds and if both pΠ (c), pΠ (d) < +∞, then pΠ (c + d) < +∞ and, therefore, pΠ (c) + pΠ (d) = −pΠ (−c) − pΠ (−d) ≤ −pΠ (−c − d) = pΠ (c + d). Condition (Π1) requires that pΠ is additive and lower semicontinuous on FΠ = {c ∈ F | pΠ (c) < +∞}, that is, the conic hull of the polar of Π in F. Condition (Π2) requires that FΠ is indeed a real-linear subspace of F and that pΠ coincides on FΠ with the evaluation of the real part of a linear functional in F ˚ . Singleton sets Π do obviously satisfy these criteria. If Π is a subcone of F ˚ , then pΠ (c) = 0 if c ∈ Π˝ and pΠ (c) = +∞ else. Thus Π satisfies (Π1), and indeed (Π2), provided it forms a real-linear subspace of F ˚ . For more elaborate settings, we refer to Example 3.1.16(c) below. We shall use the notion of 𝒞 (X)-convexity also for subsets of 𝒞V (X, E). A subset u of 𝒞V (X, E) is called 𝒞 (X)-convex (see [9]) if φ ⋅ f + (1 − φ) ⋅ g ∈ u whenever f , g ∈ u and 0 ≤ φ ≤ 1 for φ ∈ 𝒞 (X). If u ⊂ 𝒞V (X, E) is 𝒞 (X)-convex, then for every choice for F and μ ∈ F ˚ the set Ω = {f ⊗ μ | f ∈ u} is 𝒞 (X)-convex in 𝒞V (X, E) ⊗ F ˚ in the previously defined sense. If F equals ℝ or ℂ, that is, if L(𝒞V (X, E), F) = 𝒞V (X, E)˚ and Ω = {f ⊗ 1 | f ∈ u}, we observe that Ω = {f ⊗ 1 | f ∈ u}. Thus Proposition 3.1.4 applies with u, that is, the closure of u in 𝒞V (X, E), in place of Ω. We note that the function space neighborhoods in Examples 1.2.7(a) to (e) all give rise to 𝒞 (X)-convex neighborhoods in 𝒞V (X, E) . 3.1.11 Proposition. Let u be a 𝒞 (X)-convex subset of 𝒞V (X, E) and suppose that Π ⊂ F ˚ satisfies (Π1). If either 0 ∈ u or Π is a singleton set, then pΩΠu (𝒯 ) = sup{pΠ (𝒯 (f )) | f ∈ u} holds for all 𝒯 ∈ L(𝒞V (X, E), F). Consequently, ΩΠu is contained in the closed convex ˝ hull of the set {f ⊗ μ | f ∈ u, μ ∈ Π} and HΩΠu = LΠu (𝒞V (X, E), F). Proof. Let FΠ = {c ∈ F | pΠ (c) < +∞}. According to our assumption, pΠ is additive on FΠ . Let 𝒯 ∈ L(𝒞V (X, E), F). We first consider the case that 𝒯 (u) ⊂ FΠ . For any choice of fi ∈ u and 0 ≤ φi ∈ 𝒞 (X), such that ∑ni=1 φi = 1 according to Proposition 3.1.4 the function f = ∑i=1 φi ⋅ fi is contained in u. If 0 ∈ u, we have φi ⋅ fi ∈ u as well, hence T(φi ⋅ fi ) ∈ FΠ for all i = 1, . . . , n. The same holds true if FΠ = F, our second alternative. Our assumption on pΠ then renders n

n

i=1

i=1

∑ pΠ (𝒯 (φi ⋅ fi )) = pΠ (∑ 𝒯 (φi ⋅ fi )) = pΠ (T(f )). Hence

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� 181

pΩΠu (𝒯 ) = pΩΠ (𝒯 ) = sup{pΠ (𝒯 (f )) | f ∈ u} u

as claimed. In case that 𝒯 (u) ⊄ FΠ , there is f ∈ u such that 𝒯 (f ) ∉ FΠ , that is, pΠ (𝒯 (f )) = +∞. This yields pΩΠu (𝒯 ) = sup{pΠ (𝒯 (f )) | f ∈ u} = +∞ in this case as well. The last statement is an obvious consequence, since by the prior argument we have pΩΠu = pΦ , where Φ denotes the convex hull of {f ⊗μ | f ∈ u, μ ∈ Π}.

This implies that ΩΠu ⊂ Φ following the observations at the top of this section. Moreover, we have −1 Π HΩΠu = p−1 ΩΠ ((−∞, 1]) = pΦ ((−∞, 1]) = Lu (𝒞V (X, E), F). ˝

u

Hence our claim. If both u = {f } and Π = {μ} are singleton sets, Proposition 3.1.11 states that pΩ{μ} (𝒯 ) = Re μ(𝒯 (f )) {f }

holds for all 𝒯 ∈ L(𝒞V (X, E), F). 3.1.12 Corollary. Let n be a weakly lower continuous neighborhood function that defines a function space neighborhood u = {f ∈ 𝒞V (X, E) | f ≤ n}. Suppose that Π ⊂ F ˚ satisfies (Π1). If θ is the representation measure for 𝒯 ∈ L(𝒞V (X, E), F), then pΩΠu (𝒯 ) = sup(∫ n dθ)(μ). μ∈Π

X

Proof. According to Proposition 3.1.11, we have pΩΠu (𝒯 ) = sup{pΠ (𝒯 (f )) | f ∈ u} = sup{(𝒯 (f ))(μ) | f ∈ u, μ ∈ Π} = sup pΩ{μ} (𝒯 ). μ∈Π

u

Our claim follows with Proposition 2.3.25. That is, (∫ n dθ)(μ) = sup{(𝒯 (f ))(μ) | f ∈ u} = pΩ{μ} (𝒯 ) X

u

for all μ ∈ F ˚ . 3.1.13 Corollary. Let u and w be 𝒞 (X)-convex subsets of 𝒞V (X, E) and suppose that Π ⊂ F ˚ satisfies (Π1). If either 0 ∈ u ∩ w or Π is a singleton set, then pΩΠu+w = pΩΠu + pΩΠw .

182 � 3 Choquet theory Proof. We note that ΩΠu+w ⊂ ΩΠu + ΩΠw holds in general. Thus pΩΠu+w ≤ pΩΠu + pΩΠw . The reverse inequality is obvious for all operators 𝒯 ∈ L(𝒞V (X, E), F) such that pΩΠu+w (𝒯 ) = +∞. Otherwise, we have pΠ (T(f + g)) < +∞ for all f ∈ u and g ∈ w. If 0 ∈ u ∩ w, this implies that pΠ (𝒯 (f )), pΠ (𝒯 (g)) < +∞ holds for all f ∈ u and g ∈ w. If Π is a singleton set, the latter holds trivially true. Therefore, pΠ (𝒯 (f + g)) = pΠ (𝒯 (f )) + pΠ (𝒯 (g)) holds by (Π1). The assumptions of Proposition 3.1.11 apply for all of the sets u, w and u + w and we calculate that pΩΠu+w (𝒯 ) = sup{pΠ (𝒯 (f + g)) | f ∈ u, g ∈ w}

= sup{pΠ (𝒯 (f )) + pΠ (𝒯 (g)) | f ∈ u, g ∈ w} = pΩΠu (𝒯 ) + pΩΠw (𝒯 ),

our claim. As agreed earlier, we use the same symbol for a function space neighborhood and the subset of 𝒞V (X, E) determined by it. With this in mind, if u ⊂ 𝒞V (X, E) is a weakly lower continuous function space neighborhood in ℱ (X, E), we are able to calculate the value of pΩΠu (𝒯 ) in terms of the variation of the representation measure θ for 𝒯 . 3.1.14 Proposition. Let u be a measurable weakly lower continuous function space neighborhood, which is a neighborhood in 𝒞V (X, E) and let Π ⊂ F ˚ . If θ is the representation measure for an operator 𝒯 ∈ L(𝒞V (X, E), F), then n 󵄨󵄨 f ∈ u, μ ∈ Π, 0 ≤ φ ∈ 𝒞 (X) 󵄨 i i sup ‖θ‖(u,Π) = sup { 𝒯 (φi ⋅ fi )(μi ) 󵄨󵄨󵄨 i } ∑ A∩O 󵄨󵄨 such that ∑ni=1 φi ≤ χO A∈R i=1

for all open subsets O of X. In particular, pΩΠu (𝒯 ) = supA∈R ‖θ‖(u,Π) . A Proof. Let O be an open subset of X. A brief inspection of our definition of the variation of a measure in Section 2.1 together with the observation for regular measures from Proposition 2.1.14 shows that n 󵄨󵄨 󵄨󵄨 sup ‖θ‖(u,Π) A∩O = sup {∑ θAi (ai )(μi ) 󵄨󵄨󵄨 󵄨 A∈R i=1 n 󵄨󵄨 󵄨 = sup {∑ θKi (ai )(μi ) 󵄨󵄨󵄨 󵄨󵄨 i=1

Ai ∈ R disjoint, Ai ⊂ O, } ai ∈ E, χAi ⋅ ai ≤ u and μi ∈ Π Ki disjoint compact, Ki ⊂ O, }. ai ∈ E, χKi ⋅ ai ≤ u and μi ∈ Π

For i = 1, . . . , n, let fi ∈ u, let 0 ≤ φi ∈ 𝒞 (X) such that ∑ni=1 φi ≤ χO and μi ∈ Π. There is W ∈ 𝒲 such that μi ∈ W ˝ , and because V is finer that Vθ by the integral representation theorem 2.6.1, there is v ∈ V such that v ≤ v(θ,W ) . Hence |(∫X h dθ)(μi )| ≤ 1

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holds for all integrable functions h ≤ v and all μi . Since u is supposed to be a neighborhood in 𝒞V (X, E), we may assume that v ⊂ u. Let ε > 0. There is a compact subset K of X such that χX\K ⋅ fi ∈ εv for all fi . In turn, there is V ∈ 𝒱 such that χK ⋅ V ≤ εv. We find a disjoint partition of K ∩ O by subsets A1 , . . . , Am ∈ R such that both fi (x) − fi (y) ∈ V

and

φi (x)fi (x) − φi (y)fi (y) ∈ V

holds for all i = 1, . . . , n whenever x, y ∈ Ak for the same k. We fix elements xk ∈ Ak and define step functions m

si = ∑ χAk ⋅ (φi ⋅ fi )(xk ) k=1

for i = 1, . . . , n. We claim that φi ⋅ fi − si ∈ 2εv. Indeed, for x ∈ K ∩ O we have x ∈ Ak for some k = 1, . . . , m, hence (φi ⋅ fi )(x) − si (x) = (φi ⋅ fi )(x) − (φi ⋅ fi )(xk ) ∈ V and, therefore, χK∩O ⋅ (φi ⋅ fi − si ) ≤ εv by the above. Since the functions φi ⋅ fi vanish outside of O and the si vanish outside of K ∩ O, this yields φi ⋅ fi − si = χK∩O ⋅ (φi ⋅ fi − si ) + χO\K ⋅ (φi ⋅ fi − si ) = χK∩O ⋅ (φi ⋅ fi − si ) + χO\K ⋅ (φi ⋅ fi )

≤ εv + εv. Thus, since v ≤ v(θ,W ) ,

(∫ φi ⋅ fi dθ)(μi ) ≤ (∫ si )(μi ) + 2ε X

X m

= ∑ φi (xk )θAk (fi (xk ))(μi ) + 2ε. k=1

We note that ∫X φi ⋅ fi dθ = 𝒯 (φi ⋅ fi ) ∈ F. Now summing up the above inequality over i returns n

m

n

∑(𝒯 (φi ⋅ fi ))(μi ) ≤ ∑ ∑ φi (xk )θAk (fi (xk ))(μi ) + 2nε. i=1

k=1 i=1

By the above, for every i = 1, . . . , n and k = 1, . . . , m, we have fi (x) − fi (xk ) ∈ V for all x ∈ Ak , hence χAk ⋅ (fi − fi (xk )) ≤ εv ⊂ εu and, therefore, χAk ⋅ fi (xk ) ∈ (1 + ε)u. Hence from the definition of the variation ‖θ‖(u,Π) we infer that θAk (fi (xk ))(μi ) ≤ (1 + ε)‖θ‖(u,Π) . A k

184 � 3 Choquet theory Now continuing with the preceding inequalities and using that ∑ni=1 φi (xk ) ≤ 1 holds for all k, we realize that n

m

n

+ 2nε ∑(𝒯 (φi ⋅ fi ))(μi ) ≤ (1 + ε) ∑ ∑ φi (xk )‖θ‖(u,Π) A i=1

k

k=1 i=1 m

≤ (1 + ε) ∑ ‖θ‖(u,Π) + 2nε A k

k=1

≤ (1 + ε) ‖θ‖(u,Π) + 2nε O by the countable additivity of ‖θ‖(u,Π) . As the above inequality is valid for all ε > 0, we infer that n

∑(𝒯 (φi ⋅ fi ))(μi ) ≤ sup ‖θ‖(u,Π) A∩O A∈R

i=1

holds for any choice of functions fi ≤ u, 0 ≤ φi ∈ 𝒞 (X) such that ∑ni=1 φi ≤ χO and μi ∈ Π. Thus pΩΠu (𝒯 ) ≤ sup ‖θ‖(u,Π) A∩O . A∈R

For the reverse inequality, let Ki be disjoint compact subsets sets of O, let χKi ⋅ ai ∈ u and μi ∈ Π for i = 1, . . . , n. Let ε > 0. According to Lemma 1.2.4(a), since u is supposed to be weakly lower continuous there are disjoint open sets Ki ⊂ Oi ∈ O such that χOi ⋅ ai ∈ (1 + ε)u. We can therefore construct functions fi ∈ (1 + ε)u such that fi (x) = ai for all x ∈ Ki . Next, for each i = 1, . . . , n, we use Lemma 2.5.6 with the functions fi for f , the compact sets Ki for K and Oi for U to find functions φi ∈ 𝒞𝒦 (X) such that χKi ≤ φi ≤ χOi and |(𝒯 (φi ⋅ fi ) − θAi (ai ))(μi )| ≤ ε. We have ∑ni=1 φi ≤ χO since the sets Oi are disjoint subsets of O. Summing up over i then yields n

n

i=1

i=0

∑ θAi (ai )(μi ) ≤ ∑(𝒯 (φi ⋅ fi ))(μi ) + nε ≤ (1 + ε) pΩΠu (𝒯 ) + nε. We infer that ‖θ‖(u,Π) ≤ ∑ni=1 (𝒯 (φi ⋅ fi ))(μi ), thus completing our argument. Finally, for O the special case that O = X, we consider an expression n

∑(𝒯 (φi ⋅ fi ))(μi ) i=1

for fi ∈ u, 0 ≤ φi ∈ 𝒞 (X) such that ∑ni=1 φi ≤ 1 and μi ∈ Π, and set φ0 = 1 − ∑ni=1 φi ∈ n 𝒞 (X), set f0 = 0 ∈ u and choose any μ0 ∈ Π. Then ∑i=0 φi = 1 and n

n

i=0

i=1

∑ μi (𝒯 (φi ⋅ fi ))(μi ) = ∑(𝒯 (φi ⋅ fi ))(μi ).

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Hence the expression used in the proposition coincides with the one used in the definition of pΩΠu . Our final claim follows.

Elementary operators For fixed elements ν ∈ 𝒞V (X, E)˚ and e ∈ F, we define the elementary operator e

ℒν ∈ L(𝒞V (X, E), F)

as ℒeν (f ) = ν(f )e for all f ∈ 𝒞V (X, E). Finite rank operators are sums of elementary ones. In line with the conventions introduced in Section 1.3, the symbol bconv(A) stands for the balanced convex hull of a subset A of a locally convex topological vector space. Clearly, bconv(A) is bounded, whenever A is bounded. We augment this notation for subsets of 𝒞V (X, E), denoting by b𝒞X (u) the balanced 𝒞 (X)-convex hull of a subset u of 𝒞V (X, E), that is, the intersection of all balanced 𝒞 (X)-convex subsets of 𝒞V (X, E) that contain u. If all function space neighborhoods in V create 𝒞 (X)-convex neighborhoods in 𝒞V (X, E), then b𝒞X (u) is bounded whenever u is bounded. 3.1.15 Proposition. Let u be a 𝒞 (X)-convex subset of 𝒞V (X, E) and let Π ⊂ F ˚ . (a) If ν ∈ 𝒞V (X, E)˚ and e ∈ F, then 󵄨 󵄨 pΩΠu (ℒeν ) ≤ pbconv(Π) (e) sup{󵄨󵄨󵄨ν(f )󵄨󵄨󵄨 | f ∈ b𝒞X (u)}. (b) If ν ∈ 𝒞V (X, E)˚ , if e ∈ F is such that 0 ≤ μ(e) ∈ ℝ for all μ ∈ Π, and if either 0 ∈ u or Π is a singleton set, then pΩΠu (ℒeν ) = pΠ (e) sup{Re ν(f ) | f ∈ u}. (c) If b𝒞X (u) is bounded in 𝒞V (X, E) and Π is weak* bounded in F ˚ , then pΩΠu (𝒯 ) < e +∞ for every finite rank operator 𝒯 = ∑ni=1 ℒνii , where νi ∈ 𝒞V (X, E)˚ and ei ∈ F.

(d) If LΠu (𝒞V (X, E), F) is compact in the weak operator topology, then for any choice of νi ∈ b𝒞X (u)˝ , ei ∈ bconv(Π)˝ and λi ≥ 0 such that ∑∞ i=1 λi = 1 we have 𝒯 = ei ∞ ∑i=1 λi ℒνi ∈ HΩΠu . The series converges in the weak operator topology. ˝

Proof. Suppose that u ⊂ 𝒞V (X, E) is 𝒞 (X)-convex and that Π ⊂ F ˚ . Let ν ∈ 𝒞V (X, E)˚ and e ∈ F. We reformulate that n 󵄨󵄨 f ∈ u, μ ∈ Π, 0 ≤ φ ∈ 𝒞 (X) 󵄨 i i pΩΠu (ℒeν ) = sup {∑ Re μi (e)ν(φi ⋅ fi ) 󵄨󵄨󵄨 i }. 󵄨󵄨 such that ∑ni=1 φi = 1 i=1

186 � 3 Choquet theory For Part (a), we observe that pbconv(Π) (e) = sup{|μ(e)| | μ ∈ Π}. Let us consider a fixed choice of fi ∈ u, μi ∈ Π and φi ∈ 𝒞 (X) in the above expression. We set gi = γi fi ∈ b𝒞X (u), where |γi | = 1 such that Re ν(φi ⋅ gi ) = |ν(φi ⋅ fi )|. Then n

n

n

i=1

i=1

i=1

󵄨 󵄨󵄨 󵄨 ∑ Re μi (e)ν(φi ⋅ fi ) ≤ ∑󵄨󵄨󵄨μi (e)󵄨󵄨󵄨󵄨󵄨󵄨ν(φi ⋅ fi )󵄨󵄨󵄨 ≤ pbconv(Π) (e) Re ν(∑ φi ⋅ gi ). Since b𝒞X (u) is 𝒞 (X)-convex, we have ∑ni=1 φi ⋅ gi ∈ b𝒞X (u) (Proposition 3.1.4). Thus n

∑ Re μi (e)ν(φi ⋅ fi ) ≤ pbconv(Π) (e) sup{Re ν(f ) | f ∈ b𝒞X (u)} i=1

󵄨 󵄨 ≤ pbconv(Π) (e) sup{󵄨󵄨󵄨ν(f )󵄨󵄨󵄨 | f ∈ b𝒞X (u)}.

Our claim in Part (a) follows since 󵄨 󵄨 󵄨 󵄨 sup{󵄨󵄨󵄨ν(f )󵄨󵄨󵄨 | f ∈ b𝒞X (u)} = sup{󵄨󵄨󵄨ν(f )󵄨󵄨󵄨 | f ∈ b𝒞X (u)}. Now arguing Part (b), we suppose that 0 ≤ μ(e) ∈ ℝ holds for all μ ∈ Π and first assume that 0 ∈ u. For a fixed choice of fi ∈ u, μi ∈ Π and φi ∈ 𝒞 (X) we set gi = fi if Re ν(φi fi ) ≥ 0 and gi = 0 else. Then gi ∈ u and n

n

i=1

i=1

∑ Re μi (e)ν(φi ⋅ fi ) ≤ ∑ μi (e) Re ν(φi ⋅ gi ) n

≤ pΠ (e) ∑ Re ν(φi ⋅ gi ) i=1

n

= pΠ (e) Re ν(∑ φi ⋅ gi ). i=1

Since f = ∑ni=1 φi ⋅ gi ∈ u, this renders pΩΠu (ℒeν ) ≤ pΠ (e) sup{Re ν(f ) | f ∈ u}. For the reverse inequality, we recall that sup{ab | a ∈ A, b ∈ B} = (sup A)(sup B) holds for nonempty subsets A, B of ℝ such that all elements of A are nonnegative and such that sup B ≥ 0. With A = {μ(e) | μ ∈ Π}, that is, sup A = pΠ (e) and B = {Re ν(f ) | f ∈ u} we infer that pΠ (e) sup{Re ν(f ) | f ∈ u} = sup{μ(e) Re ν(f ) | f ∈ u, μ ∈ Π} ≤ pΩΠu (ℒeν ),

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hence our claim. The argument for the second alternative in (b), that is, the case that Π = {μ} is a singleton set is straightforward, because Proposition 3.1.11 returns immediately that pΩΠu (ℒeν ) = sup{pΠ (ν(f )e) | f ∈ u} = μ(e) sup{Re ν(f ) | f ∈ u} = pΠ (e) sup{Re ν(f ) | f ∈ u} holds in this case as well. Part (a) implies in particular pΩΠu (ℒeν ) ≤ 1 provided that ν ∈ e b𝒞X (u)˝ and e ∈ bconv(Π)˝ . Let 𝒯 (f ) = ∑ni=1 ℒνii be a finite rank operator, where νi ∈ 𝒞V (X, E)˚ and ei ∈ F. Suppose that b𝒞 (u) is bounded in 𝒞V (X, E) and that Π is weak* bounded in F ˚ . Since bconv(Π) is also weak* bounded, the polars b𝒞X (u)˝ and bconv(Π)˝ are absorbing in 𝒞V (X, E)˚ and F, respectively. Thus there is λ ≥ 0 such that both νi ∈ λ b𝒞X (u)˝ and ei ∈ λ bconv(Π)˝ for all i = 1, . . . , n. The former e implies that sup{|ν(f )| | f ∈ u} ≤ λ, the latter that pbconv(Π) (ei ) ≤ λ. Then pΩΠu (ℒνii ) ≤ λ2 ei for the elementary operators ℒνi . Thus n

pΩΠu (𝒯 ) ≤ ∑ pΩΠu (ℒeνii ) ≤ nλ2 , i=1

e

our claim in Part (c). Finally, since ℒνii ∈ HΩΠu whenever νi ∈ b𝒞X (u)˝ and ei ∈ bconv(Π)˝ , Part (d) follows with Proposition 1.3.10. We shall illustrate these concepts with a few examples. 3.1.16 Examples. (a) If F = ℝ or ℂ, that is, L(𝒞V (X, E), F) = 𝒞V (X, E)˚ (see Example 1.3.11(a)), then Proposition 3.1.11 applies with any 𝒞 (X)-convex neighborhood u of 0 ∈ 𝒞V (X, E) and Ψ = {1}. Then HΩΠu = u˝ . According to Corollary 1.3.9 and Proposition 3.1.10 the set HΩΠu is weak* compact and all of its extreme points are point evaluations. (b) Let u be a 𝒞 (X)-convex neighborhood of 0 ∈ 𝒞V (X, E) and let F be a semireflexive space. For Π, we choose a subset of F ˚ such that the closure of its balanced convex hull bconv(Π) is absorbing in F ˚ . Then bcore(Π˝ ) = bconv(Π)˝ is bounded, ˝ hence weakly compact in F, and according to Corollary 1.3.9 the set LΠu (𝒞V (X, E), F) is compact in the weak operator topology. Let νi ∈ b𝒞X (u)˝ , ei ∈ bconv(Π)˝ and ei ∞ λi ≥ 0 such that ∑∞ i=1 λi ≤ 1. Then 𝒯 = ∑i=1 λi ℒνi ∈ HΩΠu by Proposition 3.1.15(d). We have ∞

𝒯 (f ) = ∑ λi νi (f )ei i=1

for every f ∈ 𝒞V (X, E). Hence 𝒯 is a nuclear operator (see III.7.1 in [56]). Moreover, every nuclear operator 𝒯 ∈ L(𝒞V (X, E), F) is contained in such a subset HΩΠu for a

188 � 3 Choquet theory suitable choice of u and Π. However, not all operators in HΩΠu need to be nuclear, even if this set is compact in the weak operator topology. Indeed, according to Proposi˝ tion 3.1.10 HΩΠu contains all point evaluations in LΠu (𝒞V (X, E), F), which are generally not nuclear. For a concrete example, let F be the sequence space ℓp for 1 < p < +∞, let Π = 𝔹q be the unit ball of ℓq , where 1/p + 1/q = 1, and let ei be the i-th unit sequence in ℓp . For any choice of λi ≥ 0, such that ∑∞ i=1 λi ≤ 1 and νi ∈ 𝔹q the prior construction renders a nuclear operator 𝒯 ∈ HΩΠu such that 𝒯 (f ) = (λi νi (f ))i∈ℕ

for all f ∈ 𝒞V (X, E). However, not all operators in L(𝒞V (X, E), ℓp ) are of this type. For an example of an operator that is not absorbed by HΩΠu let X = [0, 1], let E = ℝ and let 𝒞 (X) be endowed with its usual supremum norm topology. Let u be the unit ball of 𝒞 (X). Define the operator 𝒯 ∈ L(𝒞V (X, ℝ), ℓp ) as 𝒯 (f ) = ((1/i)f (1/i))i∈ℕ . For every n ∈ ℕ, there are 0 ≤ φi ∈ 𝒞 (X) such that ∑ni=1 φi = 1, φi (1/i) = 1 and φi (1/k) = 0 for all i ≠ k. Then pΩ𝔹p (𝒯 ) ≥ ∑ni=1 1/i for all n ∈ ℕ and, therefore, pΩΠu (𝒯 ) = +∞. Hence v

𝒯 is absorbed by LΠ u (𝒞V (X, E), F), but not by HΩΠ . ˝

u

(c) Let E be an ordered topological vector space and assume that there is a neighborhood v ∈ V such that for u = v ∩ 𝒞V (X, E)+ the set b𝒞X (u) is a neighborhood of 0 ∈ 𝒞V (X, E). Let Y be a compact set and let F be the space of all regular real-valued Borel measures on Y , endowed with a polar topology of the dual pair (F, 𝒞 (Y )), where 𝒞 (Y ) is the space of all continuous functions on Y . Then F ˚ = 𝒞 (Y ) and the polar of every neighborhood in 𝒞 (Y ) is weakly compact in F. Both F and F ˚ carry their canonical order. Let ρ : Y → ℝ be a (strictly) positive-valued lower semicontinuous function and let Π = {φ ∈ 𝒞 (Y ) | φ ≤ ρ} ⊂ F ˚ . Then pΠ (η) = sup{η(φ) | φ ∈ Π} for all measures η ∈ F, that is, pΠ (η) = ∫Y ρ dη for all η ∈ F+ and pΠ (η) = +∞, else. The functional pΠ is additive but not necessarily continuous on F+ = {c ∈ F | pΠ (c) < +∞}, hence Π satisfies (Π1), but generally not (Π2). Proposition 3.1.11 applies to all operators 𝒯 ∈ L(𝒞V (X, E), F). The polar Π˝ of Π in F consists of all η ∈ F+ such that ∫Y ρ dη ≤ 1. Furthermore, Π˝ is a closed subset of (ε𝔹)˝ , where 𝔹 is the unit ball for the maximum norm in 𝒞 (Y ) and ε = min{ρ(y) | y ∈ Y } > 0. Hence Π˝ is weakly ˝ compact in F and Corollary 1.3.9 applies. The set HΩΠu = LΠu (𝒞V (X, E), F) is convex and compact in the weak operator topology; all of its elements are weakly compact operators and its extreme points are point evaluations. For a concrete example, let Y = {1/i | i ∈ ℕ} ∪ {0} endowed with the trace topology from ℝ. Then F = ℓ1 and F ˚ = c, the space of all convergent sequences in the ∞ 1 following sense: For μ = (αi )∞ i=1 ∈ c, set α0 = limi→∞ αi and for b = (βi )i=1 ∈ ℓ set ∞ μ(b) = ∑i=0 αi βi+1 . A strictly positive-valued lower semicontinuous function ρ on Y is represented by a sequence (ρi )∞ i=0 such that all ρi > 0 and ρ0 ≤ lim infn∈ℕ ρi . Then ∞ 1 pΠ ((αi )n∈ℕ ) = ∑i=0 ρi αi for all (αi )∞ i=0 ∈ ℓ+ and pΠ ((αi )n∈ℕ ) = +∞, else.

3.2 Choquet ordering for linear operators

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(d) The choice of F in Example (c) is a special case of the following: Let Q be a closed and locally compact subcone of F and Q ∩ (−Q) = {0}. Then Q has a compact μ base Bμ = μ−1 (1)∩Q for some μ ∈ F ˚ . (see Theorem II.2.6 in [1]). The convex hull B0 of Bμ and {0} is also compact. Let Γ be a convex subset of F ˚ such that Bμ is a compact μ base for Q for every μ ∈ Γ, and for all μ, ν ∈ Γ there is κ ∈ Γ such that B0κ ⊂ B0 ∩ B0ν . ˝ We set Π = Γ + Q . Then pΠ (c) = +∞ for all c ∈ F \ Q and pΠ (c) = sup{μ(c) | μ ∈ Γ} for all c ∈ Q. The above condition for Γ implies that pΠ is additive on Q. The same choice for u as in Part (c) then satisfies the requirements of Proposition 3.1.11. We observe that the concept of the variation of a measure can also be applied to its restriction with respect to a subspace G of E (see Proposition 2.6.8). That is, only elements ai ∈ G are used in the definition of the variation of the restricted measure. Proposition 3.1.14 then applies with the restriction of the corresponding operator to the subspace 𝒞V (X, G) of 𝒞V (X, E).

3.2 A Choquet ordering for linear operators on function spaces Throughout this section, we shall consider the weak operator topology on the space L(𝒞V (X, E), F), which coincides with the weak topology of its duality with 𝒞V (X, E)⊗F ˚ (see Proposition 1.3.1). Similarly, we endow 𝒞V (X, E) ⊗ F ˚ with the weak topology induced by L(𝒞V (X, E), F). Unfortunately, some of the concepts to be introduced in the beginning of this section will be less than intuitive and reveal their usefulness only later on. For a convex subset Ω of 𝒞V (X, E) ⊗ F ˚ , such that 0 ∈ Ω we say that a set Φ ⊂ 𝒞V (X, E) ⊗ F ˚ is Ω-bounded if Φ ⊂ λΩ, that is, pΦ ≤ λpΩ for the associated lower semicontinuous sublinear functionals pΦ , pΩ ∈ P, with some λ ≥ 0. Clearly, sums and positive multiples of Ω-bounded sets are again Ω-bounded. Similarly, Φ is D-bounded for a family D of such sets if it is Ω-bounded for all Ω ∈ D.

Choquet cone A Choquet core is a collection K of nonempty 𝒞 (X)-convex subsets of 𝒞V (X, E) ⊗ F ˚ , such that 0 ∈ Ω for every Ω ∈ K and such that the polar HΩ of Ω is compact in the weak operator topology of L(𝒞V (X, E), F). We also require that, given Φ, Ω ∈ K, the set Φ is contained in the union of all Ω-bounded sets in K, which are subsets of Φ, the closure of Φ in the weak* topology of 𝒞V (X, E) ⊗ F ˚ . A Choquet cone over K is a cone C of nonempty 𝒞 (X)-convex subsets of 𝒞V (X, E)⊗ F ˚ that is generated by K together with a collection of 𝒞 (X)-convex K-bounded subsets of 𝒞V (X, E) ⊗ F ˚ . Every set Φ ∈ C can be expressed as Φ = Ψ + ∑ni=1 λi Ωi , where Ψ ∈ C is K-bounded, where λi ≥ 0 and Ωi ∈ K. The sets in K themselves are generally not required to be K-bounded.

190 � 3 Choquet theory We take notice that as polars of compact convex subsets of L(𝒞V (X, E), F), according to Lemma 3.1.1(b), the closures of the sets in K absorb all bounded subsets of 𝒞V (X, E) ⊗ F ˚ , which are therefore K-bounded. We have pΩ (𝒯 ) ≥ 0 for all Ω ∈ K and 𝒯 ∈ L(𝒞V (X, E), F), and pΩ (𝒯 ) = 0 if and only if 𝒯 = 0 by Lemma 3.1.1(a). The boundedness requirement for the elements of a Choquet core is satisfied if the following more easily verifiable criterion holds: (i) The elements of sets in K are absorbed by all sets in K. (ii) Given Φ, Ω ∈ K and α ≥ 0 there is β ≥ α such that Φ ∩ (βΩ) ∈ K. Indeed, suppose that this criterion holds and let ω ∈ Φ. Since Ω absorbs ω by (i), by (ii) there is β ≥ 0 such that ω ∈ Φ ∩ (βΩ) and Φ ∩ (βΩ) ∈ K. Obviously, the set Φ ∩ (βΩ) is Ω-bounded and contained in Φ. Our claim follows. Given a Choquet core K we select a corresponding subset of operators in L(𝒞V (X, E), F) setting LK (𝒞V (X, E), F) = {𝒯 ∈ L(𝒞V (X, E), F) | pΩ (𝒯 ) < +∞ for some Ω ∈ K}. 3.2.1 Lemma. If K is not empty, then LK (𝒞V (X, E), F) is a subcone of L(𝒞V (X, E), F). Proof. Clearly, λ𝒯 ∈ LK (𝒞V (X, E), F) whenever 𝒯 ∈ LK (𝒞V (X, E), F) and λ ≥ 0. For 𝒮 , 𝒯 ∈ LK (𝒞V (X, E), F), there are Φ, Ω ∈ K such that pΦ (𝒮 ), pΩ (𝒯 ) < +∞. By our assumption on K, there is an Ω-bounded set Ψ ∈ K such that Ψ ⊂ Φ. Thus there is λ ≥ 0 such that pΨ (𝒮 + 𝒯 ) ≤ pΨ (𝒮 ) + pΨ (𝒯 ) ≤ pΦ (𝒮 ) + λpΩ (𝒯 ) < +∞. Hence 𝒮 + 𝒯 ∈ LK (𝒞V (X, E), F). Choquet ordering Corresponding to a Choquet cone C over K, we define an order relation ≺C on L(𝒞V (X, E), F) setting 𝒮 ≺C 𝒯

if pΩ (𝒮 ) ≤ pΩ (𝒯 )

(or q(−Ω) (𝒯 ) ≤ q(−Ω) (𝒮 ) )

for all

Ω ∈ C.

This order is reflexive and transitive, but not necessarily antisymmetric. It induces an equivalence relation on L(𝒞V (X, E), F), that is, 𝒮 ∼ 𝒯 if both 𝒮 ≺C 𝒯 and 𝒯 ≺C 𝒮 . It is compatible with the multiplication by nonnegative scalars, but generally not with the addition. However, Proposition 3.1.5 yields the following. 3.2.2 Proposition. (a) If 𝒮i ≺C 𝒯φi for 𝒮i , 𝒯 ∈ L(𝒞V (X, E), F) and bounded nonnegative functions φ1 , . . . , φn ∈ 𝒞 (X) such that ∑ni=1 φi is strictly positive, then ∑ni=1 𝒮i ≺C 𝒯∑ni=1 φi . (b) If 𝒮1 ≺C 𝒯1 and 𝒮2 ≺C 𝒯2 for 𝒮1 , 𝒮2 , 𝒯1 , 𝒯2 ∈ L(𝒞V (X, E), F) and if the operators 𝒯1 and 𝒯2 have mutually singular representation measures, then 𝒮1 + 𝒮2 ≺C 𝒯1 + 𝒯2 .

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Proof. (a) Let 𝒮i , 𝒯 and φi be as stated. We have n

n

n

pΩ (∑ 𝒮i ) ≤ ∑ pΩ (𝒮i ) ≤ ∑ pΩ (𝒯φi ) = pΩ (𝒯∑ni=1 φi ) i=1

i=1

i=1

for all Ω ∈ C by Proposition 3.1.5(a). Part (b) follows in similar fashion from 3.1.5(b).

Upper and lower envelopes We proceed to characterize the order ≺C in terms of upper and lower envelopes of functions in Q and P, respectively. For every q ∈ Q, we define its ℝ-valued upper envelope on L(𝒞V (X, E), F) by q̂(𝒯 ) = inf{pΩ (𝒯 ) | Ω ∈ C, q ≤ pΩ } for every 𝒯 ∈ L(𝒞V (X, E), F). Similarly, the ℝ-valued lower envelope for p ∈ P is q p(𝒯 ) = sup{q(−Ω) (𝒯 ) | Ω ∈ C, q(−Ω) ≤ p}.

We observe that −q̂(𝒯 ) = − inf{pΩ (𝒯 ) | Ω ∈ C, q ≤ pΩ } } 𝒯) = sup{q(−Ω) (𝒯 ) | Ω ∈ C, q(−Ω) ≤ −q} = (−q)(

holds for all 𝒯 ∈ L(𝒞V (X, E), F) and q ∈ Q. In this sense, both upper and lower envelopes can be equivalently used in many statements. In the sequel, we shall indicate these equivalences in brackets. Moreover, we have ? (q + r)(𝒯 ) ≤ q̂(𝒯 ) + ̂r(𝒯 ) and

̂ 𝒯 ) = αq̂(𝒯 ), (αq)(

and

} 𝒯 ) = αq (αp)( p(𝒯 ),

as well as ­ (p + s)(𝒯 ) ≥ q p(𝒯 ) + qs(𝒯 )

for 𝒯 ∈ L(𝒞V (X, E), F), α > 0 and q, r ∈ Q or p, s ∈ P, respectively, provided that the sums on the right-hand sides of the inequalities are defined in ℝ. Hence for a fixed operator 𝒯 ∈ L(𝒞V (X, E), F) the mapping p 󳨃→ q p(𝒯 ) : P → ℝ defines an extended superlinear functional on P in the sense of Section 1.4. On the subcone Q0 of Q consisting of all finitely-valued elements of Q, that is, all lower

192 � 3 Choquet theory semicontinuous superlinear functionals on L(𝒞V (X, E), F) defined by bounded convex subsets of 𝒞V (X, E) ⊗ F ˚ , the mapping q 󳨃→ q̂(𝒯 ) : Q0 → ℝ defines a sublinear functional. 3.2.3 Proposition. For 𝒮 , 𝒯 ∈ L(𝒞V (X, E), F), the following are equivalent: (i) 𝒮 ≺C 𝒯 . p(𝒯 ) ≤ q p(𝒮 ) for all p ∈ P). (ii) q̂(𝒮 ) ≤ q̂(𝒯 ) for all q ∈ Q (or q (iii) r(𝒮 ) ≤ ̂r(𝒯 ) for all r ∈ P ∩ Q (or qr(𝒯 ) ≤ r(𝒮 ) for all r ∈ P ∩ Q). Proof. (i) ⇒ (ii): If 𝒮 ≺C 𝒯 , then q̂(𝒮 ) = inf{pΩ (𝒮 ) | Ω ∈ C, q ≤ pΩ } ≤ inf{pΩ (𝒯 ) | Ω ∈ C, q ≤ pΩ } = q̂(𝒯 ) for every q ∈ Q. (ii) ⇒ (iii) is obvious since r(𝒮 ) ≤ ̂r(𝒯 ). For (iii) ⇒ (i), suppose that r(𝒮 ) ≤ ̂r(𝒮 ) holds for all r ∈ P ∩ Q and let Ω ∈ C. For every ω ∈ Ω, we have Re ω ∈ P ∩ Q and Re ω ≤ pΩ , hence ? Re ω(𝒮 ) ≤ Re ω(𝒯 ) = inf{pΦ (𝒯 ) | Φ ∈ C, Re ω ≤ pΦ } ≤ pΩ (𝒯 ) and, therefore, pΩ (𝒮 ) = sup{Re ω(𝒮 ) | ω ∈ Ω} ≤ pΩ (𝒯 ). 3.2.4 Lemma. Let q ∈ Q (or p ∈ P). For every Ω ∈ K, there is ρ ≥ 0 such that q̂ ≤ ρpΩ (or q p ≥ ρq−Ω ). In particular, if 𝒯 ∈ LK (𝒞V (X, E), F), then q̂(𝒯 ) ∈ ℝ and q p(𝒯 ) ∈ ℝ. Proof. We shall argue the case of the upper envelopes. Let q ∈ Q and Ω ∈ K. Since q is ℝ-valued and upper semicontinuous, it is bounded above by some ρ ≥ 0 on the compact set HΩ . This yields q ≤ ρpΩ . If 𝒯 ∈ LK (𝒞V (X, E), F), then pΩ (𝒯 ) < +∞ for some Ω ∈ K and, therefore, q̂(𝒯 ) ≤ ρpΩ (𝒯 ) < +∞. The argument for a functional p ∈ P is similar. 3.2.5 Lemma. Let Ω ∈ K, let q ∈ Q (or p ∈ P) and 𝒯 ∈ L(𝒞V (X, E), F) such that pΩ (𝒯 ) < +∞. Then q̂(𝒯 ) = inf{pϒ (𝒯 ) | ϒ ∈ C is Ω-bounded and q ≤ pϒ }

(or q p(𝒯 ) = sup{q(−ϒ) (𝒯 ) | ϒ ∈ C is Ω-bounded and q(−ϒ) ≤ p}). Proof. We shall prove the case for the upper envelope. Let Ω ∈ K, let q ∈ Q and 𝒯 ∈ L(𝒞V (X, E), F) such that pΩ (𝒯 ) < +∞. Clearly, q̂(𝒯 ) ≤ inf{pϒ (𝒯 ) | ϒ ∈ C is Ω-bounded and q ≤ pϒ } holds in any case. For the reverse inequality, let

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Λ = {ω ∈ 𝒞V (X, E) ⊗ F ˚ | Re ω ≥ q}. Then q = inf{Re ω | ω ∈ Λ} = qΛ = −p(−Λ) (see the introductory remarks in Section 3.1). Now let Φ ∈ C such that q ≤ pΦ , that is, 0 ≤ pΦ + p(−Λ) = pΦ−Λ . Thus p{0} ≤ pΦ−Λ , which implies that 0 ∈ Φ − Λ, the closure of the convex set Φ − Λ in 𝒞V (X, E) ⊗ F. Thus, given ε > 0, Lemma 3.1.1(c) yields that 0 ∈ (Φ − Λ) + εΩ = Φ − (Λ − εΩ). Hence we find an element ω ∈ Φ ∩ (Λ − εΩ). We have Φ = Ψ + ∑ni=1 λi Ωi , where Ψ ∈ C is K-bounded, λi ≥ 0 and Ωi ∈ K. Thus ω = ψ+∑ni=1 λi ωi with ψ ∈ Ψ and ωi ∈ Ωi . By ̃i ∈ K such that ωi ∈ Ω ̃i ⊂ Ωi . We set our assumption on K, there are Ω-bounded sets Ω n

̃i ∈ C ϒ = Ψ + ∑ λi Ω i=1

and observe that ϒ is Ω-bounded, that ω ∈ ϒ and ϒ ⊂ Ψ + ∑ni=1 λi Ωi . Thus n

n

pϒ ≤ pΨ + ∑ λi p Ω = pΨ + ∑ λi pΩi = pΦ . i=1

i

i=1

Moreover, ω ∈ ϒ ∩ (Λ − εΩ) implies that 0 ∈ ϒ + (−Λ + εΩ) and, therefore, q = −p(−Λ) ≤ p(ϒ+εΩ) = p(ϒ+εΩ) . We set Γ = ϒ+εΩ ∈ C and observe that Γ is also Ω-bounded and that q ≤ pΓ . Moreover, pΓ (𝒯 ) = pϒ (𝒯 ) + εpΩ (𝒯 ) ≤ pΦ (𝒯 ) + εpΩ (𝒯 ). Since pΩ (𝒯 ) < +∞ and since ε > 0 was arbitrarily chosen, we infer that inf{pΓ (𝒯 ) | Γ ∈ C is Ω-bounded and q ≤ pϒ } ≤ pΦ (𝒯 ). The latter holds true for all Φ ∈ C such that q ≤ pΦ . Our claim follows. 3.2.6 Proposition. For every q ∈ Q (or p ∈ P) and 𝒯 ∈ LK (𝒞V (X, E), F), we have q̂(𝒯 ) = max{q(𝒮 ) | 𝒮 ∈ L(𝒞V (X, E), F), 𝒮 ≺C 𝒯 } (or q p(𝒯 ) = min{p(𝒮 ) | 𝒮 ∈ L(𝒞V (X, E), F), 𝒮 ≺C 𝒯 }).

194 � 3 Choquet theory Proof. We shall prove the case for the lower envelope q p of a function p ∈ P. First, we observe that P is a locally convex cone (see Section 1.4, also [33] and [48]), endowed with its pointwise order and the neighborhoods VH for P, corresponding to compact (in the weak operator topology) convex subsets H of L(𝒞V (X, E), F) such that 0 ∈ H. We set g ≤ h + VH for g, h ∈ P, if g(𝒮 ) ≤ h(S) + 1 for all 𝒮 ∈ H. Every functional g ∈ P is bounded below on H, hence there is λ ≥ 0 such that 0 ≤ g + λVH , and g is seen to be bounded below as an element of P as is required for ˝ locally convex cones. The polar VH of VH in the dual cone P˚ of P consists of all linear functionals Θ on P such that Θ(g) ≤ Θ(h) + 1 whenever g ≤ h + VH . Now let p ∈ P and 𝒯 ∈ LK (𝒞V (X, E), F). Since our statement holds for positive multiples of 𝒯 whenever it holds for 𝒯 , we may assume that there is Ω ∈ K such that pΩ (𝒯 ) ≤ 1. We define ℝ-valued sub and extended superlinear functionals p and q on the cone P setting p(g) = αq p(𝒯 ) if g = αp for α ≥ 0, and p(g) = +∞ else, and q(g) = q g(𝒯 ), respectively, for g ∈ P. We shall use the sandwich theorem 1.4.1 for locally convex cones with the functionals p and q and the neighborhood VHΩ . That is, we need to verify that g ≤ h + vH for g, h ∈ P implies that q(g) ≤ p(h) + 1. For h ≠ αp, there is nothing to prove. Otherwise, g ≤ αp + VHΩ for some α ≥ 0 is equivalent to g ≤ αp + pΩ . Hence q−Φ ≤ g for Φ ∈ C implies that q−Φ − pΩ = q−Φ + q−Ω = q−(Φ+Ω) ≤ αp. Thus q−(Φ+Ω) ≤ αq p since Φ + Ω ∈ C. Hence q−Φ ≤ αq p + pΩ . The last inequality applies to all q−Φ ≤ g for Φ ∈ C and, therefore, yields q g ≤ αq p + pΩ . Thus in particular q(g) = q g(𝒯 ) ≤ αq p(𝒯 ) + pΩ (𝒯 ) ≤ p(αp) + 1, ˝ as required. We infer that there is a linear functional Θ on P such that Θ ∈ VH and Ω q ≤ Θ ≤ p. The formula

ω 󳨃→ Θ(p{ω} ) defines a real-linear functional on 𝒞V (X, E) ⊗ F ˚ , that is, in the complex case, the real part of a linear functional ψ on 𝒞V (X, E) ⊗ F ˚ . For f ⊗ μ ∈ 𝒞V (X, E) ⊗ F ˚ such that f ∈ 𝒞V (X, E)0 , we have p{f ⊗μ} = 0, because 𝒯 (f ) = 0 for all 𝒯 ∈ L(𝒞V (X, E), F). Thus ˝ Θ(p{f ⊗μ} ) = 0 and, therefore, ψ(f ⊗μ) = 0. Moreover, Θ ∈ VH implies that Θ(p{ω} ) ≤ 1, Ω

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that is, Re ψ(ω) ≤ 1, for all p{ω} ≤ VHΩ . The latter means that p{ω} (ℛ) = Re ω(ℛ) ≤ 1 holds for all ℛ ∈ VHΩ ; or equivalently, that ω ∈ Ω, the second polar of Ω. Therefore, following Lemma 3.1.1(d) there is 𝒮 ∈ L(𝒞V (X, E), F) such that φ(ω) = ω(𝒮 ) for all ω ∈ 𝒞V (X, E) ⊗ F ˚ . For every r ∈ P ∩ Q, there is ω ∈ 𝒞V (X, E) ⊗ F ˚ such that r = p{ω} and r(𝒮 ) = Re ω(𝒮 ) = Re ψ(ω) = Θ(r). Since Θ(r) ≥ q(r) = qr(𝒯 ), we infer that r(𝒮 ) ≥ qr(𝒯 ) holds for all r ∈ P ∩ Q and, therefore, that 𝒮 ≺C 𝒯 according to Proposition 3.2.3. All that is left to show is that p(𝒮 ) = q p(𝒯 ) holds true for the given functional p ∈ P. For this, let Ωp = {ω ∈ 𝒞V (X, E) ⊗ F ˚ | Re ω ≤ p}. Then p(𝒮 ) = sup{Re ω(𝒮 ) | ω ∈ Ωp } = sup{Θ(Re ω) | Re ω ≤ p} ≤ Θ(p) ≤ q p(𝒯 ). On the other hand, q p(𝒯 ) ≤ q p(ℛ) ≤ p(ℛ) follows with Proposition 3.2.3(ii) for all q ℛ ≺C 𝒯 . This implies p(𝒮 ) = p(𝒯 ) in particular and completes our argument. We note that p(𝒯 ) can take the value +∞ in our line of reasoning. Replacing p ∈ P by −q for q ∈ Q yields the statement for the upper envelopes. 3.2.7 Lemma. For every 𝒯 ∈ L(𝒞V (X, E), F), the set C 𝒯 ↓ = {𝒮 ∈ L(𝒞V (X, E), F) | 𝒮 ≺C 𝒯 }

is closed and convex. If 𝒯 ∈ LK (𝒞V (X, E), F), then 𝒯 ↓C is compact in the weak operator topology. Proof. If R, 𝒮 ≺C 𝒯 and 0 ≤ λ ≤ 1, then pΩ (λR + (1 − λ)𝒮 ) ≤ λpΩ (R) + (1 − λ)pΩ (𝒮 ) ≤ pΩ (𝒯 )

for all

Ω∈C

by the sublinearity of the functionals pΩ . Hence λR+(1−λ)𝒮 ≺C 𝒯 , and the set 𝒯 ↓C is seen to be convex. 𝒯 ↓C is closed, since it is the intersection of the inverse images of the intervals (−∞, pu (Ω)] under the lower semicontinuous mappings pΩ , for all Ω ∈ C. If 𝒯 ∈ LK (𝒞V (X, E), F), then there is Ω ∈ K such pΩ (𝒯 ) ≤ λ for some λ ≥ 0. Since 𝒯 ↓C ⊂ λHΩ , it is compact.

196 � 3 Choquet theory Minimal elements An element 𝒯 ∈ L(𝒞V (X, E), F) is minimal if 𝒮 ≺C 𝒯 for 𝒮 ∈ L(𝒞V (X, E), F) implies that 𝒮 ∼ 𝒯 . Since 𝒮 ≺C 𝒯 if and only if λ𝒮 ≺C λ𝒯 for all λ > 0, minimality of an element 𝒯 ∈ L(𝒞V (X, E), F) implies the minimality of λ𝒯 for all λ > 0. If K is not empty, then the zero operator 0 ∈ L(𝒞V (X, E), F) is minimal, since 𝒮 ≺C 0 implies that pΩ (𝒮 ) = 0 for all Ω ∈ K, hence 𝒮 = 0 by Lemma 3.1.1(a). 3.2.8 Proposition. Suppose that 𝒯 ∈ L(𝒞V (X, E), F) is minimal and that pΩ (𝒯 ) < +∞ for all Ω ∈ C. (a) If φ is a bounded nonnegative function in 𝒞 (X), then 𝒯φ is also minimal. (b) If 𝒯 = 𝒯1 + 𝒯2 such that the representation measures for 𝒯1 , 𝒯2 ∈ L(𝒞V (X, E), F) are mutually singular, then both 𝒯1 and 𝒯2 are also minimal. Proof. (a) Let 𝒯 and φ be as stated and let ρ > 0 be an upper bound for φ. We have 𝒯ρ = 𝒯φ + 𝒯ρ−φ and ρ pΩ (𝒯 ) = pΩ (𝒯ρ ) = pΩ (𝒯φ ) + pΩ (𝒯ρ−φ ) for all Ω ∈ C by Proposition 3.1.5(a). In particular, pΩ (𝒯φ ), pΩ (𝒯ρ−φ ) < +∞. We continue to verify that 𝒯φ is minimal. Let 𝒮 ≺C 𝒯φ . Then 𝒮 + 𝒯ρ−φ ≺C 𝒯φ + 𝒯ρ−φ = 𝒯ρ

by Proposition 3.2.2(a). The latter implies that 𝒯ρ ≺C 𝒮 + 𝒯ρ−φ since 𝒯ρ = ρ𝒯 is minimal. Therefore, pΩ (𝒯φ ) + pΩ (𝒯ρ−φ ) = pΩ (𝒯ρ ) ≤ pΩ (𝒮 + 𝒯ρ−φ ) ≤ pΩ (𝒮 ) + pΩ (𝒯ρ−φ ) holds for all Ω ∈ C. Because pΩ (𝒯ρ−φ ) < +∞, we infer that pΩ (𝒯φ ) ≤ pΩ (𝒮 ), hence 𝒯φ ≺C 𝒮 . Our claim follows. Part (b) is argued in a similar fashion using Propositions 3.1.5(b) and 3.2.2(b). 3.2.9 Proposition. For every operator 𝒯 ∈ LK (𝒞V (X, E), F) there is a minimal element 𝒯 ∈ LK (𝒞V (X, E), F) such that 𝒯 ≺C 𝒯 . Proof. Let 𝒯 ∈ LK (𝒞V (X, E), F). Given a totally ordered subset Λ of the compact convex set 𝒯 ↓C , the intersection of the sets 𝒮 ↓C , for all 𝒮 ∈ Λ, cannot be empty, hence contains a lower bound for Λ in 𝒯 ↓C . By Zorn’s lemma, there is a minimal element in 𝒯 ↓C. 3.2.10 Proposition. For 𝒯 ∈ LK (𝒞V (X, E), F), the following are equivalent: (i) 𝒯 is minimal. (ii) q̂(𝒯 ) = max{q(𝒮 ) | 𝒮 ∼ 𝒯 } for all q ∈ Q (or q p(𝒯 ) = min{p(𝒮 ) | 𝒮 ∼ 𝒯 } for all p ∈ P).

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(iii) q̂ −Ω (𝒯 ) = q−Ω (𝒯 ) for all Ω ∈ C (or p|Ω (𝒯 ) = pΩ (𝒯 ) for all Ω ∈ C). (iv) For every Ω ∈ C and ε > 0, there is Φ ∈ C such that 0 ∈ Ω + Φ and p(Ω+Φ) (𝒯 ) ≤ ε. Proof. Clearly, (ii) implies (iii), since q−Ω ∈ Q for every Ω ∈ C, and 𝒮 ∼ 𝒯 implies that q−Ω (𝒮 ) = q−Ω (𝒯 ). Now suppose that (iii) holds and let 𝒮 ≺C 𝒯 . Then for every Ω ∈ C we have ̂ q−Ω (𝒮 ) ≤ q̂ −Ω (𝒮 ) ≤ q −Ω (𝒯 ) = q−Ω (𝒯 ) by Proposition 3.2.3(ii). This yields 𝒯 ≺C 𝒮 , hence the minimality of 𝒯 and, therefore, (i). If 𝒯 is minimal and q ∈ Q, then q̂(𝒯 ) = max{q(𝒮 ) | 𝒮 ≺C 𝒯 } = max{q(𝒮 ) | 𝒮 ∼ 𝒯 } by Proposition 3.2.6. Therefore, (i) implies (ii) as well. Finally, (iv) is just a reformulation of (iii), since q−Ω ≤ pΦ for Ω, Φ ∈ C is equivalent to 0 ≤ pΩ + pΦ = p(Ω+Φ) , that is, 0 ∈ Ω + Φ, and pΦ (𝒯 ) ≤ q−Ω (𝒯 ) + ε is equivalent to p(Ω+Φ) (𝒯 ) ≤ ε. The Choquet boundary As in classical Choquet theory, we try to characterize minimal operators in L(𝒞V (X, E), F) and their representation measures in terms of supporting subsets of X. For this, we introduce an analogue to the classical Choquet boundary. For a set Ω ∈ K, we define 󵄨󵄨 x ∈ X, T ∈ L(E, F), 𝒟T is nonzero, 󵄨 x ΔΩC = {𝒟xT ∈ L(𝒞V (X, E), F) 󵄨󵄨󵄨 } 󵄨󵄨 Ω-extremal and minimal and 𝜕CΩ = {x ∈ X | 𝒟xT ∈ ΔΩC for some T ∈ L(E, F)}. Furthermore, we denote ΔC = ⋃{ΔΩC | Ω ∈ K} and

𝜕C = ⋃{𝜕CΩ | Ω ∈ K}.

We realize that Corollary 3.1.8 for the characterization of Ω-extremal point evaluations applies. We shall say that a point evaluation 𝒟xT is K-extremal if it is Ω -extremal for some Ω ∈ K. Thus ΔC consists of all K-extremal point evaluations in L(𝒞V (X, E), F). Note that our general settings allow that ΔC is empty. Indeed, choose C = K = {𝒞V (X, E) ⊗ F ˚ }. Then pΩ (𝒯 ) = +∞ for all 𝒯 ≠ 0 and Ω ∈ K. Hence there are no nonzero point evaluations in LK (𝒞V (X, E), F). We shall further elaborate on this in Corollary 3.2.12 below.

198 � 3 Choquet theory The main tool for advancing the theory is an analogue to the classical case (Proposition I.4.10 in [1]). However, the presence of unbounded and indeed infinity-valued functions defining the Choquet ordering will complicate matters. Let Ω ∈ K. We shall say that a function p ∈ P is Ω-bounded above if p ≤ λpΩ , that is, if Ωp = {ω ∈ 𝒞V (X, E) ⊗ F ˚ | Re ω ≤ p} ⊂ λΩ for some λ ≥ 0, that is, if the set Ωp is Ω-bounded. Similarly, q ∈ Q is Ω-bounded below on HΩ if q ≥ λq−Ω , that is, if {ω ∈ 𝒞V (X, E) ⊗ F ˚ | Re ω ≥ q} ⊂ −λΩ for some λ ≥ 0. A sequence (pn )n∈ℕ in P or (qn )n∈ℕ in Q is Ω-bounded above or below if the respective property holds with the same λ for all n ∈ ℕ. 3.2.11 Theorem. Let Ω ∈ K and let (qn )n∈ℕ be an Ω-bounded below sequence in Q (or (pn )n∈ℕ an Ω-bounded above sequence in P ). Let α ≥ 0. If lim inf q̂n (𝒟xT ) ≥ αq−Ω (𝒟xT ) n∈ℕ

(or lim sup q pn (𝒟xT ) ≤ αpΩ (𝒟xT ) ) n∈ℕ

for all 𝒟xT ∈ ΔΩC , then lim inf q̂n (𝒯 ) ≥ αq−Ω (𝒯 ) n∈ℕ

(or lim sup q pn (𝒯 ) ≤ αpΩ (𝒯 ) ) n∈ℕ

for all 𝒯 ∈ L(𝒞V (X, E), F) such that q−Ω (𝒯 ) > −∞ (or pΩ (𝒯 ) < +∞). Proof. We will to some extent follow the outlines of the proof of Theorem 2.5 in [47]. Our more general approach will however require some substantial adaptations. We shall verify the case of the lower envelopes. Let (pn )n∈ℕ in P, 𝒯 ∈ L(𝒞V (X, E), F), Ω ∈ K and α ≥ 0 be as stated. If pΩ (𝒯 ) = 0, then 𝒯 = 0 by Lemma 3.1.1(a). But q pn (0) ≤ pn (0) = 0 for all n ∈ ℕ, hence our claim is true in this case. We may therefore assume that pΩ (𝒯 ) = 1, hence 𝒯 ∈ HΩ . The sequence (pn )n∈ℕ is supposed to be Ω-bounded above, that is, pn ≤ λpΩ holds with some λ ≥ 0 for all n ∈ ℕ. This implies that q pn ≤ λpΩ holds as well and, in particular, q pn (𝒯 ) ≤ λ. Using Lemma 3.2.5, we construct a corresponding sequence (Φn )n∈ℕ of sets in C such that Ω absorbs Φn and such that q−Φn ≤ pn

and

q−Φn (𝒯 ) ≥ q pn (𝒯 ) − 1/n.

We note that the sequence (q−Φn )n∈ℕ is bounded above by λ on HΩ and that each of the functions q−Φn is bounded below on HΩ since Φn ⊂ λn Ω for some λn ≥ 0 implies that q−Φn = −pΦn ≥ −λn qΩ . Hence q−Φn ≥ −λn on HΩ . As in the proof for the classical case, we shall use a representation of HΩ in the space ℝℕ and apply Choquet’s theorem for metrizable compact sets. For this, define Θ : HΩ → ℝℕ by Θ(𝒮 ) = (q−Φn (𝒮 ))n∈ℕ

for 𝒮 ∈ HΩ .

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Since all of the functions q−Φn are bounded on HΩ , the sets q−Φn (HΩ ) are all bounded in ℝ. The Cartesian product of their compact closures is therefore compact in the metric space ℝℕ , and so is the set K = Θ(HΩ ), as it is a closed subset of this product. Now let C be the sup-stable point-separating cone of continuous real-valued functions on K, generated by the constants and the projections πm : ℝℕ → ℝ, for all m ∈ ℕ. Let 𝜕C ⊂ K denote the classical Choquet boundary of C in K (see I.5 in [1]). Recall that 𝜕C consists of those points in K whose point evaluation measure is maximal among all probability measures on K with respect to the classical Choquet ordering defined by the functions in C. We shall first establish that for every nonzero (αn )n∈ℕ ∈ 𝜕C there is 𝒟xT ∈ ΔΩC such that Θ(𝒟xT ) = (αn )n∈ℕ . Indeed, for 0 ≠ (αn )n∈ℕ ∈ 𝜕C set Z = {𝒮 ∈ HΩ | q−Φn (𝒮 ) ≥ αn for all n ∈ ℕ}. Because the functions q−Φn are superlinear and upper semicontinuous on the space L(𝒞V (X, E), F), Z is a compact convex subset of HΩ . Let us verify that it is not empty. There is a net (𝒮i )i∈ℐ in HΩ such that both 𝒮i → 𝒮 in HΩ and Θ(𝒮i ) → (αn )n∈ℕ in ℝℕ . From the upper semicontinuity of the functions q−Φn , we infer that αn = lim q−Φn (𝒮i ) ≤ q−Φn (lim 𝒮i ) = q−Φn (𝒮 ) i∈ℐ

i∈ℐ

holds for all n ∈ ℕ. Hence 𝒮 ∈ Z. Let ℒ be the collection of all nonempty compact convex subsets L of Z with the following property: If q−Φ (𝒮 ) ≤ λq−Φ (𝒮1 ) + (1 − λ)q−Φ (𝒮2 ) for all Φ ∈ C,

some 𝒮 ∈ L, 𝒮1 , 𝒮2 ∈ HΩ and 0 ≤ λ ≤ 1, then 𝒮1 , 𝒮2 ∈ L. Let us verify that Z ∈ ℒ. Let q−Φ (𝒮 ) ≤ λq−Φ (𝒮1 ) + (1 − λ)q−Φ (𝒮2 ) for all Φ ∈ C, some 𝒮 ∈ Z, and 𝒮1 , 𝒮2 ∈ HΩ . Set βn = q−Φn (𝒮1 ) and γn = q−Φn (𝒮2 ). Then αn ≤ q−Φn (𝒮 ) ≤ λq−Φn (𝒮1 ) + (1 − λ)q−Φn (𝒮2 ) = λβn + (1 − λ)γn for all n ∈ ℕ. If δ(αn ) , δ(βn ) and δ(γn ) denote the point evaluations on K at the points (αn )n∈ℕ , (βn )n∈ℕ and (γn )n∈ℕ , respectively, then the above shows that the measure λδ(βn ) + (1 − λ)δ(γn ) dominates the measure δ(αn ) in the classical Choquet ordering generated by C on K. Hence δ(αn ) = δ(βn ) = δ(γn ) , and αn = βn = γn , since (αn )n∈ℕ ∈ 𝜕C is maximal. Thus 𝒮1 , 𝒮2 ∈ Z. This shows in particular that the collection ℒ is not empty. Next, we observe that every set L ∈ ℒ is decreasing with respect to the order ≺C . Indeed, suppose that ℛ ≺C 𝒮 for 𝒮 ∈ L and ℛ ∈ L(𝒞V (X, E), F). Then ℛ ∈ HΩ since pΩ (ℛ) ≤ pΩ (𝒮 ) ≤ 1, and q−Φ (𝒮 ) ≤ q−Φ (ℛ) for all Φ ∈ C, implies that ℛ ∈ L. Furthermore, every extreme point 𝒮 of L is some Ω-extremal point evaluation 𝒟xT , because otherwise, according to Corollary 3.1.7, 𝒮 is not an extreme point of HΩ and

200 � 3 Choquet theory can be expressed as 𝒮 = λ𝒮1 +(1−λ)𝒮2 with 𝒮1 , 𝒮2 ∈ HΩ distinct from 𝒮 and 0 ≤ λ ≤ 1 and such that q−Φ (𝒮 ) = λq−Φ (𝒮1 ) + (1 − λ)q−Φ (𝒮2 ) holds for all Φ ∈ C. Thus 𝒮1 , 𝒮2 ∈ F by the definition of ℒ, a contradiction. The family ℒ is ordered by set inclusion, the intersection of any subfamily of its elements is again in L and, therefore, every chain has a lower bound. Thus ℒ contains a minimal element L0 . We shall verify that all operators in L0 are equivalent and minimal with respect to the order ≺C in L(𝒞V (X, E), F). Indeed, let Φ0 ∈ C and G0 = {𝒮 ∈ L0 | q−Φ0 (𝒮 ) = max{p−Φ0 (L0 )}}. Because the functional q−Φ0 is upper semicontinuous and superlinear, G0 is a nonempty compact convex subset of L0 and q−Φ (𝒮 ) ≤ λq−Φ (𝒮1 ) + (1 − λ)q−Φ (𝒮2 ) for all Φ ∈ C, some 𝒮 ∈ G0 , 𝒮1 , 𝒮2 ∈ HΩ and 0 ≤ λ ≤ 1, implies that 𝒮1 , 𝒮2 ∈ L0 , and consequently, p−Φ0 (𝒮1 ) = p−Φ0 (𝒮2 ) = max{p−Φ0 (L0 )}. We infer that 𝒮1 , 𝒮2 ∈ G0 , and that G0 is also an element of ℒ. Now the minimality of L0 implies that G0 = L0 and, therefore, that all the functionals p−Φ , for Φ ∈ C, are constant on L0 , our claim. Moreover, all elements of L0 are minimal because L0 is decreasing. Since (αn )n∈ℕ ≠ 0, since L0 ≠ {0}, and every nonzero extreme point of L0 is a minimal Ω-extremal point evaluation by the above and, therefore, an element of ΔΩC , we conclude that Θ−1 ((αn )n∈ℕ ) ∩ ΔΩC ≠ H. Next, we choose any 𝒟xT ∈ Θ−1 ((αn )n∈ℕ ) ∩ ΔΩC and observe that q−Φm ≤ pm implies that q−Φm ≤ q pm by the definition of the lower envelope. Thus lim sup πm ((αn )n∈ℕ ) = lim sup αm m∈ℕ

m∈ℕ

= lim sup q−Φm (𝒟xT ) m∈ℕ

≤ lim sup q pm (𝒟x𝒯 ) m∈ℕ

≤ αpΩ (𝒟xT ) ≤ α by our assumption. This argument applies to all elements (αn )n∈ℕ of the Choquet boundary 𝜕C ⊂ K, that is, lim supm∈ℕ πm ≤ α holds pointwise on 𝜕C. Choquet’s

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theorem for compact convex metric spaces guarantees the existence of a probability measure θ on K, which is supported by 𝜕C and on the cone C of continuous real-valued functions on K dominates the point evaluation at Θ(𝒯 ) ∈ K. That is to say, ∫ πm dθ = ∫ πm dθ ≥ πm (Φ(𝒯 )) = q−Φm (𝒯 )

K

𝜕C

holds for all m ∈ ℕ. The sequence (πm )m∈ℕ in C is uniformly bounded above on K, because this holds for the sequence (q−Φm )m∈ℕ on HΩ . Thus we may use Fatou’s lemma for lim sup ∫ πm dθ ≤ ∫ lim sup πm dθ ≤ α. m∈ℕ

𝜕C

𝜕C

m∈ℕ

Moreover, our selection of the functionals q−Φm renders that lim sup q pm (𝒯 ) = lim sup q−Φm (𝒯 ). m

m∈ℕ

Now combining all of the above, we have lim sup q pm (𝒯 ) ≤ α = αpΩ (𝒯 ). m

This completes our argument. 3.2.12 Corollary. Let Ω ∈ K. The Choquet boundary ΔΩC is nonempty if and only if there is an operator 𝒯 ∈ L(𝒞V (X, E), F), such that pΩ (𝒯 ) < +∞ and pΦ (𝒯 ) < 0 for some Φ ∈ C. Proof. If 𝒟xT ∈ ΔΩx , then pΩ (𝒟xT ) < +∞. Since 𝒟xT ≠ 0, we have pΩ (𝒟xT ) > 0 by Lemma 3.1.1(a). Thus 𝒟xT ≁ 0 since pΩ (0) = 0 and, therefore, 0 ≺/ C 𝒟xT since 𝒟xT is minimal. The latter means that pΦ (𝒟xT ) < pΦ (0) = 0 for some Φ ∈ C, our claim. For the converse statement, let 𝒯 ∈ L(𝒞V (X, E), F) such that pΩ (𝒯 ) < +∞ and pΦ (𝒯 ) < 0 for Φ ∈ C and assume that ΔΩC = H. We apply the bracketed case of Theorem 3.2.11 with the constant sequence pn = pΩ and α = 0. Since the condition involving ΔΩC is void, the theorem renders that q pΩ (𝒯 ) ≤ 0. On the other hand, q{0} ≤ pΩ implies q q that pΩ (𝒯 ) ≥ 0 and, therefore, pΩ (𝒯 ) = 0. Thus according to Proposition 3.2.6 there is S ∈ L(𝒞V (X, E), F) such that S ≺C T and pΩ (S) = 0. By Lemma 3.1.1(a), the latter implies that S = 0. Hence pΦ (0) ≤ pΦ (T) < 0, a contradiction since pΦ (0) = 0. In special cases, the expressions in Theorem 3.2.11 involving limits and upper (or lower) envelopes of functions can be reformulated using the statements of Proposition 3.2.6. 3.2.13 Lemma. Let 𝒯 ∈ LK (𝒞V (X, E), F) and let (qn )n∈ℕ be a sequence in Q that is increasing (or (pn )n∈ℕ a sequence in P that is decreasing) on 𝒯 ↓C . Then

202 � 3 Choquet theory lim inf q̂n (𝒯 ) = sup {sup qn (𝒮 )} n∈ℕ

𝒮≺C 𝒯 n∈ℕ

(or lim sup q pn (𝒯 ) = inf { inf pn (𝒮 )}). 𝒮≺C 𝒯 n∈ℕ

n∈ℕ

If the sequence (qn )n∈ℕ is increasing (or the sequence (pn )n∈ℕ is decreasing) on all of L(𝒞V (X, E), F) and if one of the functions qn (or pn ) is continuous at 0 ∈ LK (𝒞V (X, E), F), then the suprema (or infima) with respect to 𝒮 ≺C 𝒯 in the above expressions are in fact maxima (or minima). Proof. We shall prove the case of the upper envelopes. Let 𝒯 ∈ LK (𝒞V (X, E), F) and (qn )n∈ℕ be as stated. The sequence (q̂n (𝒯 ))n∈ℕ is increasing in ℝ and we have lim inf q̂n (𝒯 ) = sup q̂n (𝒯 ) = sup{ max qn (𝒮 )} = sup {sup qn (𝒮 )} n∈ℕ

n∈ℕ

n∈ℕ 𝒮≺C 𝒯

𝒮≺C 𝒯 n∈ℕ

by Proposition 3.2.6. If the sequence (qn )n∈ℕ is increasing on the whole space L(𝒞V (X, E), F) and if the function qn0 is continuous at 0 ∈ LK (𝒞V (X, E), F), then it is real-valued and so are all qn for n ≥ n0 . Under these assumptions, we continue our argument as follows: There are 𝒮n ≺C 𝒯 such that qn (𝒮n ) = max qn (𝒮 ) = q̂n (𝒯 ), 𝒮≺C 𝒯

and using the compactness of 𝒯 ↓C we find an accumulation point 𝒮 ≺C T of the sequence (𝒮n )n∈ℕ . We claim that sup qn (𝒮 ) = sup qn (𝒮n ). n∈ℕ

n∈ℕ

Indeed, since qn (𝒮 ) ≤ qn (𝒮n ) for all n ∈ ℕ, we have sup qn (𝒮 ) ≤ sup qn (𝒮n ). n∈ℕ

n∈ℕ

On the other hand, given ε > 0, by the continuity of the function qn0 at 0 ∈ L(𝒞V (X, E), F) there is n1 ≥ n0 such that qn (𝒮 − 𝒮n ) ≥ qn0 (𝒮 − 𝒮n ) ≥ −ε for all n ≥ n1 . Now using the superlinearity of the functionals qn , we infer that qn (𝒮 ) ≥ qn (𝒮n ) + qn (𝒮 − 𝒮n ), hence qn (𝒮n ) ≤ qn (𝒮 ) − qn (𝒮 − 𝒮n ) ≤ qn (𝒮 ) + ε

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for all n ≥ n1 . As qn (𝒮n ) = q̂n (𝒯 ) ≤ q̂m (𝒯 ) = qm (𝒮m ) whenever n ≤ m, the sequence (qn (𝒮n ))n∈ℕ is increasing in ℝ and we have sup qn (𝒮n ) = sup qn (𝒮n ) ≤ sup qn (𝒮 ) + ε. n≥n1

n∈ℕ

n∈ℕ

The latter holds for all ε > 0, and since sup qn (𝒮n ) = sup q̂n (𝒯 ) = lim inf q̂n (𝒯 ), n∈ℕ

n∈ℕ

n∈ℕ

our claim follows. The next corollary relates to the classical Choquet–Bishop–DeLeeuw theorem (see Corollary I.4.12 in [1]). It establishes a localization for the support of the representation measures of minimal operators. Recall that a Gδ -subset of X is defined to be a countable intersection of open sets. An Fσ -set is a countable union of closed sets. All Gδ - and Fσ -sets are obviously contained in the σ-field A. 3.2.14 Corollary. Let u be a function space neighborhood, which is a neighborhood in 𝒞V (X, E) and let Π ⊂ F ˚ such that ΩΠ u ∈ K. Let 𝒯 ∈ LK (𝒞V (X, E), F) be minimal and ΩΠ

pΩΠu (𝒯 ) < +∞. If B is a Gδ -subset of X disjoint from 𝜕Cu , then inf sup {‖θ‖(u,Π) A∩B | θ represents 𝒮 } = 0.

𝒮∼𝒯 A∈R

ΩΠ

Proof. Let On be open subsets of X such that B = ⋂n∈ℕ On ∈ A is disjoint from 𝜕Cu . We may assume that the sequence (On )n∈ℕ is decreasing. The subsets n 󵄨󵄨 f ∈ u, μ ∈ Π, 0 ≤ φ ∈ 𝒞 (X) 󵄨 i i Ωn = {∑(φi ⋅ fi ) ⊗ μi 󵄨󵄨󵄨 i } 󵄨󵄨 such that ∑ni=1 φi ≤ χO n i=1

of 𝒞V (X, E) ⊗ F ˚ are 𝒞 (X)-convex and the functionals pn = pΩn are contained in P. We have n 󵄨󵄨 f ∈ u, μ ∈ Π, 0 ≤ φ ∈ 𝒞 (X) 󵄨 i i pn (𝒯 ) = sup {∑(𝒯 (φi ⋅ fi ))(μi ) 󵄨󵄨󵄨 i } 󵄨󵄨 such that ∑ni=1 φi ≤ χO n i=1

for all 𝒯 ∈ LK (𝒞V (X, E), F). Proposition 3.1.14 states that (u,Π) sup ‖θ‖(u,Π) A∩B ≤ sup ‖θ‖A∩O = pn (𝒯 ) ≤ pΩΠu (𝒯 ),

A∈R

A∈R

n

where θ denotes the representation measure for 𝒯 . The sequence (pn )n∈ℕ is decreasΩΠ

ing and since (⋂n∈ℕ On ) ∩ 𝜕Cu = H, we infer that

204 � 3 Choquet theory lim sup q pn (𝒟xη ) ≤ lim sup pn (𝒟xη ) = 0 n∈ℕ

η

n∈ℕ

ΩΠ

for all 𝒟x ∈ ΔCu . If pΩΠu (𝒯 ) < +∞ for 𝒯 ∈ L(𝒞V (X, E), F), we may therefore apply the bracketed case in Theorem 3.2.11 with α = 0 and obtain pn (𝒯 ) = inf q pn (𝒯 ) = 0. lim sup q n∈ℕ

n∈ℕ

If 𝒯 is indeed minimal, then the above together with Proposition 3.2.10 shows that q inf {‖θ‖(u,Π) A∩B | θ represents 𝒮 } ≤ inf pn (𝒮 ) = pn (𝒯 )

𝒮∼𝒯

𝒮∼𝒯

holds for all n ∈ ℕ. Our claim follows. For the next corollary, we recall from Section 2.1 that a subset Π of F ˚ is called θ-dense if for all A ∈ R and a ∈ E we have θA (a) = 0 whenever θA (a)(μ) = 0 for all μ ∈ Π. For this condition to hold for an L(E, F ˚˚ )-valued representation measure θ, it suffices that the linear span of Π is dense in of F ˚ in its weak topology. If θ is indeed L(E, F)-valued, then density in the weak* topology is enough. 3.2.15 Corollary. Let u be a function space neighborhood, which is a neighborhood in (u,Π) 𝒞V (X, E), and let Π ⊂ F ˚ such that ΩΠ = ‖ϑ‖(u,Π) whenever θ u ∈ K and that ‖θ‖ and ϑ are the representation measures for equivalent operators. If 𝒯 ∈ LK (𝒞V (X, E), F) is minimal and pΩΠu (𝒯 ) < +∞, and if Π is θ-dense in F ˚ for the representation measure Π

θ for 𝒯 , then θ is supported by every Fσ -subset of X that contains 𝜕Ωu . C

Proof. Suppose that the operator 𝒯 ∈ L(𝒞V (X, E), F) is minimal and that pΩΠu (𝒯 ) < +∞. Let θ be the representation measure for 𝒯 . The complement of an Fσ -subset of ΩΠ

ΩΠ

X that contains 𝜕Cu is a Gδ -subset B ∈ A disjoint from 𝜕Cu . Under our assumptions, Corollary 3.2.14 yields that ‖θ‖(u,Π) = 0 for all subsets A ∈ R of B. Our claim follows A with Proposition 2.1.11(b).

C-superharmonic sets Theorem 3.2.11 and its corollaries suggest the benefit of having small or indeed singleton equivalence classes in LK (𝒞V (X, E), F) while preserving the given Choquet boundary. We shall say that a 𝒞 (X)-convex subset Φ of 𝒞V (X, E) ⊗ F ˚ is C-superharmonic if pΦ (𝒯 ) ≤ pΦ (𝒟xT ) whenever 𝒯 ≺C 𝒟xT for 𝒯 , 𝒟xT ∈ LK (𝒞V (X, E), F) such that 𝒟xT is a K-extremal point evaluation. Clearly, sums and positive multiples of C-superharmonic sets are again C-superharmonic. ̃ of a Choquet cone C over K as the cone 𝒞 (X)-convex We define the completion C ˚ subsets of 𝒞V (X, E) ⊗ F , which is generated by K and the collection of all nonempty

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̃ can be K-bounded C-superharmonic subsets Φ of 𝒞V (X, E) ⊗ F ˚ . Every set Φ ∈ C n expressed as Φ = Ψ + ∑i=1 λi Ωi , where Ψ ∈ C is K-bounded and C-superharmonic, ̃ are C-superharmonic. where λi ≥ 0 and Ωi ∈ K. All elements of C ̃ of a Choquet cone C is itself a Choquet cone over 3.2.16 Proposition. The completion C K that contains C and ΔC ̃ ⊂ ΔC . ̃ Let us verify that Δ̃ ⊂ ΔC . Proof. Obviously, C is a Choquet cone over K and C ⊂ C. C T T T T If 𝒟x ∈ ΔC ̃ and 𝒯 ≺C 𝒟x for 𝒯 , 𝒟x ∈ LK (𝒞V (X, E), F), then 𝒯 ≺C ̃ 𝒟x holds as ̃ are C-superharmonic. Thus 𝒟T ≺̃ 𝒯 and, therefore, well, since all elements of C x C T T 𝒟x ≺C 𝒯 . Hence 𝒟x is ≺C -minimal, and consequently, an element of ΔC . ̃ of a Choquet cone C is not necOur construction reveals that the completion C essarily the largest Choquet cone over K containing C without increasing its Choquet boundary. But it serves our purpose since the preceding results are applicable with the ̃ of C in place of C. We proceed to recognize some C-superharmonic completion C subsets of 𝒞V (X, E) ⊗ F ˚ . If K ≠ H, we consider the closed 𝒞 (X)-convex subcone K0 = ⋂{εΩ | Ω ∈ K, ε > 0} of 𝒞V (X, E) ⊗ F ˚ . For every 𝒯 ∈ LK (𝒞V (X, E), F), there is Ω ∈ K such that pΩ (𝒯 ) < +∞. Thus pΦ (𝒯 ) ≤ 0 for every 𝒞 (X)-convex subset Φ of K0 , and pK0 (𝒯 ) = 0 in particular, since 0 ∈ K0 . Moreover, if ω ∈ K0 and if 0 ≤ φ ∈ 𝒞 (X) is nonnegative and bounded, then φ ⋅ ω ∈ K0 . 3.2.17 Proposition. Let C be a Choquet cone over K ≠ H. ̃ (a) If K ≠ H, then K0 ∈ C. ̃ for i = 1, . . . , n with bounded functions φi ∈ 𝒞 (X), (b) If ω ∈ K0 and if {φi ⋅ ω} ∈ C ̃ where φ = ∨n φi . then {φ ⋅ ω} ∈ C, i=1 Proof. (a) Clearly, K0 is C-superharmonic since pK0 (𝒯 ) = 0 for all operators 𝒯 ∈ LK (𝒞V (X, E), F), and K0 is K-bounded. For (b), let ω, φi and φ be as stated. The singleton sets {φi ⋅ω} and {φ⋅ω} are obviously 𝒞 (X)-convex. All the elements (φ − φi ) ⋅ ω are contained in K0 and, therefore, p{(φ−φi )⋅ω} (𝒯 ) ≤ 0, that is, p{φ⋅ω} (𝒯 ) ≤ p{φi ⋅ω} (𝒯 ) holds for all 𝒯 ∈ LK (𝒞V (X, E), F) by the above. For a point evaluation 𝒟xT and a bounded function ψ ∈ 𝒞 (X), we evaluate p{ψ⋅ω} (𝒟xT ) = Re(ψ ⋅ ω)(𝒟xT ) = ψ(x)Re ω(𝒟xT ) = ψ(x)p{ω} (𝒟xT ). Thus 𝒯 ≺C 𝒟xT for 𝒯 , 𝒟xT ∈ LK (𝒞V (X, E), F) such that 𝒟xT is a K-extremal point evaluation implies that p{φi ⋅ω} (𝒯 ) ≤ φi (x)Re ω(𝒟xT )

206 � 3 Choquet theory for all i = 1, . . . , n. Now using that Re ω(𝒟xT ) ≤ 0 we calculate that n

p{φ⋅ω} (𝒯 ) ≤ inf{φi (x)Re ω(𝒟xT )} i=1

n

= Re ω(𝒟xT ) sup{φi (x)} = =

i=1 φ(x)Re ω(𝒟xT ) p{φ⋅ω} (𝒟xT ).

̃ The singleton set {φ⋅ω} is therefore indeed C-superharmonic and contained in C. Proposition 3.2.17 is of course meaningful only if K0 contains nonzero elements. This is not always the case. We shall therefore provide some further criteria, which do not rely on this assumption. 3.2.18 Proposition. Let C be a Choquet cone and let D be a family of C-superharmonic subsets of 𝒞V (X, E) ⊗ F ˚ . If Ψ = ⋂Φ∈D Φ is not empty and if pΨ (𝒟xT ) = infΦ∈D pΦ (𝒟xT ) for all K-extremal point evaluations 𝒟xT ∈ LK (𝒞V (X, E), F), then Ψ is also C-superharmonic. Proof. Let D be as stated and let Ψ denote the nonempty intersection of its elements. Clearly, pΨ (𝒯 ) ≤ infΦ∈D pΦ (𝒯 ) for every 𝒯 ∈ L(𝒞V (X, E), F). Let 𝒯 ≺C 𝒟xT for 𝒯 , 𝒟xT ∈ LK (𝒞V (X, E), F) such that 𝒟xT is a K-extremal point evaluation. Then pΨ (𝒯 ) ≤ inf pΦ (𝒯 ) ≤ inf pΦ (𝒟xT ) = pΨ (𝒟xT ) Φ∈D

u∈D

by our assumption, hence our claim. As an immediate consequence of Corollary 3.1.13, we notice the following. 3.2.19 Proposition. Let C be a Choquet cone, let u and w be 𝒞 (X)-convex subsets of Π 𝒞V (X, E), suppose that Π ⊂ F ˚ satisfies (Π1) and that both ΩΠ u and Ωw are C-superΠ harmonic. If either 0 ∈ u ∩ w or Π is a singleton set, then Ωu+w is also C-superharmonic. The following criteria are rather technical, yet useful in many cases. For a convex subset U of E, such that 0 ∈ U and a lower semicontinuous ℝ+-valued function ρ : X → ℝ we set ρ

uU = {f ∈ 𝒞V (X, E) | f (x) ∈ ρ(x)U for all x ∈ X}. Recall from Section 1.1 that (+∞)U is defined to be the conic hull of U in E. Clearly, ρ uU is a 𝒞 (X)-convex subset of 𝒞V (X, E). For the following lemma, also recall that we generally set 0(+∞) = 0 and (+∞)(+∞) = +∞ (see Section 1.1).

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3.2.20 Lemma. Let U be a convex subset of E such that 0 ∈ U, let ρ : X → ℝ be a lower semicontinuous ℝ+ -valued function and let Π ⊂ F ˚ . Then pΩΠ ρ (𝒟xT ) = ρ(x) sup pΠ (T(a)) u

a∈U

U

for every point evaluation 𝒟xT ∈ L(𝒞V (X, E), F). Proof. Let U, ρ and Π be as stated and let 𝒟xT ∈ L(𝒞V (X, E), F) be a point evaluation. We abbreviate Φ = ΩΠuρ and β = supa∈U pΠ (T(a)). According to Proposition 3.1.9, we U have ρ

pΦ (𝒟xT ) = sup{pΠ (T(f (x))) | f ∈ uU } ρ

If ρ(x) = 0, there is nothing to prove since f (x) = 0 for every f ∈ uU , and both sides in the equation in our lemma equal zero. If β = 0, then pΠ (T(a)) ≤ 0 holds for all a ∈ U, and indeed for all elements of the conic hull of U. Since T(f (x)) is an element ρ of this subcone for any f ∈ uU , we conclude again that both sides in the above equation equal zero. We may therefore assume that both ρ(x) > 0 and β > 0 and continue as ρ follows. We have f (x) ∈ ρ(x)U for all f ∈ uU . Thus pΦ (𝒟xT ) ≤ ρ(x)β. For the reverse inequality, let 0 < γ < ρ(x). There is a neighborhood 𝒩 ∈ R of x such that ρ(y) > γ for all y ∈ 𝒩 . There is ψ ∈ 𝒞 (X), such that 0 ≤ ψ ≤ 1, ψ(x) = 1 and ψ(y) = 0 for ρ all y ∈ X \ 𝒩 . Then for any choice of a ∈ U the function γ(ψ ⋅ a) is contained in uU , hence pΦ (𝒟xT ) ≥ pΠ (T(γa)) for all a ∈ U, hence pΦ (𝒟xT ) ≥ γβ. The latter holds true for all choices of 0 < γ < ρ(x) and yields pΦ (𝒟xT ) ≥ ρ(x)β. Using the notation from above, we formulate our next criterion. 3.2.21 Proposition. Let C be a Choquet cone. Let U be a closed convex subset of E such that 0 ∈ U and let ρ : X → ℝ be a strictly positive lower semicontinuous function. Let Γ be a multiplicative subgroup of the unit sphere of ℝ or ℂ and suppose that γU ⊂ U for all γ ∈ Γ. For i = 1, . . . , n, let φi be continuous real-or complex-valued functions such that |φi |(x) < ρ(x)

and

󵄨 󵄨 φi (x)/󵄨󵄨󵄨φi (x)󵄨󵄨󵄨 ∈ Γ

for all x ∈ X such that φi (x) ≠ 0. Suppose that Π ⊂ F ˚ satisfies (Π2). If for all a ∈ U and i = 1, . . . , n the sets ΩΠwaφ , i

ρ

where

waφi = uU − φi ⋅ a,

where

σ = ρ − ∨ni=1 |φi |,

are C-superharmonic, then the set ΩΠuσ , U

is also C-superharmonic.

208 � 3 Choquet theory ρ

Proof. Let U, ρ, φi and Γ be as stated. For a ∈ U, we denote waφi = uU − φi ⋅ a and w = ⋂{waφi | a ∈ U, i = 1, . . . , n}. We claim that uσU ⊂ w, where σ = ρ − ∨ni=1 |φi |. We note that the function σ : X → ℝ is also strictly positive and lower semicontinuous. To verify our claim, let f ∈ uσU , a ∈ U, i ∈ {1, . . . , n} and x ∈ X. We shall establish that f (x) + φi (x)a ∈ ρ(x)U. We have f (x) ∈ (ρ(x) − |φi (x)|) U. Our assumptions on U and φi guarantee that there is γi ∈ Γ such that φi (x) = γi |φi (x)| and γi a ∈ U. Thus 󵄨 󵄨 󵄨 󵄨 φi (x)a = 󵄨󵄨󵄨φi (x)󵄨󵄨󵄨(γi a) ∈ 󵄨󵄨󵄨φi (x)󵄨󵄨󵄨U and 󵄨 󵄨 󵄨 󵄨 f (x) + φi (x)a ∈ (ρ(x) − 󵄨󵄨󵄨φi (x)󵄨󵄨󵄨) U + 󵄨󵄨󵄨φi (x)󵄨󵄨󵄨U = ρ(x)U. ρ

This demonstrates that f +φi ⋅a ∈ uU holds for all a ∈ U and i = 1, . . . , n and, therefore, that f ∈ w. For the second step of our argument, we consider the family D = {ΩΠwaφ | a ∈ U, i = 1, . . . , n} i

of subsets of 𝒞 (X, E) ⊗ F ˚ . By our assumption, all sets in D are C-superharmonic. Using the first step of our argument, we realize that Ψ = ⋂{ΩΠwaφ ∈ D} ⊃ ΩΠuσ . U

i

Thus pΩΠ σ ≤ pΨ ≤ inf{pΩΠ a | ΩΠwaφ ∈ D} u

wφ

U

i

i

holds pointwise on L(𝒞V (X, E), F). We proceed to verify that these expressions coincide at all point evaluations 𝒟xT ∈ L(𝒞V (X, E), F). For this, we abbreviate β = sup{pΠ (T(a)) | a ∈ U} and recall from Lemma 3.2.20 that pΩΠ σ (𝒟xT ) = σ(x)β. u

U

If β = +∞, this shows pΨ (𝒟xT ) = +∞ since σ(x) > 0 and proves the reverse of the above inequality. If β < +∞, we continue as follows. Since ΩΠwaφ ⊂ ΩΠuρ + ΩΠuρ , we i

have

pΩΠ a (𝒟xT ) ≤ pΩΠ ρ (𝒟xT ) + pΩΠ wφ

i

u

U

{−φi .a}

(𝒟xT ) = ρ(x)β + pΩΠ

for all i = 1, . . . , n and use Proposition 3.1.9 for

{−φi .a}

U

(𝒟xT )

{−φi .a}

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3.2 Choquet ordering for linear operators

pΩΠ

{−φi ⋅a}

(𝒟xT ) = pΠ (φi (x)T(−a)).

For every i, there is γi ∈ Γ such that φi (x) = γi |φi (x)|. Hence 󵄨 󵄨 pΠ (φi (x)T(−a)) = 󵄨󵄨󵄨φi (x)󵄨󵄨󵄨pΠ (T(−γi a)). As γi a ∈ U if and only if a ∈ U, for every i we calculate 󵄨 󵄨 inf pΩΠ a (𝒟xT ) = ρ(x)β + 󵄨󵄨󵄨φi (x)󵄨󵄨󵄨 inf{pΠ (T(−a)) | a ∈ U}.

a∈U

wφ

i

Because pΠ (T(a)) ≤ β < +∞ for all a ∈ U, our assumption on the functional pΠ yields that pΠ (T(−a)) = −pΠ (T(a)) and, therefore, inf{pΠ (T(−a)) | a ∈ U} = − sup{pΠ (T(a)) | a ∈ U} = −β. Now taking the infimum over all the elements of D, that is, n

inf inf pΩΠ a (𝒟xT ) i=1 a∈U

wφ

i

we obtain 󵄨 󵄨 inf pΩΠ a (𝒟xT ) = ρ(x)β − ∨ni=1 󵄨󵄨󵄨φi (x)󵄨󵄨󵄨β = σ(x)β = pΩΠ σ (𝒟xT ),

ΩΠwa φ

i

∈D

wφ

u

i

U

thus demonstrating that pΨ (𝒟xT ) = inf{pΩΠ a (𝒟xT ) | ΩΠwaφ ∈ D} = pΩΠ σ (𝒟xT ). wφ

i

i

u

U

Now according to Proposition 3.2.18 the set Ψ is C-superharmonic. So is ΩΠuσ , because U

𝒯 ≺C 𝒟xT for 𝒯 , 𝒟xT ∈ LK (𝒞V (X, E), F) such that 𝒟xT is a K-extremal point evaluation

implies that

pΩΠ σ (𝒯 ) ≤ pΦ (𝒯 ) ≤ pΦ (𝒟xT ) = pΩΠ σ (𝒟xT ). u

u

U

U

We shall formulate the next criterion for the special case that E is a Hilbert space over ℝ or ℂ with the inner product ⟨ , ⟩ and unit ball 𝔹. For functions f , g ∈ ℱ (X, E), we denote the real- or complex-valued function x 󳨃→ ⟨f (x), g(x)⟩ on X by ⟨f , g⟩. 3.2.22 Proposition. Let C be a Choquet cone. Let E be a Hilbert space with unit ball 𝔹 ρ and let ρ : X → ℝ be a strictly positive lower semicontinuous function such that u𝔹 is {μ} ˚ a neighborhood in 𝒞V (X, E). Let μ ∈ F such that Ω ρ ∈ C. If for 𝒯 ∈ L(𝒞V (X, E), F) there is a function g ∈ uρ𝔹 such that

u𝔹

210 � 3 Choquet theory Ω{−g} ∈ C {μ}

and Re μ(𝒯 (g)) = pΩ{μ} (𝒯 ) < +∞, u

ρ 𝔹

then for every 𝒮 ∈ L(𝒞V (X, E), F) such that 𝒮 ≺C 𝒯 we have Re μ(𝒮 (f )) ≤ 0 for all f ∈ 𝒞V (X, E) such that Re⟨f (x), g(x)⟩ ≤ 0 for all x ∈ Y , where Y is the compact set {x ∈ X | ‖g(x)‖ = ρ(x)}. Proof. Unfortunately, the verification of our claim will be rather laborious. We shall formulate our argument for the complex case. Let ρ, g, μ and 𝒮 ≺C 𝒯 be as stated. We shall begin with a few preliminary observations. For every f ∈ 𝒞V (X, E), we have −Re μ(𝒮 (g)) = Re μ(𝒮 (−g)) = pΩ{μ} (𝒮 ) {−g}

≤ pΩ{μ} (T) = Re μ(𝒯 (−g)) = −Re μ(𝒯 (g)). {−g}

That is, Re μ(𝒯 (g)) ≤ Re μ(𝒮 (g)). We shall make use of the following facts: ρ (i) If for f ∈ 𝒞V (X, E), there is α ≥ 1 such that f + g ∈ αu𝔹 , then Re μ(𝒮 (f )) ≤ (α − 1)pΩΠ ρ (𝒯 ). u

𝔹

ρ

Indeed, if f + g ∈ αu𝔹 , then pΩ{μ} (𝒮 ) ≤ αpΩ{μ} (𝒮 ) ≤ αpΩ{μ} (𝒯 ) = αRe μ(𝒯 (g)), {f +g}

u

ρ 𝔹

u

ρ 𝔹

that is, Re μ(𝒮 (f )) + Re μ(𝒮 (g)) ≤ αRe μ(𝒯 (g)) ≤ αRe μ(𝒮 (g)). ρ

Hence our claim. We note that g ∈ u𝔹 implies that ‖g(x)‖ ≤ ρ(x) for all x ∈ X. We define the nonnegative lower semicontinuous function σ : X → ℝ setting σ(x) = ρ(x) − ‖g(x)‖ and observe that Y = {x ∈ X | σ(x) = 0} is a closed subset of X. Next we shall establish the following: (ii) If for f ∈ 𝒞V (X, E), there are α, β ≥ 0 such that 󵄩󵄩 󵄩 󵄩󵄩f (x)󵄩󵄩󵄩 ≤ ασ(x) + βρ(x) for all x ∈ X, then |Re μ(𝒮 (f ))| ≤ β pΩΠ ρ (𝒯 ). u

𝔹

3.2 Choquet ordering for linear operators

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Indeed, we have ‖f (x)‖ ≤ α(ρ(x) − ‖g(x)‖) + βρ(x), that is, ‖(f + αg)(x)‖ ≤ (α + β)ρ(x) ρ for all x ∈ X. If α = 0, this shows f ∈ βu𝔹 . Hence Re μ(𝒮 (f )) ≤ β pΩ{μ} (𝒮 ) ≤ β pΩ{μ} (𝒯 ). u

ρ 𝔹

u

ρ 𝔹

If α > 0, the above shows that α+β ρ 1 f +g ∈( )u𝔹 , α α hence α+β 1 Re μ(𝒮 (f )) ≤ ( − 1)pΩ{μ} (𝒯 ) ρ α α u 𝔹 and, therefore, Re μ(𝒮 (f )) ≤ β pΩΠ ρ (𝒯 ) by (i). These arguments also apply to the funcu

𝔹

tion −f and, therefore, yield (ii). Let f ∈ 𝒞V (X, E). We initially assume that Re⟨f (x), g(x)⟩ ≤ 0 holds for all x ∈ ρ X and shall demonstrate that Re μ(𝒮 (f )) ≤ 0. Indeed, since u𝔹 is supposed to be a ρ neighborhood in 𝒞V (X, E), hence absorbing, we may assume that f ∈ u𝔹 . For any λ ≥ 1, we claim that ρ

h = (λ − 1)g + √λ − (1/2) f ∈ λu𝔹 . For this, we shall verify that ‖h(x)‖ ≤ λρ(x) holds for all x ∈ X. If g(x) = 0 or λ = 1, this is evident since ‖h(x)‖ = √λ − (1/2) ‖f ‖ ≤ λρ(x) in this case. If g(x) ≠ 0 and λ > 1, we set m(x) = g(x)/‖g(x)‖ and a(x) = ⟨h(x), m(x)⟩ m(x) = (λ − 1)g(x) + (√λ − (1/2) ⟨f (x), m(x)⟩)m(x), that is, the orthogonal projection of h(x) onto the linear span of g(x) in E, and b(x) = h(x) − a(x) = √λ − (1/2)(f (x) − ⟨f (x), m(x)⟩ m(x)). We have ‖h(x)‖2 = ‖a(x)‖2 + ‖b(x)‖2 . It is now straightforward to establish that 󵄩󵄩 󵄩2 󵄩 󵄩2 2 󵄩󵄩b(x)󵄩󵄩󵄩 ≤ (λ − (1/2))󵄩󵄩󵄩f (x)󵄩󵄩󵄩 ≤ (λ − (1/2))ρ(x) . Furthermore, since g(x) = ‖g(x)‖ m(x) and since Re⟨f (x), m(x)⟩ ≤ 0, we calculate 󵄩󵄩 󵄩2 󵄨 󵄩 󵄩 󵄨2 󵄩󵄩a(x)󵄩󵄩󵄩 = 󵄨󵄨󵄨(λ − 1)󵄩󵄩󵄩g(x)󵄩󵄩󵄩 + √λ − (1/2)⟨f (x), m(x)⟩󵄨󵄨󵄨 󵄩 󵄩 󵄩 󵄩2 = (λ − 1)2 󵄩󵄩󵄩g(x)󵄩󵄩󵄩 + 2(λ − 1)󵄩󵄩󵄩g(x)󵄩󵄩󵄩√λ − (1/2) Re⟨f (x), m(x)⟩

212 � 3 Choquet theory 󵄨2 󵄨 + (λ − (1/2))󵄨󵄨󵄨⟨f (x), m(x)⟩󵄨󵄨󵄨 󵄩2 󵄩 󵄩2 󵄩 ≤ (λ − 1)2 󵄩󵄩󵄩g(x)󵄩󵄩󵄩 + (λ − (1/2))󵄩󵄩󵄩f (x)󵄩󵄩󵄩 ≤ (λ − 1)2 ρ(x)2 + (λ − (1/2))ρ(x)2 ≤ (λ2 − λ + (1/2))ρ(x)2 . ρ

Thus indeed, ‖h(x)‖2 ≤ λ2 ρ(x)2 . This verifies that h ∈ λu𝔹 and ρ

(√λ − (1/2)/(λ − 1))f + g ∈ (λ/(λ − 1))u𝔹 . Using statement (i), we infer that √λ − (1/2) λ Re μ(𝒮 (f )) ≤ ( − 1)pΩΠ ρ (𝒯 ), (λ − 1) λ−1 u 𝔹 that is, √λ − (1/2) Re μ(𝒮 (f )) ≤ pΩΠ (𝒯 ). ρ u

𝔹

The latter holds for all λ ≥ 1 and, therefore, renders Re μ(𝒮 (f )) ≤ 0. This completes the initial step in the argument for our claim. ρ Now in the second step of the argument we suppose that f ∈ u𝔹 and that Re⟨f (x), g(x)⟩ ≤ 0 holds for all x ∈ Y . Both functions f and g vanish at infinity relative to the neighρ borhood u𝔹 . Thus there is a compact subset K of X such that both ‖f (x)‖ ≤ (1/2)ρ(x) and ‖g(x)‖ ≤ (1/2)ρ(x) for all x ∈ X \ K. Let Z be another compact subset of X whose topological interior U contains K, and let α > 0 be the minimum value of the lower semicontinuous function ρ, and β the maximum value of the continuous function x 󳨃→ ‖f (x)‖ on Z. Let B = {x ∈ Z | σ(x) ≤ α/2}

and

󵄩 󵄩 O = {x ∈ U | 󵄩󵄩󵄩g(x)󵄩󵄩󵄩 > α/4}.

Clearly, B is compact, O is open and Y ⊂ B ⊂ O ⊂ Z. The latter follows since for any x ∈ B we have ρ(x) − ‖g(x)‖ ≤ α/2, hence ‖g(x)‖ ≥ ρ(x) − α/2 ≥ α/2. The above implies in particular that Y is compact, one of our assertions. There is a function φ ∈ 𝒞𝒦 (X) such that χB ≤ φ ≤ χO . Setting h = φ ⋅ f and k = (1 − φ) ⋅ f , both of these functions are contained in 𝒞V (X, E), we have Re μ(S(f )) = Re μ(S(h)) + Re μ(S(k)) and claim that Re μ(S(k)) = 0. Indeed, k(x) = 0 if x ∈ B,

3.2 Choquet ordering for linear operators

� 213

󵄩 󵄩 󵄩 󵄩 󵄩 󵄩󵄩 󵄩󵄩k(x)󵄩󵄩󵄩 ≤ 󵄩󵄩󵄩f (x)󵄩󵄩󵄩 ≤ (1/2)ρ(x) ≤ ρ(x) − 󵄩󵄩󵄩g(x)󵄩󵄩󵄩 = σ(x) if x ∈ X \ K and 󵄩 󵄩 󵄩 󵄩󵄩 󵄩󵄩k(x)󵄩󵄩󵄩 ≤ 󵄩󵄩󵄩f (x)󵄩󵄩󵄩 ≤ β = (2β/α)(α/2) ≤ (2β/α)σ(x) if x ∈ K \ B. Setting λ = max{1, (2β/α)}, we have ‖k(x)‖ ≤ λσ(x) for all x ∈ X, and our claim follows from (ii). We infer that Re μ(S(f )) = Re μ(S(h)) and observe that χX\O ⋅ h = 0. Next, we define the function l setting l(x) =

Re⟨h(x), g(x)⟩ ∨ 0 g(x) ‖g(x)‖2

for x ∈ O, and l(x) = 0 else.

Since ‖l(x)‖ ≤ ‖h(x)‖ holds for all x ∈ X, the function l is continuous at all points x ∈ X where h(x) = 0, in particular, at all x ∈ X \ O. Moreover, since ‖g(x)‖ > 0 holds for are x ∈ O, the function l is also continuous at the points in O. Its support is contained in the compact set Z, and l is seen to be an element of 𝒞V (X, E). Because f and h coincide on Y , we note that Re⟨h(x), g(x)⟩ ≤ 0, hence l(x) = 0 holds for all x ∈ Y . For the function h − l, we realize that Re⟨(h − l)(x), g(x)⟩ = Re⟨h(x), g(x)⟩ − (Re⟨h(x), g(x)⟩ ∨ 0) ≤ 0 for all x ∈ X. This implies Re μ(S(h − l)) ≤ 0 by the first step of our argument. Hence Re μ(S(f )) = Re μ(S(h)) ≤ Re μ(S(l)). All left to show is that Re μ(S(l)) ≤ 0. Given ε > 0, we claim that for every x ∈ Z there is λx ≥ 0 such that ‖l(y)‖ < λx σ(y) + ερ(y) holds for all y in an open neighborhood Ux of x. We note that the mapping x 󳨃→ ‖l(x)‖ is continuous. Our claim is obvious for x ∈ Y since ‖l(x)‖ = σ(x) = 0 and ρ(x) > 0 in this case. If x ∈ Z \ Y , then σ(x) > 0, and we find λx ≥ 0 such that ‖l(x)‖ < λx σ(x). Since σ is lower semicontinuous, this relation holds for all elements in an open neighborhood Ux of x as stated. The compact set Z is covered by a finite number Uxi , for i = 1, . . . , n, and if we set λ = ∨ni=1 λxi , then ‖l(x)‖ ≤ λσ(x)+ερ(x) holds for all x ∈ Z, and indeed for all x ∈ X since l is supported by Z. Now statement (ii) from above renders that 󵄨󵄨 󵄨 󵄨󵄨Re μ(𝒮 (l))󵄨󵄨󵄨 ≤ ε pΩ{μ} (𝒯 ). ρ u

𝔹

The latter holds true for all ε > 0 and shows that Re μ(𝒮 (f )) ≤ Re μ(𝒮 (l)) = 0. Hence our claim. Let Π be a subset of F ˚ such that its linear span is weak*-dense in F ˚ . This condition implies that c = 0 for c ∈ F whenever Re μ(γc) ≤ 0 for all μ ∈ Π and γ ∈ ℂ, because otherwise we could find μ ∈ Π such that μ(c) ≠ 0 and γ ∈ ℂ such that

214 � 3 Choquet theory ρ

Re μ(γc) > 0. Suppose that for every μ ∈ Π there is a function gμ ∈ u𝔹 as in the assumptions of Proposition 3.2.22, that is,

{μ} Ω{−g } μ

∈ C and

Re μ(𝒯 (gμ )) = pΩ{μ} (𝒯 ) < +∞, u

ρ 𝔹

and let Y be the closure in X of the set 󵄩 󵄩 { x ∈ X | 󵄩󵄩󵄩gμ (x)󵄩󵄩󵄩 = ρ(x) for some μ ∈ Π }. Let 𝒮 ≺C 𝒯 and let θ be the representation measure for 𝒮 . If f ∈ 𝒞V (X, E) is supported by X \ Y , then ⟨f (x), g(x)⟩ = 0 holds for all x ∈ Y , and we have Re μ(γS(f )) = Re μ(S(γf )) ≤ 0 for all μ ∈ Π and all γ ∈ ℂ by Proposition 3.2.22. Hence S(f ) = 0 by the above. Now using Proposition 2.5.9 we conclude that θ is supported by Y . We formulate this observation as a corollary. 3.2.23 Corollary. Let C be a Choquet cone. Let E be a Hilbert space with unit ball 𝔹 ρ and let ρ : X → ℝ be a strictly positive lower semicontinuous function such that u𝔹 is a neighborhood in 𝒞V (X, E). Let Π ⊂ F ˚ such that its linear span is weak*-dense in F ˚ {μ} and Ω ρ ∈ C for all μ ∈ Π and let 𝒯 ∈ L(𝒞V (X, E), F). If for every μ ∈ Π there is a u𝔹

ρ

function gμ ∈ u𝔹 such that Ω{−g } ∈ C {μ}

μ

and Re μ(𝒯 (gμ )) = pΩ{μ} (𝒯 ) < +∞, u

ρ 𝔹

then the representation measure for any operator 𝒮 ∈ L(𝒞V (X, E), F) such that 𝒮 ≺C 𝒯 is supported by the closure of the set 󵄩 󵄩 { x ∈ X | 󵄩󵄩󵄩gμ (x)󵄩󵄩󵄩 = ρ(x) for some μ ∈ Π }.

3.3 Special cases and examples This final section provides a range of examples and applications of the concepts developed in the preceding sections. We begin with the classical origins of Choquet theory (see [1] and [41]). The case that X is compact and that E = F = ℝ Let X be a compact Hausdorff space and let E = F = ℝ. We endow 𝒞 (X) with the topology of uniform convergence with the unit ball 𝔹. With these insertions, we have

3.3 Special cases and examples

� 215

L(𝒞V (X, E), F) = 𝒞 (X)˚ endowed with its weak* topology, and 𝒞V (X, E) ⊗ F ˚ = 𝒞 (X) endowed with its weak topology. In order to model classical Choquet theory, we use the Choquet core K = {u}, where u = {f ∈ 𝒞 (X) | f ≤ 1}. The polar of u is weak*-compact (see Corollary 1.3.9 with S = [−∞, 1)). Let H be a subspace of 𝒞V (X, E) that contains the constant functions and separates the points of X. The suitable choice for our Choquet cone is C = {αu + f | α ≥ 0, f ∈ H}. The operators in LK (𝒞V (X, E), F) are positive linear functionals in this case, and 𝒮 ≺C 𝒯 for 𝒮 , 𝒯 ∈ LK (𝒞V (X, E), F) means that 𝒮 (f ) = 𝒯 (f ) for all f ∈ H. Let n

C = {⋀ fi | fi ∈ L} . i=1

It is straightforward to check that all the sets {f } for f ∈ C are C-superharmonic. Indeed, let f = ∧ni=1 fi ∈ C and let 𝒯 , 𝒟x ∈ LK (𝒞V (X, E), F) such that 𝒯 ≺C 𝒟x . Then 𝒯 (fi ) = fi (x) for all i = 1, . . . , n hence n

n

i=1

i=1

𝒯 (f ) ≤ ⋀ T(fi ) ≤ ⋀ fi (x) = f (x),

that is, p{f } (𝒯 ) ≤ p{f } (𝒟x ). The order ≺C̃ then leads to the classical approach to Choquet theory.

Sample settings in the general case In this sample setting, the Choquet core K is constructed in the following way. Let U be a nonempty family of 𝒞 (X)-convex, but not necessarily balanced neighborhoods of 0 in 𝒞V (X, E), such that for all u, w ∈ U and α ≥ 0 there is β ≥ α such that u ∩ (βw) ∈ U, and let Π be a subset of F ˚ such that the balanced core bcore(Π˝ ) of its polar in F is weakly compact. We set K = {ΩΠu | u ∈ U} and verify the conditions for a Choquet core from the beginning of Section 3.2. First, ˝ for every u ∈ U according to Corollary 1.3.9 the set LΠu (𝒞V (X, E), F) is compact in the weak operator topology of L(𝒞V (X, E), F). Hence, keeping with Proposition 3.1.10 the set HΩΠu , that is, the polar of ΩΠu in L(𝒞V (X, E), F), is also compact. Next, for the boundedness condition let ΩΠu , ΩΠw ∈ K and let ω ∈ ΩΠu , that is, ω = ∑ni=1 (φi ⋅ fi ) ⊗ μi

216 � 3 Choquet theory for fi ∈ u, μi ∈ Π and 0 ≤ φi ∈ 𝒞 (X) such that ∑ni=1 φi = 1. There is α ≥ 0 such that fi ∈ αw for all i = 1, . . . , n. Hence by our condition on U there is β ≥ α such that s = u ∩ (βw) ∈ U. Then ΩΠs ∈ K,

ω ∈ ΩΠs ,

ΩΠs ⊂ ΩΠu

and

ΩΠs ⊂ βΩΠw .

This proves our claim. We remark that the weak*-closure of the balanced convex hull bconv(Π) of Π is the polar of the balanced core bcore(Π˝ ) of Π˝ in F, which is supposed to be weakly compact. But the polar of a weakly compact set is known to be absorbing. This implies in particular that the linear span of Π is weak*-dense in F ˚ as required in some of the results of the preceding sections. Let C0 be a collection of nonempty subsets ΩΓw of 𝒞V (X, E) ⊗ F ˚ such that Γ ⊂ Π and w ⊂ 𝒞V (X, E) is 𝒞 (X)-convex and U-bounded, that is, for every u ∈ U there is λ ≥ 0 such that w ⊂ λu. Every such set ΩΓw is obviously K-bounded. For the Choquet cone C, we choose the cone of subsets of 𝒞V (X, E) ⊗ F ˚ generated by K and C0 . We notice that all operators 𝒯 ∈ LK (𝒞V (X, E), F) are weakly compact in this setting. Hence their representation measures are L(E, F)-valued (Theorem 2.6.1). Indeed, if pΩΠu (𝒯 ) < +∞ for ΩΠu ∈ K, then Π˝

𝒯 ∈ λHΩΠ ⊂ λLu (𝒞V (X, E), F) u

for some λ ≥ 0, and Corollary 1.3.9 applies. As a consequence, all representation measures of operators in LK (𝒞V (X, E), F) are L(E, F)-valued and the set Π is θ-dense in F ˚ with respect to any such measure θ. According to Corollary 3.1.8, a point evaluation 𝒟xT ∈ L(𝒞V (X, E), F) is an element of the Choquet boundary ΩΠ

ΔCu

for

ΩΠu ∈ K

if and only if it is minimal, if pΩΠu (𝒟xT ) < +∞ and if T is an extreme point of the set {S ∈ L(E, F) | pΩΠu (𝒟xS ) ≤ pΩΠu (𝒟xT )}. We continue to formulate our observations for the case that both E and F are vector spaces over ℂ. The real case is similar. 3.3.1 Proposition. Let K and C be as specified and suppose in addition that all sets in U are balanced and that ΩΓγw ∈ C0 for all |γ| = 1 whenever ΩΓw ∈ C0 . Then the following hold: (a) If Ω ∈ C, then αΩ ∈ C for all α ∈ ℂ. (b) If 𝒮 ≺C 𝒯 for 𝒮 , 𝒯 ∈ L(𝒞V (X, E), F), then α𝒮 ≺C α𝒯 for all α ∈ ℂ. (c) If Φ ⊂ 𝒞V (X, E) ⊗ F ˚ is C-superharmonic, then αΦ is C-superharmonic for all α ∈ ℂ. (d) LK (𝒞V (X, E), F) is a subspace of L(𝒞V (X, E), F).

3.3 Special cases and examples

� 217

Proof. According to Lemma 3.2.1, LK (𝒞V (X, E), F) is a subcone of the operator space L(𝒞V (X, E), F). Thus for Part (d), we only have to validate that α𝒯 ∈ LK (𝒞V (X, E), F) whenever 𝒯 ∈ LK (𝒞V (X, E), F) and α ∈ ℂ. This, as well as the statements in (a), (b) and (c), is obvious for α = 0. If α ≠ 0, we set γ = α/|α| and argue as follows. For Part (a), let Ω ∈ C, that is, n

m

i=1

k=1

Ω = ∑ κi ΩΠui + ∑ λk ΩΓwk k Γ

for ΩΠui ∈ K, Ωwk k ∈ C0 and κi , λk ≥ 0. We observe that αΩΠui = |α|ΩΠui

and

k k αΩΓwk k = ΩΓαw = |α|ΩΓγw k k

since the sets ui are supposed to be balanced. Thus n

m

i=1

k=1

k αΩ = |α|(∑ κi ΩΠui + ∑ λk ΩΓγw )∈C k

by our assumption on C0 . For Part (b), suppose that 𝒮 ≺C 𝒯 for 𝒮 , 𝒯 ∈ L(𝒞V (X, E), F) and let α ∈ ℂ. Then for every Ω ∈ C we have pΩ (α𝒮 ) = pαΩ (𝒮 ) ≤ pαΩ (𝒯 ) = pΩ (α𝒯 ). Thus α𝒮 ≺C α𝒯 . For Part (d), let 𝒯 ∈ LK (𝒞V (X, E), F). There is ΩΠu ∈ K such that pΩΠu (𝒯 ) < +∞. Since pγΩΠu (α𝒯 ) = |α|pγΩΠu (γ𝒯 ) = |α|pΩΠu (𝒯 ) < +∞, we infer that α𝒯 ∈ LK (𝒞V (X, E), F). For Part (c), suppose that Φ ⊂ 𝒞V (X, E) ⊗ F ˚ is C-superharmonic and let 𝒯 ≺C 𝒟xT for 𝒯 , 𝒟xT ∈ LK (𝒞V (X, E), F) such that 𝒟xT is a K-extremal point evaluation. Then α𝒯 , α𝒟xT ∈ LK (𝒞V (X, E), F) by Part (d) and α𝒟xT is also a K-extremal point evaluation. We have α𝒯 ≺C α𝒟xT by Part (b) and, therefore, pαΦ (𝒯 ) = pΦ (α𝒯 ) ≤ pΦ (α𝒟xT ) = pαΦ (𝒟xT ) since Φ is C-superharmonic. Hence our claim. We prepare to further specify our settings for C0 , hence for C. Let ℋ be a subcone of 𝒞V (X, E). Let V ∈ 𝒱 and let ρ be a strictly positive ℝ+ -valued lower semicontinρ uous function on X, such that uV is a U-bounded neighborhood in 𝒞V (X, E). For a subspace G of E, we denote HG = {φ ∈ 𝒞 (X, ℂ) | φ ⋅ a ∈ ℋ for all a ∈ G} and

218 � 3 Choquet theory n

CG = {⋁ |φi | | φi ∈ HG }. i=1

Clearly, HG is a subspace of 𝒞 (X, ℂ), CG is a subcone and CG − CG , that is, the subspace spanned by CG , is a sublattice of 𝒞 (X). Let μ ∈ Π. We consider the {μ} 𝒞 (X)-convex subsets Ω ρ of 𝒞V (X, E) ⊗ F ˚ , which are K-bounded, and the funcuV ∩G

tionals on L(𝒞V (X, E), F) that are defined by these sets. If θ is the representation measure for an operator 𝒯 ∈ L(𝒞V (X, E), F), we recall from Proposition 2.6.8 that its L(G, F ˚˚ )-valued restriction θ‖G to G represents the restriction of 𝒯 to 𝒞V (X, G). The latter coincides with θ on all sets A ∈ R and all elements of the subspace G of E. (See Proposition 2.6.8 and the remarks at the end of Section 3.1.) Using Propositions 2.3.11 and 3.1.14 and Corollary 3.1.12, we calculate that ρ

pΩ{μ} (𝒯 ) = sup{Re μ(𝒯 (f )) | f ∈ uV ∩G } = ∫ ρ d[|θ‖G |(V ∩ G)(μ)]. u

ρ V ∩G

X

Since the integral is additive, the above implies in particular that pΩ{μ} (𝒯 ) = pΩ{μ} (𝒯 ) + pΩ{μ} (𝒯 ) u

ρ+σ V ∩G

u

ρ V ∩G

uσ V ∩G

holds for strictly positive ℝ+ -valued lower semicontinuous functions ρ and σ. Employing these concepts, we shall specify our choice for the family C0 of subsets of 𝒞V (X, E) ⊗ F ˚ to be used in our arrangement. Let 𝒢 be a family of subspaces of E and for each G ∈ 𝒢 we choose a neighborhood VG ∈ 𝒱 and a function ρG as above. For the sake of readability, we shall omit the space G in the notation for V and ρ in the sequel. This dependency will be evident from the context. We are now ready for the completion of our setup. For C0 , we choose the following subsets of 𝒞V (X, E) ⊗ F ˚ : {μ} (i) all sets Ω{f } for f ∈ ℋ and μ ∈ Π and (ii) all sets Ω

{μ} ρ uV ∩G

for G ∈ 𝒢 and μ ∈ Π.

We take note that pΩ{μ} (𝒯 ) = Re μ(𝒯 (f )) = (𝒯 (f ))(μ) {f }

holds for all 𝒯 ∈ L(𝒞V (X, E), F) (Proposition 3.1.11). We shall use these settings throughout the remainder of this subsection. 3.3.2 Proposition. Suppose that 𝒮 ≺C 𝒯 for 𝒮 , 𝒯 ∈ L(𝒞V (X, E), F). (a) If ℋ is a subspace of 𝒞V (X, E), then 𝒮 (f ) = 𝒯 (f ) for all f ∈ ℋ. ρ ρ (b) If Π absorbs all elements of F ˚ , then 𝒮 (uV ∩G ) ⊂ 𝒯 (uV ∩G ) for all G ∈ 𝒢 .

3.3 Special cases and examples

� 219

Proof. Suppose that 𝒮 ≺C 𝒯 for 𝒮 , 𝒯 ∈ L(𝒞V (X, E), F). (a) For f ∈ ℋ, we have Re αμ(𝒮 (f )) = pΩ{μ} (𝒮 ) ≤ pΩ{μ} (𝒯 ) = Re αμ(𝒯 (f )) α{αf }

{αf }

for all α ∈ ℂ and μ ∈ Π, that is, μ(𝒮 (f )) = μ(𝒯 (f )) for all μ ∈ Π and, therefore, ρ

ρ

𝒮 (f ) = 𝒯 (f ) since the linear span of Π is weak*-dense in F ˚ . (b) If 𝒮 (uV ∩G ) ⊄ 𝒯 (uV ∩G ) ρ for G ∈ 𝒢 , then there are f ∈ uV ∩G and ν ∈ F ˚ such that ρ

Re ν(𝒮 (f )) > sup{Re ν(𝒯 (g)) | g ∈ uV ∩G } = pΩ{μ} (𝒯 ). u

ρ V ∩G

By our assumption, there are μ ∈ Π and λ ≥ 0 such that ν = λμ. Then pΩ{μ} (𝒮 ) ≤ pΩ{μ} (𝒯 ) u

ρ V ∩G

u

ρ V ∩G

implies that

pΩ{ν} (𝒮 ) ≤ pΩ{ν} (𝒯 ), u

ρ V ∩G

u

ρ V ∩G

contradicting the above. Propositions 3.2.21 and 3.2.22 permit to identify certain C-superharmonic subsets of 𝒞V (X, E). We begin with a technical lemma. 3.3.3 Lemma. Let θ be the representation measure for 𝒯 ∈ L(𝒞V (X, E), F). Let G ∈ 𝒢 , let μ ∈ Π and suppose that pΩ{μ} (𝒯 ) < +∞. u

ρ V ∩G

(a) If V ∩ G = G, then the ℝ+ -valued measure [|θ‖G |(V ∩ G)(μ)] equals 0. (b) If V ∩ G ≠ G, then for every φ ∈ HG and ε > 0 there is A ∈ R and λ ≥ 0 such that |φ(x)| ≤ λρ(x) for all x ∈ X and |φ(x)| ≤ ερ(x) for all x ∈ X \ A. ρ

Proof. (a) If V ∩ G = G, then uV ∩G = 𝒞V (X, G), and pΩ{μ} (𝒯 ) < +∞ implies that u

μ(𝒯 (f )) = 0 for all f ∈ 𝒞V (X, G). Hence

ρ V ∩G

pΩ{μ} (𝒯 ) = sup{Re μ(𝒯 (f )) | f ∈ 𝒞V (X, G)} uκ V ∩G

= ∫ κ d[|θ‖G |(V ∩ G)(μ)] = 0 X

for every strictly positive lower semicontinuous ℝ+ -valued function κ. As for every A ∈ R, there is such a function such that χA ≤ κ, we have 0 ≤ [|θ‖G |(V ∩ G)(μ)]A ≤ ∫ κ d[|θ‖G |(V ∩ G)(μ)] = 0 X

as claimed. (b) If V ∩ G ≠ G, then there is a ∈ G \ V . Consequentially, αa ∈ βV for α ∈ ℂ and β ≥ 0 implies that either α = 0 or a ∈ (β/α)V and, therefore, |β/α| > 1. ρ Thus |α| < β holds in any case. Let φ ∈ HG . Since φ ⋅ a ∈ 𝒞V (X, E) and since uV is

220 � 3 Choquet theory ρ

λρ

supposed to be a neighborhood in 𝒞V (X, E), there is λ ≥ 0 such that φ ⋅ a ∈ λuV = uV . By the above, this implies that |φ(x)| < λρ(x) for all x ∈ X. Now, given ε > 0, since φ⋅a ρ vanishes at infinity, there is A ∈ R such that χ(X\A) ⋅ (φ ⋅ a) ∈ εuV . Thus |φ(x)| ≤ ερ(x) for all x ∈ X \ A, our claim. 3.3.4 Proposition. Let G ∈ 𝒢 and τ ∈ CG . {μ} (a) If τ(x) < ρ(x) for all x ∈ X, then for all μ ∈ Π the sets Ωuσ , where σ = ρ − τ, V ∩G ̃ are C-superharmonic and elements of C.

(b) If 𝒮 ≺C ̃ 𝒯 for 𝒮 , 𝒯 ∈ L(𝒞V (X, E), F) and ϑ and θ are the representation measures for 𝒮 and 𝒯 , respectively, and if pΩ{μ} (𝒮 ) = pΩ{μ} (𝒯 ) < +∞ for μ ∈ Π, then u

ρ V ∩G

u

ρ V ∩G

∫ τ d[|ϑ‖G |(V ∩ G)(μ)] ≥ ∫ τ d[|θ‖G |(V ∩ G)(μ)]. X

X

Proof. Let τ = ∨ni=1 φi ∈ CG for φi ∈ 𝒞 (X, ℂ) be as stated. For Part (a), in Proposition 3.2.21 we set U = VG , Γ the full unit sphere of ℂ and Π = {μ}. Since the sets {μ} {μ} Ω ρ and Ω{φ ⋅a} for all i and a ∈ G are contained in C, Proposition 3.2.19 applies uV ∩G

i

and states that the sets

where

Ωwa , {μ}

φi

ρ

waφi = uV ∩G − φi .a,

are C-superharmonic. Proposition 3.2.21 now yields that the set Ωuσ

{μ}

harmonic. As a subset of

{μ} Ω ρ uV ∩G

V ∩G

is also C-super-

̃ For Part (b), is K-bounded, and thus an element of C.

suppose that 𝒮 ≺C ̃ 𝒯 for 𝒮 , 𝒯 ∈ L(𝒞V (X, E), F). We distinguish two cases. If V ∩G = G, then [|θ‖G |(V ∩ G)(μ)] = [|ϑ‖G |(V ∩ G)(μ)] = 0 by Lemma 3.3.3(a). Our claim holds trivially true in this case. In the second case, if V ∩G ≠ G following Lemma 3.3.3(b) there is λ ≥ 0 such that |φi (x)| < λρ(x) for all x ∈ X, and we may apply Part (a) with λρ in place of ρ. We abbreviate σ = λρ − τ and infer that pΩ{μ} (𝒮 ) ≤ pΩ{μ} (𝒯 ) uσ V ∩G

uσ V ∩G

holds by Part (a). Using this, we calculate pΩ{μ} (𝒮 ) − pΩ{μ} (𝒮 ) = pΩ{μ} (𝒮 ) u

λρ V ∩G

uτ V ∩G

uσ V ∩G

≤ pΩ{μ} (𝒯 ) = pΩ{μ} (𝒯 ) − pΩ{μ} (𝒯 ), uσ V ∩G

hence

u

λρ V ∩G

uτ V ∩G

3.3 Special cases and examples

� 221

∫ τ d[|ϑ‖G |(V ∩ G)(μ)] = pΩ{μ} (𝒮 ) uτ V ∩G

X

≥ pΩ{μ} (𝒯 ) = ∫ τ d[|θ‖G |(V ∩ G)(μ)] uτ V ∩G

X

since pΩ{μ} (𝒮 ) = pΩ{μ} (𝒯 ) < +∞. Hence our claim. u

λρ V ∩G

u

λρ V ∩G

3.3.5 Corollary. Let G ∈ 𝒢 and suppose that for every x ∈ X there is φ ∈ HG such that φ(x) ≠ 0, and for distinct points x, y ∈ X there is φ ∈ HG such that φ(x) = 0 ≠ φ(y). If ϑ and θ are the representation measures for 𝒮 , 𝒯 ∈ L(𝒞V (X, E), F), respectively, if 𝒮 ∼ 𝒯 with respect to the order ≺C ̃ and if pΩ{μ} (𝒯 ) < +∞ for μ ∈ Π, then u

ρ V ∩G

[|ϑ‖G |(V ∩ G)(μ)] = [|θ‖G |(V ∩ G)(μ)]. If 𝒯 is minimal in this order and if there is a set u ∈ U that is defined by a function space ρ neighborhood and such that uV ∩G ≤ u and pΩΠu (𝒯 ) < +∞, then θ‖G is supported by ΩΠ

any Fσ -subset of X that contains 𝜕Cu .

Proof. Let G ∈ 𝒢 be as stated. If V ∩ G = G, then [|θ‖G |(V ∩ G)(μ)] = [|ϑ‖G |(V ∩ G)(μ)] = 0 holds for all μ ∈ Π by Lemma 3.3.3(a). Otherwise, we shall use a suitable space 𝒞Vℝ (X) of real-valued functions and an appropriate version of the Stone–Weierstrass theorem to establish our argument. For this, we define the function space neighborhood w in ℱ (X) by the single neighborhood function x 󳨃→ [−ρ(x), +ρ(x)], that is, 󵄨 󵄨 w = {φ ∈ ℱ (X) | 󵄨󵄨󵄨φ(x)󵄨󵄨󵄨 ≤ ρ(x) for all x ∈ X}, and the function space neighborhood system Vℝ = {αw | α > 0} for ℱ (X). Lemma 3.3.3(b) then shows that all functions in HG , hence all functions in CG vanish at infinity and are therefore contained in 𝒞Vℝ (X). The linear span of CG , that is, D = CG − CG , forms a sublattice of 𝒞Vℝ (X). Applying a version of the Stone–Weierstrass theorem from [49] we can demonstrate that D is dense 𝒞V (X). We use Corollaries 2.2 and 2.3 in [49], which require that (i) for every x ∈ X there is φ ∈ D such that φ(x) ≠ 0 and (ii) for all x ≠ y ∈ X there is φ ∈ D such that φ ≥ 0, φ(x) = 0 and φ(y) > 0. Both conditions are immediately implied by our assumptions on HG . Now let 𝒮 , 𝒯 ∈ L(𝒞V (X, E), F) such that 𝒮 ∼ 𝒯 with respect to the order ≺C ̃ and let μ ∈ Π such that pΩ{μ} (𝒮 ) = pΩ{μ} (𝒯 ) < +∞. u

ρ V ∩G

Then Proposition 3.3.4(b) applies. We have

u

ρ V ∩G

222 � 3 Choquet theory ∫ τ d[|ϑ‖G |(V ∩ G)(μ)] = ∫ τ d[|θ‖G |(V ∩ G)(μ)] X

X

for all τ ∈ CG . For φ ∈ 𝒞Vℝ (X), such that |φ| ≤ ρ and μ ∈ Π we realize that 󵄨󵄨 󵄨󵄨 󵄨 󵄨󵄨 󵄨󵄨∫ φ d[|θ‖G |(V ∩ G)(μ)]󵄨󵄨󵄨 ≤ ∫ ρ d[|θ‖G |(V ∩ G)(μ)] = pΩ{μ} (𝒯 ) < +∞. ρ 󵄨󵄨 󵄨󵄨 u V ∩G X

X

Thus the mapping φ 󳨃→ ∫ φ d[|θ‖G |(V ∩ G)(μ)] : 𝒞Vℝ (X) → ℝ X

is a continuous linear functional on 𝒞Vℝ (X). In particular, the measure [|θ‖G |(V ∩ G)(μ)] is finitely valued on R and, therefore, regular by Proposition 2.1.15. The same considerations apply with ϑ in place of θ. Both functionals coincide on the dense subspace D and, therefore, on all of 𝒞Vℝ (X). Now using Proposition 2.5.8, we infer that [|ϑ‖G |(V ∩ G)(μ)] = [|θ‖G |(V ∩ G)(μ)], that is, the first part of our claim. Now suppose that 𝒯 ∈ L(𝒞V (X, E), F) is minimal in Π the order ≺C ̃ and that there is a function space neighborhood u ∈ U, that is, Ωu ∈ K, such that uV ∩G ≤ u and pΩΠu (𝒯 ) < +∞. (Recall that we agreed to use the same symbols for function space neighborhoods and their intersections with 𝒞V (X, E).) Since pΩ{μ} (𝒯 ) ≤ pΩΠu (𝒯 ) u

ρ V ∩G

for all μ ∈ Π, our first claim applies to all such μ. Let B be a Gδ -subset of X disjoint ΩΠ

from 𝜕Cu and let A ∈ R be a subset of B. Since A is relatively compact and ρ is strictly positive and lower semicontinuous, there is λ ≥ 0 such that χA ⋅ V ≤ λu. Hence using Lemma 2.1.5(a), we infer that |θ‖G |A (V ∩ G)(μ) ≤ ‖θ‖G ‖A

(λu,{μ})

≤ λ sup ‖θ‖G ‖(u,Π) A∩B A∈R

holds for all μ ∈ Π. The same holds true with ϑ in place of θ. Next, we use Corollary 3.2.14, the first part of the current corollary and the prior observation for |θ‖G |A (V ∩ G)(μ) = inf {|ϑ‖G |A (V ∩ G)(μ) | ϑ represents 𝒮 } 𝒮∼𝒯

≤ inf {‖ϑ‖(u,Π) | ϑ represents 𝒮 } A 𝒮∼𝒯

3.3 Special cases and examples

� 223

≤ inf sup {‖ϑ‖(u,Π) A∩B | ϑ represents 𝒮 } = 0. 𝒮∼𝒯 A∈R

Thus |θ‖G |A (V ∩ G)(μ) = 0 for all μ ∈ Π. Finally, since the operator 𝒯 is weakly compact, according to Theorem 2.6.1 its representation measure θ is L(E, F)-valued. Hence for a fixed V ∈ 𝒱 and A ∈ R the mapping μ 󳨃→ |θ‖G |A (V ∩ G)(μ) is sublinear and lower semicontinuous with respect to the weak* topology of F ˚ (see the definition of the modulus of a measure in Section 2.1). Thus we have |θ‖G |A (V ∩ G)(μ) = 0 for all μ in the weak*-closure of the balanced convex hull bconv(Π) of Π, which was understood to be absorbing. This demonstrates that |θ‖G |A (V ∩ G)(μ) = 0 holds for all μ ∈ F ˚ , our claim. The application of Proposition 3.2.22 in this standard setting requires that the family 𝒢 of subspaces of E contains E itself. 3.3.6 Proposition. Let E be a Hilbert space with unit ball 𝔹 and suppose that E ∈ 𝒢 , that VE = 𝔹 and ρ = ρE . Let 𝒮 , 𝒯 ∈ L(𝒞V (X, E), F) such that 𝒮 ≺C 𝒯 . If for μ ∈ Π ρ there is gμ ∈ (−ℋ) ∩ u𝔹 such that Re μ(𝒯 (gμ )) = pΩ{μ} (𝒯 ), then u

ρ 𝔹

Re μ(𝒮 (f )) ≤ 0 for all f ∈ 𝒞V (X, E) such that Re⟨f (x), gμ (x)⟩ ≤ 0 for all x ∈ Yμ , where Yμ is the compact set {x ∈ X | ‖gμ (x)‖ = ρ(x)}. If for every μ ∈ Π there is such a function gμ ∈ ℋ ∩ u, then the representation measure for S is supported by the closure of the set ⋃μ∈Π Yμ . Proof. Since Ω ρ , Ω{−g} ∈ C in our sample settings, the first part of our claim follows di{μ} u𝔹

{μ}

rectly from Proposition 3.2.22. The second part is a reformulation of Corollary 3.2.23.

We go forward to consider a few special cases using the preceding sample settings.

The case that F = ℝ or F = ℂ In this setting, L(E, F) is the dual E ˚ of E, endowed with its weak* topology, L(𝒞V (X, E), F) = 𝒞V (X, E)˚ , that is, the dual of 𝒞V (X, E) endowed with its weak* topology, and 𝒞V (X, E) ⊗ F ˚ = 𝒞V (X, E) endowed with its weak topology. We shall use the previously established sample settings with Π = {1}. That is, K = U, where U is a family of (not necessarily balanced) 𝒞 (X)-convex neighborhoods of 0 in 𝒞V (X, E)

224 � 3 Choquet theory ρ

with the required properties, ℋ is a subcone of 𝒞V (X, E) and 𝒢 and the sets uV are as stated in the general setting. Then C0 ⊂ 𝒞V (X, E) consists of: (i) all sets {f } for f ∈ ℋ and ρ (ii) all sets uV ∩G for G ∈ 𝒢 . We notice that according to Proposition 3.1.11 we have pw (𝒯 ) = sup{Re 𝒯 (f ) | f ∈ w} for every 𝒞 (X)-convex subset w of 𝒞V (X, E) containing 0 and 𝒯 ∈ 𝒞V (X, E)˚ . The measures |θ‖G |(VG ) are ℝ+ -valued. The formulations in Propositions 3.3.1 and 3.3.4, Lemma 3.3.3 and Corollary 3.3.5 simplify accordingly: There is no mention of Π or the functional μ ∈ Π in the statements, and the sets ΩΠuρ , V ∩G

ρ

Ω

{μ} ρ uV ∩G

and

Ω{f }

{μ}

ρ

read as uV , uV ∩G and {f }, respectively. Proposition 3.3.6 requires that E ∈ ℋ and simplifies accordingly.

The case that E = ℝ or E = ℂ In this setting, we have L(E, F) = F, endowed with its weak topology, and 𝒞V (X, E)⊗F ˚ equals 𝒞V (X) ⊗ F ˚ or 𝒞V (X, ℂ) ⊗ F ˚ . We shall formulate the case that E = ℂ and use the previously established sample settings. The real case is similar. That is, we have K = {ΩΠu | u ∈ U}, where U is a family of (not necessarily balanced) 𝒞 (X)-convex neighborhoods of 0 in 𝒞V (X, ℂ) with the required properties, and Π ⊂ F ˚ is such that bcore(Π˝ ) is weakly compact in F, ℋ is a subcone of 𝒞V (X, ℂ) and {0} and ℂ are the only possible ρ elements of 𝒢 . If G = ℂ is in ℋ, then it is associated with a set u𝔹 in 𝒞V (X, ℂ), whereby 𝔹 denotes the unit ball of ℂ. The corresponding subspace HG of 𝒞 (X, ℂ) and subcone CG of 𝒞 (X) are HG = bcore(ℋ) = ⋂ γℋ |γ|=1

Then C0 ⊂ 𝒞V (X, ℂ) ⊗ F ˚ consists of {μ} (i) all sets Ω{f } for f ∈ ℋ and μ ∈ Π and (ii) if 𝒢 = {ℂ}: all sets Ω

{μ} ρ u𝔹

for μ ∈ Π.

n

and CG = {⋁ |φi | | φi ∈ ℋ}. i=1

3.3 Special cases and examples

� 225

The measures |θ|(𝔹)(μ) are ℝ+ -valued. The formulations in Propositions 3.3.4 and Corollary 3.3.5 simplify only in the sense that they do not involve any subspaces G of E. Proposition 3.3.6 requires that ℂ ∈ ℋ and simplifies accordingly. We continue to investigate this case and relate the given Choquet ordering of L(𝒞V (X, ℂ), F) to a corresponding Choquet ordering in L(𝒞V (X, ℂ), ℂ) in the following way. We employ a parallel Choquet cone Cℂ for L(𝒞V (X, ℂ), ℂ) is spanned by the Choquet kernel Kℂ = U and the collection (Cℂ )0 = {w ⊂ 𝒞V (X, ℂ) | Ω{μ} w ∈ C0 for some μ ∈ Π}. That is to say, (Cℂ )0 = {{f } | f ∈ ℋ} in case that 𝒢 = H, and ρ

(Cℂ )0 = {{f } | f ∈ ℋ} ∪ {u𝔹 } in case that 𝒢 = {ℂ}. We choose Πℂ = {1}. With an operator 𝒯 ∈ L(𝒞V (X, ℂ), ℂ), that is, 𝒯 ∈ 𝒞V (X, ℂ)˚ , and an element e ∈ F we associate the elementary operator ℒe𝒯 ∈ L(𝒞V (X, ℂ), F) (see Section 3.1) defined by ℒe𝒯 (f ) = 𝒯 (f ) e for f ∈ 𝒞V (X, ℂ). We observe that all point evaluations in L(𝒞V (X, ℂ), F) are elementary operators 𝒟xe = ℒe𝒟x in this sense, where x ∈ X and 𝒟x = 𝒟x1 is a point evaluation in L(𝒞V (X, ℂ), ℂ). We shall say that an element e ∈ F is Π-positive if 0 ≤ μ(e) ∈ ℝ holds for all μ ∈ Π. Note that pΠ (e) = 0 for a Π-positive element e ∈ F implies that e = 0. 3.3.7 Proposition. Let 𝒮 , 𝒯 ∈ L(𝒞V (X, ℂ), ℂ) and let e ∈ F be Π-positive. (a) If w is a 𝒞 (X)-convex subset of 𝒞V (X, ℂ) and Γ ⊂ Π such that either 0 ∈ w or that Γ = {μ} is a singleton set, then pΩΓw (ℒe𝒯 ) = pΓ (e) pw (𝒯 ). (b) If 𝒮 ≺Cℂ 𝒯 , then ℒe𝒮 ≺C ℒe𝒯 . (c) If ℒe𝒮 ≺C ℒe𝒯 , if e ≠ 0 and pΠ (e) < +∞, then 𝒮 ≺Cℂ 𝒯 . (d) If ℒe𝒯 ≠ 0, then ℒe𝒯 ∈ LK (𝒞V (X, ℂ), F) if and only if 𝒯 ∈ LKℂ (𝒞V (X, ℂ), ℂ) and pΠ (e) < +∞. Proof. Recall that pw (𝒯 ) = sup{Re 𝒯 (f ) | f ∈ w} for a 𝒞 (X)-convex subset w of 𝒞V (X, ℂ) and 𝒯 ∈ L(𝒞V (X, ℂ), ℂ). Thus our claim in (a) follows from Proposition 3.1.15(b). For (b), in order to establish that 𝒮 ≺Cℂ 𝒯 we only have to verify that pw (𝒮 ) ≤ pw (𝒯 ) holds for all w ∈ Kℂ ∪ (Cℂ )0 . A similar assertion applies to the operators in L(𝒞V (X, ℂ), F). Thus if 𝒮 ≺Cℂ 𝒯 , since all elements of K ∪ C0 satisfy one of its alternative requirements, Part (a) yields that ℒe𝒮 ≺C ℒe𝒯 also holds. For (c), suppose that pΠ (e) < +∞ for e ≠ 0 and that ℒe𝒯 ≺C ℒe𝒯 . Since the linear span of Π is weak*-dense in F ˚ , there is μ ∈ Π such that μ(e) > 0 and, therefore, pΠ (e) > 0. Thus Part (a) yields that pu (𝒮 ) ≤ pu (𝒯 ) holds for all u ∈ U. Furthermore, for every {μ} w ∈ (Cℂ )0 and μ ∈ Π the set Ωw is contained in C0 , and Part (a) with Γ = {μ} renders that pw (𝒮 ) ≤ pw (𝒯 ). We infer that 𝒮 ≺Cℂ 𝒯 as claimed. For (d), first suppose that 𝒯 ∈ LKℂ (𝒞V (X), ℂ), that is, pu (𝒯 ) < +∞ for some u ∈ U, and that pΠ (e) < +∞. Then pΩΠu (ℒe𝒯 ) < +∞ by Part (a) and, therefore, ℒe𝒯 ∈ LK (𝒞V (X, ℂ), F). Conversely,

226 � 3 Choquet theory if 0 ≠ ℒe𝒯 ∈ LK (𝒞V (X, ℂ), F), there is u ∈ U such that pΩΠu (ℒe𝒯 ) = pΠ (e) pu (𝒯 ) < +∞. Since e ≠ 0, we have pΠ (e) > 0 and pu (𝒯 ) > 0 since u is a neighborhood in 𝒞V (X) and 𝒯 ≠ 0. Thus both pΠ (e) < +∞ and pu (𝒯 ) < +∞, that is, 𝒯 ∈ LKℂ (𝒞V (X, ℂ), ℂ), our claim. 3.3.8 Proposition. Suppose that for every u ∈ U the linear span of the set ⋂{γu | γ ∈ ℂ, Re γ > 0} is dense in 𝒞V (X, ℂ). (a) Let 𝒯 ∈ L(𝒞V (X, ℂ), ℂ) and e ∈ F. If ℒe𝒯 ∈ LK (𝒞V (X, ℂ), F) is not zero, then there is α ≠ 0 such that αe is Π-positive and pΠ (αe) = 1. (b) We have 𝜕C ⊂ 𝜕Cℂ . Proof. For (a), let 0 ≠ ℒe𝒯 ∈ LK (𝒞V (X, ℂ), F). There is u ∈ U such that pΩΠu (ℒe𝒯 ) < +∞. Since 𝒯 ≠ 0, by our assumption there is f ∈ ⋂{γu | γ ∈ ℂ, Re γ > 0} such that β = −𝒯 (f ) ≠ 0. For every μ ∈ Π, we have Re μ(e)𝒯 (g) ≤ pΩΠu (ℒe𝒯 ) < +∞ for all g ∈ u. Since γf ∈ u for all γ ∈ ℂ such that Re γ > 0, this implies that Re γμ(e)(−β) ≤ 0 for all such γ. We conclude that 0 ≤ μ(βe) ∈ ℝ. The element βe ∈ F is therefore Π-positive. Moreover, since βe

e

ℒ(1/β)𝒯 = ℒ𝒯 ∈ LK (𝒞V (X, ℂ), F),

Proposition 3.3.7(d) renders that 0 < pΠ (βe) < +∞. Our claim follows with α = ΩΠ

β/pΠ (βe). For (b), let x ∈ 𝜕C . Then there is ΩΠu ∈ K and a point evaluation 𝒟xe ∈ ΔCu . We shall establish that 𝒟x(1/α) ∈ ΔuCℂ with some α ≠ 0. Since 𝒟xe = ℒe𝒟x , Part (a) demonstrates that there is α ≠ 0 such that αe is Π-positive and pΠ (αe) = 1. We write 𝒟xe = ℒαe(1/α) and infer using Proposition 3.3.7(d) that 𝒟x(1/α) ∈ LKℂ (𝒞V (X, ℂ), ℂ) is a 𝒟x point evaluation. Now applying 3.3.7(a) for every choice of β ∈ ℂ, we calculate (β/α) pΩΠu (𝒟xβe ) = pΩΠu (ℒαe ). (β/α) ) = pu (𝒟x 𝒟 x

We proceed to verify that 𝒟x(1/α) is an element of ΔuCℂ . First, we establish that it is

minimal in L(𝒞V (X, ℂ), ℂ). For this, let 𝒯 ∈ L(𝒞V (X, ℂ), ℂ) such that 𝒯 ≺Cℂ 𝒟x(1/α) . Then αe

αe

e

ℒ𝒯 ≺C ℒ𝒟(1/α) = 𝒟x x

e e αe by 3.3.7(b) and, therefore, 𝒟xe ≺C ℒαe 𝒯 because 𝒟x is minimal. Since 𝒟x = ℒ𝒟 (1/α) , x

Proposition 3.3.7(c) yields that 𝒟x(1/α) ≺Cℂ 𝒯 , and the point evaluation 𝒟x(1/α) is seen to be minimal as claimed. All that is left to show is that 𝒟x(1/α) is u-extremal. For this, let 1/α = α1 + α2 for α1 , α2 ∈ ℂ such that

3.3 Special cases and examples

� 227

pu (𝒟x(1/α) ) = pu (𝒟xα1 ) + pu (𝒟xα1 ). Then e = α1 αe + α2 αe, and we have pΩΠu (𝒟xe ) = pu (𝒟x(1/α) ),

pΩΠu (𝒟xα1 αe ) = pu (𝒟xα1 )

and pΩΠu (𝒟xα2 αe ) = pu (𝒟xα2 ) by the above. Thus pΩΠu (𝒟xe ) = pu (𝒟x(1/α) )

= pu (𝒟xα1 ) + pu (𝒟xα1 )

= pΩΠu (𝒟xα1 αe ) + pΩΠu (𝒟xα2 αe ). Because 𝒟xe is ΩΠu -extremal, the latter implies that pΩΠu (𝒟xe ) α1 e = pΩΠu (𝒟xα1 αe )(1/α)e and pΩΠu (𝒟xe )α2 e = pΩΠu (𝒟xα2 αe )(1/α)e, hence pu (𝒟x(1/α) )α1 = pu (𝒟xα1 )(1/α) and pu (𝒟x(1/α) )α2 = pu (𝒟xα2 )(1/α). Thus 𝒟x(1/α) is indeed u-extremal, and we conclude that 𝒟x(1/α) ∈ ΔuCℂ and therefore that x ∈ 𝜕Cℂ . Part 3.3.8(b) is of course the most consequential of these statements. We shall illustrate it in the following example using the real case. 3.3.9 Example. Let X be a compact convex set and suppose that 𝒞V (X) = 𝒞 (X) is endowed with the topology of uniform convergence. Let U = {u}, where u = {f ∈ 𝒞 (X) | f ≤ 1}. Then ⋂{γu | Re γ > 0} is the cone of all negative functions, and the assumptions of Corollary 3.3.8 hold. For ℋ, we choose the cone of all concave functions in 𝒞 (X). We opt for 𝒢 = {ℝ} and the neighborhood u1𝔹 = { f ∈ 𝒞 (X) | |f | ≤ 1} corresponding to ℝ ∈ 𝒢 . Then 𝜕Cℝ is the set of extreme points of X according to classical Choquet theory

228 � 3 Choquet theory (see [1] or [41]). Moreover, Hℝ is the subspace of all affine functions in 𝒞 (X) and Corollary 3.3.5 applies. For F, we choose a sequence space ℓp for 1 < p < +∞ furnished with its pointwise order and for Π the positive elements of the unit ball in F ˚ = ℓq , where 1/p + 1/q = 1. The Π-positive elements of ℓp are its positive sequences. The requirements of Propositions 3.3.7 and 3.3.8 are easily verified. Thus 3.3.8(b) applies, but the main results of Section 3.2 lend themselves only to operators 𝒯 ∈ L(𝒞V (X), ℓp ) such that pΩΠu (𝒯 ) < +∞. Since u contains all negative functions in 𝒞V (X), the latter requires that the operator 𝒯 is positive. We observe that pΠ (c) = ‖c‖p ≤ ‖c‖1 holds for positive sequences c ∈ ℓp and that the norm ‖ ‖1 is monotone and additive on the positive cone of ℓp . Using this, for a positive operator 𝒯 ∈ L(𝒞V (X), ℓp ) we calculate n 󵄨 󵄩 󵄩 󵄨󵄨 f ∈ u, 0 ≤ φi ∈ 𝒞 (X) pΩΠu (𝒯 ) = sup {∑󵄩󵄩󵄩𝒯 (φi ⋅ fi )󵄩󵄩󵄩p 󵄨󵄨󵄨 i } 󵄨󵄨 such that ∑ni=1 φi = 1 i=1 n 󵄨 󵄩 󵄩 󵄨󵄨 0 ≤ φi ∈ 𝒞 (X) ≤ sup {∑󵄩󵄩󵄩𝒯 (φi )󵄩󵄩󵄩1 󵄨󵄨󵄨 } 󵄨󵄨 such that ∑ni=1 φi = 1 i=1 󵄩 󵄩 = 󵄩󵄩󵄩𝒯 (1)󵄩󵄩󵄩1 .

This provides a sufficient condition for pΩΠu (𝒯 ) to be finite. It is however not necessary, as a simple example can show: Let x ∈ X and define 𝒯 ∈ L(𝒞V (X), ℓp ) by 𝒯 (f ) = f (x)c, where c = (1/i)i∈ℕ ∈ ℓp . That is, 𝒯 is the elementary operator ℒcδx (see Section 3.1). Then pΩΠu (𝒯 ) = pΠ (c) sup{f (x) | f ∈ u} = ‖c‖p < +∞ by Proposition 3.1.15(b), but ‖𝒯 (1)‖1 = +∞. If pΩΠu (𝒯 ) < +∞ for an operator 𝒯 ∈ L(𝒞V (X), ℓp ), then the results of Section 3.2 apply. The Choquet boundary 𝜕C is contained in the set of extreme points by Proposition 3.3.8(b) and Proposition 3.2.9 guarantees that there is an operator 𝒮 ≺C ̃ 𝒯 , which is minimal with respect to the order ≺C ̃, and according to Corollary 3.3.5 the representation measure for 𝒮 is supported by any Fσ -subset of X that contains all extreme points of X. (Recall that 𝜕C ̃ ⊂ 𝜕C .) For a concrete example of this setting, let X = [0, 1] and consider the operator 𝒯 ∈ L(𝒞V (X), ℓp ) defined by 2

𝒯 (f ) = ((1/i )f (1/i))i∈ℕ .

Then ∞

󵄩 󵄩 pΩΠu (𝒯 ) ≤ 󵄩󵄩󵄩𝒯 (1)󵄩󵄩󵄩1 = ∑ 1/i2 < +∞ i=1

by the above, and the results of Section 3.2 apply. The Choquet boundary 𝜕C ̃ is contained in the set {0, 1}, and in this example we can easily construct a minimal operator c d 𝒮 ≺C ̃ 𝒯 , which is supported by this set. We have 𝒮 = 𝒟0 + 𝒟1 , where c = (γi )i∈ℕ and p d = (δi )i∈ℕ , both contained in ℓ and easily determined by the fact that 𝒮 and 𝒯

3.3 Special cases and examples

� 229

coincide on all affine functions on [0, 1]. For the function f (x) = x, this requires that δi = 1/i3 for all i ∈ ℕ and for the constant function f (x) = 1 that γi + δi = 1/i2 . Thus γi = (i − 1)/i3 . We recall from Proposition 1.2.9 that the assumptions of this subsection also apply if E is a function space 𝒞U (Y , G), where G is a normed space, Y is compact and U renders the topology of uniform convergence on Y . The space 𝒞V (X, 𝒞U (Y , G)) was understood to be isomorphic to a space 𝒞V ̃ (X × Y , G) in this case. The case that both E and F are either ℝ or ℂ Again we formulate the complex case. In this setting, L(E, F) equals ℂ, the operator space L(𝒞V (X, E), F) is the dual 𝒞V (X, ℂ)˚ of 𝒞V (X, ℂ) endowed with its weak* topology, and the dual 𝒞V (X, ℂ) ⊗ F ˚ of L(𝒞V (X, E), F) equals 𝒞V (X, ℂ) endowed with its weak topology. We shall use the previously established sample settings with Π = {1} and consider the balanced case. That is, K = U, where U is a family of balanced 𝒞 (X)-convex neighborhoods of 0 in 𝒞V (X, E) with the required properties, ℋ is a subspace of 𝒞V (X, ℂ) and either 𝒢 = {ℂ} or 𝒢 = H. For u = Ω{1} u ∈ C and 𝒯 ∈ 𝒞V (X, ℂ)˚ , we have pu (𝒯 ) = sup{Re 𝒯 (f ) | f ∈ u}. Proposition 3.3.1 applies and Proposition 3.3.4 and Corollary 3.3.5 simplify in obvious ways. This special case allows for the extension and further investigation of some of the concepts from the general case. The representation measure θ for an operator 𝒯 ∈ 𝒞V (X, ℂ)˚ is complex-valued. We abbreviate [θ] for the ℝ+ -valued measure [|θ|(𝔹)(1)], which is the usual absolute value of a complex-valued measure. (See Example 2.1.8(f).) Boundedness means that [θ]A < +∞ for all A ∈ R, that is, that [θ] is real-valued. Corollary 3.3.5 applies with [θ] in place of [|θ‖G |(V ∩ G)(μ)]. Moreover, it is well known that there is a measurable function ω such that |ω(x)| = 1 and θ = ω[θ] (see Theorem 6.12 in [55]), that is, θA = ∫A ω d[θ] for all A ∈ R. The measure [θ] defines and represents a functional [𝒯 ] ∈ 𝒞V (X, ℂ)˚ , called the absolute value of 𝒯 and defined by [𝒯 ](f ) = ∫X f d[θ] for f ∈ 𝒞V (X, ℂ). We have [𝒯 ](f ) = sup{Re 𝒯 (g) | g ∈ 𝒞V (X, ℂ), |g| ≤ f } = puf (𝒯 ) 𝔹

for nonnegative functions f ∈ 𝒞V (X) (see also Propositions 2.3.25 and 3.1.11). The statement of Proposition 3.3.6 can be developed further in this particular situation. We extend the notation of the composition of an operator with a bounded continuous real- or complex-valued function from Section 2.6 in the following way: Let 𝒯 ∈ 𝒞V (X, ℂ)˚ and let φ be a bounded measurable complex-valued function. It is straightforward to argue that for all f ∈ 𝒞V (X, ℂ) the function φ ⋅ f is integrable over all sets A ∈ R with respect to the representation measure θ for 𝒯 . We shall

230 � 3 Choquet theory argue that this function is also integrable over X. Indeed, the measure θ is continuous relative to some neighborhood v ∈ V (Theorem 2.6.1), that is, | ∫X g dθ| ≤ 1 for all integrable functions g ≤ v. Then, given ε > 0, since f vanishes at infinity, there is A ∈ R such that χX\A ⋅ f ∈ εv. Thus for A, B, C ∈ R such that A ⊂ C ⊂ B we have χB\C ⋅ f ∈ εv and, therefore, 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 󵄨 󵄨 󵄨󵄨 󵄨󵄨∫ φ ⋅ f dθ − ∫ φ ⋅ f dθ󵄨󵄨󵄨 = 󵄨󵄨󵄨 ∫ φ ⋅ f dθ󵄨󵄨󵄨 < λε, 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 B

C

B\C

where λ ≥ 0 is a bound for the function |φ|. Thus the net (∫A φ ⋅ f )A∈R is indeed convergent and its limit is ∫X φ⋅f . Furthermore, since ∫X φ⋅f ≤ λ for all f ≤ v, the formula 𝒯φ (f ) = ∫X φ ⋅ f dθ defines a functional in 𝒞V (X, ℂ)˚ . Moreover, if θ is the complexvalued representation measure for 𝒯 , then φθ is the representation measure for 𝒯φ . A unimodular function is a measurable function ω ∈ ℱ (X, ℂ) such that |ω| takes only the values 0 and 1 on X. We set Xω = {x ∈ X | |ω(x)| = 1}. For a function f ∈ ℱ (X, ℂ), we define f (x)/|f (x)|, if f (x) ≠ 0

ωf (x) = {

0,

if f (x) = 0.

Hence f = |f | ⋅ ωf . For the following, we assume that one of the neighborhoods used ρ in the setup for U equals u𝔹 , where ρ is a strictly positive lower semicontinuous ℝ-valued function. Proposition 3.3.6 returns the following. ρ

ρ

3.3.10 Proposition. Suppose that u𝔹 ∈ U. If for 𝒯 ∈ 𝒞V (X, ℂ)˚ there is g ∈ ℋ ∩ u𝔹 such that Re 𝒯 (g) = puρ (𝒯 ), then the representation measure for every 𝒮 ∈ 𝒞V (X, ℂ)˚ 𝔹 such that 𝒮 ≺C 𝒯 is supported by the compact set Y = {x ∈ X | |g(x)| = ρ(x)}. We have [𝒮 ] = 𝒮ωg and 𝒮 = [𝒮 ]ωg and puρ (𝒮 ) = puρ (𝒯 ). 𝔹

𝔹

Proof. If the functions ρ and g and the linear functionals 𝒮 , 𝒯 ∈ 𝒞V (X, ℂ)˚ are as stated, we may utilize Proposition 3.3.6 with μ = 1, which states that the representation measure ϑ for 𝒮 is supported by the compact set Y = {x ∈ X | |g(x)| = ρ(x)}. Because g(x) ≠ 0 for all x ∈ Y , the function ωg is continuous on Y and admits a continuous extension φ ∈ 𝒞V (X, ℂ) with compact support and such that |φ| ≤ 1. For every f ∈ 𝒞V (X, ℂ) and x ∈ Y , we have Re (φ ⋅ f )(x)g(x) = |g(x)|Re f (x). Thus according to Proposition 3.3.6, if Re f (x) ≤ 0 for all x ∈ Y , then Re 𝒮 (φ ⋅ f ) ≤ 0. We infer that Re 𝒮 (φ ⋅ f ) = 0 whenever Re f = 0 and Im 𝒮 (φ ⋅ f ) = 0 whenever Im f = 0. Hence 𝒮φ (f ) is real-valued for real-valued functions f and 𝒮φ (f ) ≤ 0 whenever f ≤ 0. Since φ ⋅ f = φ ⋅ Re f + iφ ⋅ Im f , this yields Re 𝒮φ (f ) = 𝒮φ (Re f )

and

Im 𝒮φ (f ) = 𝒮φ (Im f )

3.3 Special cases and examples

� 231

for all f ∈ 𝒞V (X, ℂ). Moreover, because ϑ is supported by Y and (φ ⋅ φ)(x) = 1 for all x ∈ Y , we observe that 𝒮 (f ) = 𝒮φ (f ⋅ φ) holds for all f ∈ 𝒞V (X, ℂ). Using this, for a nonnegative real-valued function f ∈ 𝒞V (X, ℂ), we calculate [𝒮 ](f ) = sup{Re 𝒮 (h) | h ∈ 𝒞V (X, ℂ), |h| ≤ f } = sup{Re 𝒮φ (h ⋅ φ) | h ∈ 𝒞V (X, ℂ), |h| ≤ f }

= sup{𝒮φ (Re(h ⋅ φ)) | h ∈ 𝒞V (X, ℂ), |h| ≤ f } = 𝒮φ (f ),

that is, [𝒮 ] = 𝒮φ = 𝒮ωg . The last equality arises since φ and ωg coincide on Y . Conversely, since 𝒮 (f ) = 𝒮φ (f ⋅ φ) = [𝒮 ](f ⋅ φ) = [𝒮 ]φ (f )

holds for all f ∈ 𝒞V (X, E), we have 𝒮 = [𝒮 ]φ = [𝒮 ]ωg as well. Moreover, 𝒮 ≺C 𝒯 implies that puρ (𝒮 ) ≤ puρ (𝒯 ) and 𝒮 (g) = 𝒯 (g). Thus 𝔹

𝔹

puρ (𝒮 ) ≥ Re 𝒮 (g) = Re 𝒯 (g) = puρ (𝒯 ). 𝔹

𝔹

Hence our claim. η

Following Proposition 3.3.1, a point evaluation 𝒟x for x ∈ X and 0 ≠ η ∈ ℂ is minimal if and only if 𝒟x = 𝒟x1 is minimal. Moreover, if 󵄨 󵄨 0 < pu (𝒟xη ) = |η| sup{󵄨󵄨󵄨f (x)󵄨󵄨󵄨 | f ∈ u} < +∞ η

η

η

for some u ∈ U and if η = η1 + η2 and pu (𝒟x ) = pu (𝒟x 1 ) + pu (𝒟x 2 ) holds for 0 ≠ η1 , η2 ∈ ℂ, then η/|η| = η1 /|η1 | = η2 /|η2 | and, therefore, pu (𝒟xη ) η1 = pu (𝒟xη1 ) η

and

pu (𝒟xη ) η2 = pu (𝒟xη2 ) η. η

This demonstrates that any point evaluation operator 𝒟x ∈ 𝒞V (X, ℂ)˚ is K-extremal, provided that 0 < pu (𝒟x ) < +∞ for some u ∈ U. We summarize with the following. η

3.3.11 Lemma. A point evaluation 𝒟x for x ∈ X and 0 ≠ η ∈ ℂ is an element of ΔC if and only if 𝒟x is minimal and 0 < pu (𝒟x ) < +∞ for some u ∈ U. A 𝒞 (X)-convex subset w of 𝒞V (X, ℂ) is C-superharmonic if and only if pu (𝒯 ) ≤ pu (𝒟x ) whenever 𝒯 ≺C 𝒟x for 𝒯 , 𝒟x ∈ 𝒞V (X, ℂ)˚ , where 𝒟x is a point evaluation such that 0 < pu (𝒟x ) < +∞ for some u ∈ U. 3.3.12 Proposition. Let Y be a closed subset of X. Suppose that for every y ∈ Y there is ρ ρ u𝔹y ∈ U and gy ∈ ℋ ∩ u𝔹y such that |gy (y)| = ρy (y) and |gy (x)| < ρy (x) for all x ∈ X \ Y . Then the following hold: (a) If 𝒯 ≺C 𝒟y for 𝒯 , 𝒟y ∈ 𝒞V (X, ℂ)˚ , where 𝒟y is a point evaluation such that y ∈ Y , then the representation measure for 𝒯 is supported by Y and 𝒯 = [𝒯 ]γωg , where γ = ωgy (y).

y

232 � 3 Choquet theory ˚ (b) If C contains an absorbing U-bounded set, if 𝒮 ≺C ̃ 𝒯 for 𝒮 , 𝒯 ∈ 𝒞V (X, ℂ) , and if the representation measure for 𝒯 is supported by Y , then the representation measure for 𝒮 is also supported by Y . ρ

Proof. (a) Let y ∈ Y . We employ Proposition 3.3.10 with u𝔹y ∈ U, the function g = ρ γgy ∈ u𝔹y ∩ L, where γ = ωgy (y) = gy (y)/|gy (y)| and the point evaluation 𝒟y in place of 𝒯 . We have 󵄨 󵄨 󵄨 󵄨 g(y) = 󵄨󵄨󵄨g(y)󵄨󵄨󵄨 = 󵄨󵄨󵄨gy (y)󵄨󵄨󵄨 = ρy (y), and puρy (𝒟y ) = ρy (x) = g(x) = 𝒟y (g), 𝔹

as required. Thus, if 𝒯 ≺C 𝒟y for 𝒯 ∈ 𝒞V (X, ℂ)˚ , then according to 3.3.10 the representation measure for 𝒯 is supported by Y and 𝒯 = 𝒯ωg = [𝒯 ]γωg For Part (b), y suppose that C contains an absorbing U-bounded set u. Then the set ũ = {f ∈ w | f |Y = 0} is also U-bounded, and 𝒞 (X)-superharmonic as we go on to demonstrate. For this, let 𝒯 ≺C 𝒟x for x ∈ X such that 0 < pw (𝒟x ) < +∞ for some w ∈ U. If x ∈ Y , then 𝒯 is supported by Y by Part (a) and we have pũ (𝒯 ) = pũ (𝒟x ) = 0. If x ∈ X \ Y , we find a function φ ∈ 𝒞𝒦 (X) such that 0 ≤ φ ≤ χX\Y and φ(x) = 1. Then φ ⋅ f ∈ ũ whenever f ∈ u and 󵄨 󵄨 pu (𝒟x ) = sup{󵄨󵄨󵄨f (x)󵄨󵄨󵄨 | f ∈ u} 󵄨 󵄨 = sup{󵄨󵄨󵄨(φ ⋅ f )(x)󵄨󵄨󵄨 | f ∈ u} 󵄨 󵄨 = sup{󵄨󵄨󵄨f (x)󵄨󵄨󵄨 | f ∈ ũ} = pũ (𝒟x ). Thus pũ (𝒯 ) ≤ pu (𝒯 ) ≤ pu (𝒟x ) = pũ (𝒟x ). ̃ Hence We infer that ũ is indeed 𝒞 (X)-superharmonic and, therefore, an element of C. ˚ we have pũ (𝒮 ) ≤ pũ (𝒯 ) = 0 whenever 𝒮 ≺C ̃ 𝒯 for 𝒮 , 𝒯 ∈ 𝒞V (X, ℂ) such that the representation measure for 𝒯 is supported by Y . That is, pũ (𝒮 ) = 0. Now Proposition 2.5.9 yields that the representation measure for 𝒮 is also supported by Y . Indeed, let φ ∈ 𝒞𝒦 (X) and a ∈ E such that φ is supported by X \ Y . Since u is absorbing and balanced, there is ε > 0 such that γφ⋅a ∈ ũ for all |γ| ≤ ε. Hence |𝒮 (φ⋅a)| ≤ pũ (𝒮 ) = 0. Our claim in Part (b) follows.

3.3 Special cases and examples

� 233

In preparation for our next result, for a closed subset Y of X and f ∈ ℱ (X, ℂ), we abbreviate f |Y for χY ⋅ f . For a 𝒞 (X)-convex subset u of 𝒞V (X, ℂ) and Y ⊂ X we set u|Y = { f ∈ 𝒞V (X, ℂ) | f |Y = g|Y for some g ∈ u} and observe the following. 3.3.13 Lemma. Let u be a 𝒞 (X)-convex subset of 𝒞V (X, ℂ) and let Y be a closed subset of X. Let 𝒮 , 𝒯 ∈ 𝒞V (X, ℂ)˚ , and let ϑ and θ be their respective representation measures. (a) If pu|Y (𝒯 ) < +∞, then θ is supported by Y . (b) If θ is supported by Y , then pu|Y (𝒯 ) = pu (𝒯 ). ̃ if pu|Y (𝒯 ) < +∞, and if 𝒮 ≺̃ 𝒯 , then ϑ is supported by Y . (c) If u|Y ∈ C, C (d) If Y is the union of a family {Yi }i∈ℐ of closed subsets of X and if all the sets u|Yi are C-superharmonic, then u|Y is also C-superharmonic. Proof. For (a), suppose that there is x ∈ supp θ for x ∉ Y . There is g ∈ 𝒞V (X, ℂ) such that g|Y = 0 and 𝒯 (g) = ∫X g dθ ≠ 0. Thus f + αg ∈ u|Y whenever f ∈ u and α ∈ ℂ. Hence pu|Y (𝒯 ) = +∞, our claim. For (b), suppose that θ is supported by Y . Then 󵄨 pu|Y (𝒯 ) = sup{Re ∫ f dθ 󵄨󵄨󵄨 f ∈ u|Y } Y

󵄨 = sup{Re ∫ f dθ 󵄨󵄨󵄨 f ∈ u} = pu (𝒯 ). Y

̃ if 𝒮 ≺̃ 𝒯 and if pu|Y (𝒯 ) < +∞, then pu|Y (𝒮 ) ≤ pu|Y (𝒯 ) < +∞. For (c), if u|Y ∈ C, C Hence by (a) the representation measure for 𝒮 is supported by Y . We use Lemma 3.3.11 for Part (d). Suppose that the sets u|Yi are C-superharmonic and let 𝒮 ≺C 𝒟x for 𝒮 ∈ 𝒞V (X, ℂ)˚ and x ∈ X such that 0 < pw (𝒟x ) < +∞ for some w ∈ U. There is nothing to prove if pu|Y (𝒟x ) = +∞. Otherwise, we have x ∈ Y by Part (a), that is, x ∈ Yi for some i ∈ ℐ . Then pu|Yi (𝒟x ) = pu (𝒟x ) = pu|Y (𝒟x ) < +∞ by Part (b). Now, since 𝒮 ≺C 𝒟x implies that pu|Yi (𝒮 ) ≤ pu|Yi (𝒟x ) < +∞, Part (a) renders that the representation measure ϑ for 𝒮 is supported by Yi . Thus pu|Y (𝒮 ) = pu|Yi (𝒮 ) ≤ pu|Yi (𝒟x ) = pu|Y (𝒟x ). Given a unimodular function ω, for every f ∈ ℱ (X, ℂ) we define its ω-real and ω ω ω-imaginary parts as the real-valued functions Re f = Re(f ⋅ ω) and Im f = Im(f ⋅ ω). Then

234 � 3 Choquet theory ω

ω

Re f ⋅ ω + i Im f ⋅ ω = f |Xω , where Xω = {x ∈ X | |ω(x)| = 1}. If both f and ω are continuous on a subset Y of ω ω X, so are both Re f and Im f . The following criterion is formulated in very general, therefore, somewhat complicated terms. 3.3.14 Proposition. Let ω ∈ ℱ (X, ℂ) be a unimodular function and let Y be a closed ρ subset of Xω . Suppose that for every y ∈ Y there is u𝔹y ∈ U and a continuous nonnegaρ tive real-valued function φy such that φy ⋅ ω ∈ ℋ ∩ u𝔹y and such that φy (y) = ρy (y) and φy (x) < ρy (x) for all x ∈ X \ Y . Then the following hold: (a) If u ⊂ 𝒞V (X, ℂ) is C-superharmonic, then u|Y is C-superharmonic. ω (b) If u ⊂ 𝒞V (X, ℂ) and u|Y is C-superharmonic, then {Re f ⋅ ω | f ∈ u}|Y is C-superharmonic. (c) If f1 , . . . , fn and g1 , . . . , gm are functions in ℋ, then both sets {h ⋅ ω | h ∈ 𝒞 (X), ∨ni=1 Re fi |Y ≤ h|Y }|Y ω

and {h ⋅ ω | h ∈ 𝒞 (X), h|Y ≤ ∧m k=1 Re gk |Y }|Y ω

are C-superharmonic, so is their intersection, provided it is not empty. ˚ (d) If {0}|Y is U-bounded, if 𝒮 ≺C ̃ 𝒯 for 𝒮 , 𝒯 ∈ 𝒞V (X, ℂ) , and if the representation measure for 𝒯 is supported by Y , then the representation measure for 𝒮 is also supported by Y . Proof. Our claims in Parts (a) to (c) are concerned with C-superharmonicity of sets u|Y , where u is a 𝒞 (X)-convex subset of 𝒞V (X, ℂ). For this, let 𝒮 ≺C 𝒟x for x ∈ X such that 0 < pw (𝒟x ) < +∞ for some w ∈ U. We need to establish that pu|Y (𝒮 ) ≤ pu|Y (𝒟x ). If x ∉ Y , then pu|Y (𝒮 ) ≤ pu|Y (𝒟x ) = +∞ by Lemma 3.3.13(a) for every 𝒞 (X)-convex subset u of 𝒞V (X, ℂ). Thus if investigating 𝒞 (X)-superharmonicity of the set u|Y we only need to consider point evaluations supported by Y . If x ∈ Y , we employ ρ Proposition 3.3.12(a) with the functions gy = φy ⋅ ω ∈ u𝔹y ∩ L for all y ∈ Y . Then |gy (y)| = φy (y) = ρy (y) and |gy (x)| < ρy (x) for all x ∈ X \ Y . Consequently, if 𝒮 ≺C 𝒟x according to 3.3.12(a) the representation measure for 𝒮 is supported by Y and we have [𝒮 ] = 𝒮γω and 𝒮 = [𝒮 ]γω , where γ = ω(x). The latter follows since ωgy = ω. From this, we infer that pu|Y (𝒮 ) = pu (𝒮 ) by Lemma 3.3.13(b). Therefore, if u ⊂ 𝒞V (X, ℂ) is C-superharmonic as supposed in Part (a), we have pu|Y (𝒮 ) ≤ pu|Y (𝒟x ) and the set u|Y is seen to be also C-superharmonic. In Parts (b) and (c), we claim C-superharmonicity for sets w|Y , where f ⋅ ω is a real-valued function for all f ∈ w. For function of this type, we observe that Re 𝒮 (f ) = Re [𝒮 ](γf ⋅ ω) = Re γ [𝒮 ](f ⋅ ω) = (Re γ)[𝒮 ](f ⋅ ω)

3.3 Special cases and examples

� 235

Using this, we calculate pw|Y (𝒮 ) = pw (𝒮 ) = sup{Re 𝒮 (f ) | f ∈ w} = sup{(Re γ) [𝒮 ](f ⋅ ω) | f ∈ w}. If Re γ = 0, this shows pw|Y (𝒮 ) = pw (𝒮 ) = pw|Y (𝒟x ) = 0. Thus we may assume that ω Re γ ≠ 0. Furthermore, we notice that if f = Re h ⋅ ω for some h ∈ 𝒞V (X, E), then ω

[𝒮 ](f ⋅ ω) = [𝒮 ](Re h) = [𝒮 ](Re(h ⋅ ω)) = Re [𝒮 ](h ⋅ ω) = Re [𝒮 ](γh ⋅ (γω)) = Re 𝒮 (γh). Now for Part (b) let u be a C-superharmonic subset of C and let ω

w = {Re h ⋅ ω | h ∈ u}. ω

Then for f = Re h ⋅ ω ∈ w we have (Re γ)[𝒮 ](f ⋅ ω) = (Re γ)Re 𝒮 (γh) by the preceding. We set ζ = γ/Re γ and evaluate pw|Y (𝒮 ) = sup{(Re γ) Re 𝒮 (γh) | h ∈ u} = sup{Re 𝒮 (h) | h ∈ ζ u} = pζ u (𝒮 ). According to Proposition 3.3.1(c), the set ζ u is also C-superharmonic. Hence pw|Y (𝒮 ) = pζ u (𝒮 ) ≤ pζ u (𝒟x ) = pw|Y (𝒟x ). This argument demonstrates that w|Y is indeed C-superharmonic as claimed in Part (b). For Part (c), let f1 , . . . , fn ∈ L and let w = {h ⋅ ω | h ∈ 𝒞 (X), ∨ni=1 Re fi |Y ≤ h|Y }|Y . ω

We abbreviate f = ∨ni=1 Re fi . Let ϑ be the representation measure for 𝒮 , which is supported by Y . For f = h ⋅ ω ∈ w, we have [𝒮 ](f ⋅ ω) = [𝒮 ](h) = ∫Y h d[ϑ] and ω

󵄨 pw|Y (𝒮 ) = sup{Re γ ∫ h d[ϑ] 󵄨󵄨󵄨 h ∈ 𝒞 (X), f ≤ h|Y } Y

by the above. Thus pw|Y (𝒟x ) = sup{h(x) | h ∈ 𝒞 (X), f ≤ h|Y } = +∞, if Re γ > 0, and our requirement is satisfied. If Re γ < 0, then 󵄨 pw|Y (𝒮 ) = Re γ inf{∫ h d[ϑ] 󵄨󵄨󵄨 h ∈ 𝒞 (X), f ≤ h|Y } = Re γ ∫ f d[ϑ]. Y

Y

236 � 3 Choquet theory ω

In particular, pw|Y (𝒟x ) = Re γf (x). We remark that ∫Y f d[ϑ] ≥ ∫Y Re fi d[ϑ], hence ω

ω

pw|Y (𝒮 ) ≤ Re γ ∫ Re fi d[ϑ] = Re γ [𝒮 ](Re fi ) Y

for all i = 1, . . . , n. For the functions fi ∈ L, we have 𝒮 (fi ) = 𝒟x (fi ). Thus ω

ω

ω

[𝒮 ](Re fi ) = Re 𝒮 (γfi ) = Re 𝒟x (γfi ) = 𝒟x (Re fi ) = Re fi (x) by the above and pw|Y (𝒮 ) ≤ ∧ni=1 Re γ [𝒮 ](Re fi ) = Re γ ∨ni=1 (Re fi )(x) ω

ω

= Re γ(f )(x) = pw|Y (𝒟x ).

Hence pγu (𝒮 ) ≤ pγu (𝒟x ) holds in any case, and the set w|Y is seen to be C-superharmonic as claimed in Part (c). For the second statement in (c), we set u = {h ⋅ ω | h ∈ 𝒞 (X) and h|Y ≤ ∧m k=1 Re gk |Y }|Y ω

and observe that u = −{h ⋅ ω | h ∈ 𝒞 (X) and ∨m k=1 Re (−gk )|Y ≤ h|Y }|Y . ω

Our claim follows from the first statement with Proposition 3.3.1(c). If the intersection of ω ω u and w is not empty, that is, if Re fi |Y ≤ Re gk |Y for all i, k, then Proposition 3.2.18 yields that the set u ∩ w = {h ⋅ ω | h ∈ 𝒞 (X), ∨ni=1 Re fi |Y ≤ h|Y ≤ ∧m k=1 Re gk |Y }|Y ω

ω

is also C-superharmonic. For this, we abbreviate g = ∧m k=1 Re gk |Y . For x ∈ Y , a point evaluation 𝒟x and γ = ω(x) using the above we have pu (𝒟x ) = pw (𝒟x ) = pu∩w (𝒟x ) = 0 if Re γ = 0, we have pw (𝒟x ) = Re γf (x), pu (𝒟x ) = +∞ and pu∩w (𝒟x ) = Re γf (x) if Re γ > 0, and we have pw (𝒟x ) = +∞, pu (𝒟x ) = Re γg(x) and pu∩w (𝒟x ) = Re γg(x) if Re γ < 0. Thus pu∩w (𝒟x ) = pu (𝒟x ) ∧ pw (𝒟x ) holds in any case, as is required in 3.2.18. Finally, for Part (d) suppose that 𝒮 ≺C ̃ 𝒯 and that 𝒯 is supported by Y . The set ̃ We have {0}|Y is C-superharmonic by Part (a), and if it is U-bounded, an element of C. p{0}|Y (𝒯 ) = p{0} (𝒯 ) = 0 by Lemma 3.3.14(b). Thus the support of the representation measure for 𝒮 is contained in Y by 3.3.14(d). ω

Given a unimodular function ω ∈ ℱ (X, ℂ), for an operator 𝒯 ∈ 𝒞V (X, ℂ)˚ we ω ω define its ω-real and ω-imaginary parts Re 𝒯 and Im 𝒯 as the operators ω

Re 𝒯 (f ) = Re 𝒯ω (Re f ) + i Re 𝒯ω (Im f )

3.3 Special cases and examples

� 237

and ω

ω

Im 𝒯 (f ) = Im 𝒯ω (Re f ) + i Im 𝒯ω (Im f ) ω

ω

for f ∈ 𝒞V (X, ℂ). We observe that 𝒯ω = Re 𝒯 + i Im 𝒯 , that is, ω

ω

𝒯χX = (Re μ)ω + i μ(Im μ)ω , ω

ω

ω

where Xω = {x ∈ X | |ω(x)| = 1}. Both operators Re 𝒯 , Im 𝒯 ∈ 𝒞V (X, ℂ)˚ take real values at real-valued functions, hence their respective representation measures are real-valued (see Corollary 2.6.2 with the insertion of ℝ for both E+ and F+ ). Operators ℛ ∈ 𝒞V (X, ℂ)˚ of this type can be split into positive and negative parts that are represented by positive measures, that is, ℛ = ℛ+ − ℛ− , where ℛ+ (f ) = sup{ℛ(g) | 0 ≤ g ≤ f }

and ℛ− (f ) = sup{ℛ(g) | −f ≤ g ≤ 0}

for nonnegative functions f ∈ 𝒞V (X). The representation measures for ℛ+ and ℛ− are mutually singular (see Proposition 11.22 in [53]). We use Proposition 3.3.14 for the following corollary. 3.3.15 Corollary. Let ω ∈ ℱ (X, ℂ) be a unimodular function and let Y be a closed subset ρ of Xω such that {0}|Y is U-bounded. Suppose that for every y ∈ Y there is u𝔹y ∈ K and ρy a continuous nonnegative real-valued function φy such that φy ⋅ ω ∈ u𝔹 ∩ L and such ˚ that φy (y) = ρy (y) and φy (x) < ρy (x) for all x ∈ X \Y . If 𝒮 ≺C ̃ 𝒯 for 𝒮 , 𝒯 ∈ 𝒞V (X, ℂ) and if the representation measure for 𝒯 is supported by Y , then ω

ω

(Re 𝒮 )ω ≺C ̃ (Re 𝒯 )ω

ω

ω

and (Im 𝒮 )ω ≺C ̃ (Im 𝒯 )ω .

If f1 , . . . , fn are functions in ℋ such that f = ∧ni=1 Re fi |Y ≥ 0, then ω

ω

ω

(Re 𝒮 )+ (f ) ≤ (Re 𝒯 )+ (f )

ω

ω

and (Re 𝒮 )− (f ) ≤ (Re 𝒯 )− (f )

as well as ω

ω

(Im 𝒮 )+ (f ) ≤ (Im 𝒯 )+ (f )

ω

and (Im 𝒮 )− (f ) ≤ (Im 𝒯 )− (f ).

Proof. Since u|Y = u + {0}|Y for every 𝒞 (X)-convex subset u of 𝒞V (X, ℂ), the set ̃ provided that it u|Y is U-bounded whenever u is U-bounded and an element of C, is C-superharmonic in addition. If 𝒮 ≺C ̃ 𝒯 and if the representation measure for 𝒯 is supported by Y , following Proposition 3.3.14(d) the representation measure for 𝒮 ω ω is also supported by Y . Hence also the representation measures for Re 𝒮 and Re 𝒯 and their respective positive and negative parts. The functionals pu and pu|Y therefore ̃ and set w = {Reω f ⋅ ω | f ∈ u}. For a coincide for these (Lemma 3.3.13(b)). Let u ∈ C function f ∈ 𝒞V (X, E), we observe that

238 � 3 Choquet theory ω

(Re 𝒯 )ω (f ) = Re 𝒯ω (Re f ⋅ ω) + i Re 𝒯ω (Im f ), hence ω

ω

Re ((Re 𝒯 )ω (f )) = Re 𝒯ω (Re f ⋅ ω) = Re 𝒯 (Re f ⋅ ω). Using this, we evaluate ω

ω

pu ((Re 𝒯 )ω ) = sup{Re((Re 𝒯 )ω (f )) | f ∈ u} ω

= sup{Re 𝒯 (Re f ⋅ ω) | f ∈ u} = pw (𝒯 ) = pw|Y (𝒯 ) ω

by the above, and likewise pu ((Re 𝒮 )ω ) = pw|Y (𝒮 ). Since w|Y is C-superharmonic ̃ we infer that by 3.3.14(a) and U-bounded by the above, hence an element of C, ω

ω

pu ((Re 𝒮 )ω ) = pw|Y (𝒮 ) ≤ pw|Y (𝒯 ) = pu ((Re 𝒯 )ω ). ω

ω

Thus (Re 𝒮 )ω ≺C ̃ (Re 𝒯 )ω . The argument for the imaginary parts is similar. For the subsequent statements, we use the set u = {h ⋅ ω | h ∈ 𝒞 (X), 0 ≤ h|Y ≤ ∧ni=1 Re fi |Y }|Y , ω

where f1 , . . . , fn ∈ ℋ such that f = ∧ni=1 Re fi |Y ≥ 0. This set is U-bounded by our asω sumptions and C-superharmonic by Proposition 3.3.14(c). We infer that pu ((Re 𝒮 )ω ) ≤ ω pu ((Re 𝒯 )ω ) holds by our first argument, and continue to calculate that ω

ω

ω

ω

pu ((Re 𝒯 )ω ) = sup{Re 𝒯 (h) | 0 ≤ h|Y ≤ f } = (Re 𝒯 )+ (f ), ω

ω

ω

ω

and likewise pu ((Re 𝒮 )ω ) = (Re 𝒮 )+ (f ). This shows (Re 𝒮 )+ (f ) ≤ (Re 𝒯 )+ (f ) as claimed. The remaining arguments for the negative and imaginary parts are similar. 3.3.16 Examples. (a) Let X = ℂ and let V be the neighborhood system of compact ρ convergence (see Example 1.2.7(b)) generated by the neighborhoods uK = u𝔹K , where 𝔹 is the unit ball of ℂ, K is a compact subset of X = ℂ and ρK (x) = 1 for x ∈ K and ρK (x) = +∞, else. We have 𝒞V (ℂ, ℂ) = 𝒞 (ℂ, ℂ) in this case. The dual of 𝒞V (ℂ, ℂ) is represented by the space of all regular complex-valued Borel measures on ℂ with compact support (see Theorem 2.6.1). In the settings of this subsection, let K be a fixed compact subset of X, let K = {uK }, 𝒢 = H, and let ℋ be the subspace of all anã of C for the results of this lytic functions in 𝒞 (ℂ, ℂ). We shall use the completion C subsection. The assumptions of Corollary 3.3.5, Proposition 3.3.14 and Corollary 3.3.15 are satisfied with Y = K, with ω = 1, the constant function x 󳨃→ 1 for φy and ρy = ρK for all y ∈ K. For 𝒯 ∈ 𝒞 (ℂ, ℂ)˚ , we have puK (𝒯 ) < +∞ if and only if the representation measure for 𝒯 is supported by K. In particular, the Coquet boundary u u 𝜕̃K is a subset of K. Moreover, if x is in the topological interior of K, then x ∉ 𝜕̃K . C

C

3.3 Special cases and examples

� 239

Indeed, there is ε > 0 such that the ball Bε (x) = {z ∈ 𝔹 | |z − x| ≤ ε} is contained in K. Let 0 ≠ η ∈ ℂ and consider the functional 𝒯 ∈ 𝒞 (ℂ, ℂ)˚ defined by 2π

it

𝒯 (f ) = (η/2π) ∫ f (x + εe ) dt. 0 η

Then 𝒯 (f ) = 𝒟x (f ) for all f ∈ ℋ by the Cauchy integral formula. Furthermore, η η η since puK (𝒟x ) = puK (𝒯 ) = |η|, we infer that 𝒯 ≺C 𝒟x , hence 𝒯 ≺C ̃ 𝒟x . Because η η Corollary 3.3.5 applies and [𝒯 ] ≠ |𝒟x |, we conclude that 𝒯 ≁ 𝒟x , and the point evalη ˚ uation 𝒟x is not minimal in the ordering ≺C ̃ . If puK (𝒯 ) < +∞ for 𝒯 ∈ 𝒞 (ℂ, ℂ) , then by Corollary 3.2.9 there exists a minimal (with respect to the order ≺C ̃ ) operator 𝒯 ∈ 𝒞 (ℂ, ℂ)˚ , which coincides with 𝒯 on ℋ, such that puK (ℐ ) ≤ puK (𝒯 ), and according to Corollary 3.3.5 the representation measure for 𝒯 is supported by the topological boundary of K. (b) Let X = ℝ and consider the neighborhood system V generated by the function space neighborhood vρ , which is defined by the neighborhood function n = ρ⋅𝔹 on ℝ, where 𝔹 = [−1, +1] is the unit ball of ℝ and ρ(x) = (1 + x 2 ). Then 𝒞V (ℝ, ℂ) consists of all continuous complex-valued functions f on ℝ such that for every ε > 0 there is a compact subset K of ℝ such that |f (x)| ≤ ερ(x) for all x ∈ ℝ \ K. In the settings ρ of this subsections, let U = {u𝔹 }, let 𝒢 + H and let ℋ be the subspace of 𝒞V (ℝ, ℂ) generated by the functions f1 (x) = eiπx and f2 (x) = xeiπx . We shall establish that u

ρ

𝜕C = 𝜕C𝔹 = [−1, +1]. Following Lemma 3.3.11, it suffices to investigate point evaluations 𝒟y in order to establish this claim. Let 𝒮 ≺C 𝒟y for y ∈ ℝ. First, suppose that |y| > 1. We shall demonstrate that 𝒟y ∉ ΔC . For this, we set 𝒮 = λ𝒟+1 + κ𝒟−1 , where λ = −(1/2)eiπy (y + 1) and κ = (1/2)eiπy (y − 1). It is straightforward to verify that 𝒟y and 𝒮 coincide on the functions in ℋ and that puρ (𝒮 ) = |λ|ρ(1) + |κ|ρ(−1) = |y + 1| + |y − 1| < 1 + y2 = puρ (𝒟y ). 𝔹

𝔹

Thus 𝒮 ≺C 𝒟y , but 𝒮 ≁ 𝒟y since puρ (𝒮 ) ≠ puρ (𝒟y ). The point evaluation 𝒟y is 𝔹 𝔹 therefore not minimal and not an element of ΔC . We infer that y ∉ 𝜕C . Now suppose that |y| ≤ 1 and consider the function gy ∈ ℋ defined by gy (x) = (2yx + (1 − y2 ))eiπ(x−y) for all x ∈ ℝ. We realize that 󵄨󵄨 󵄨 2 2 2 2 󵄨󵄨gy (x)󵄨󵄨󵄨 ≤ 2|y||x| + (1 − y ) ≤ (x + y ) + (1 − y ) = ρ(x) ρ

holds for all x ∈ ℝ. Hence gy ∈ ℋ ∩ u𝔹 . Moreover, if x = y, then |gy (x)| = ρ(x). Conversely, if |gy (x)| = ρ(x), then |x| = |y| and indeed x = y in case that |y| < 1. In case that |y| = 1, we have |gy (1)| = ρ(1) and |gy (−1)| = ρ(−1). Now we use Propo-

240 � 3 Choquet theory sition 3.3.10 with 𝒟y in place of 𝒯 and gy in place of g for our argument. Then puρ (𝒟y ) = ρ(y) = gy (y) = 𝒟y (g). In case that |y| < 1, we have 𝔹

󵄨 󵄨 Y = {x ∈ X | 󵄨󵄨󵄨gy (x)󵄨󵄨󵄨 = ρ(x)} = {y} by the above. Proposition 3.3.10 returns that 𝒮 is supported by {y} and that 𝒮 = [𝒮 ]ω , where ω = χY ⋅ gy /|gy |, that is, ω(y) = 1. We conclude that 𝒮 = λ𝒟y with some λ ≥ 0, and indeed λ = 1 since puρ (𝒮 ) = puρ (𝒟y ) ≠ 0, also by 3.3.10. Hence 𝒮 = 𝒟y and, 𝔹 𝔹 therefore, 𝒟y ∈ ΔC and y ∈ 𝜕C . In case that |y| = 1, say y = +1, we have 󵄨 󵄨 Y = {x ∈ X | 󵄨󵄨󵄨g(x)󵄨󵄨󵄨 = ρ(x)} = {−1, +1}. Thus 𝒮 is supported by Y and 𝒮 = λ𝒟+1 − κ𝒟−1 with κ, λ ≥ 0 because ω(1) = 1 and ω(−1) = −1. For the functions f1 , f2 ∈ ℋ, we calculate −1 = 𝒟+1 (f1 ) = 𝒮 (f1 ) = −(λ − κ) and

−1 = 𝒟+1 (f2 ) = 𝒮 (f2 ) = −(λ + κ).

Thus λ = 1 and κ = 0 and, therefore, 𝒮 = 𝒟+1 . The case that y = −1 is similar. This yields [−1, +1] ⊂ 𝜕C , and our claim follows. The assumptions of Propositions 3.3.12 and 3.3.14(a) to (c) are satisfied with Y = [−1, +1], the unimodular function ω(x) = eiπx and the functions ρ for ρy and φy = |gy |. The case that both E and F are ordered topological vector spaces In this setting, we assume that both E and F are ordered topological vector spaces over ℝ or ℂ with closed positive cones E+ and F+ , respectively. (See Sections 1.2 and 2.6.) We abbreviate E− for −E+ and E+˚ for the positive cone of E ˚ , that is, the polar of E− . We note that a ≤ b holds for a, b ∈ E if and only if Re μ(a) ≤ Re μ(b) for all μ ∈ E+˚ . Similar notation applies to F. The order relation of E transfers to 𝒞V (X, E), and its positive cone is the closure of the subcone of all E+ -valued functions. We intend to investigate positive operators in L(𝒞V (X, E), F), that is, operators that map E+ -valued functions into F+ (or equivalently, E− -valued functions into F− ). For this, we shall use the previously formulated sample settings in the following way. Let U be family of 𝒞 (X) -convex neighborhoods of 0 in 𝒞V (X, E) with the prescribed properties and require that each u ∈ U contains all E− -valued functions in 𝒞V (X, E). Let Π be a subset of F+˚ such that the weak* closure of its conic hull is all of F+˚ and such that the balanced core bcore(Π˝ ) of its polar in F is weakly compact. This construction requires that ⋂|γ|=1 γF+ = {0}. Indeed, if γc ∈ F+ for all |γ| = 1, then μ(c) = 0 for all μ ∈ Π and c = 0 since the linear span of Π is weak*-dense in F ˚ . Moreover, the first condition on Π guarantees that c ≤ d holds for elements c, d ∈ F whenever Re μ(c) ≤ Re μ(d)

3.3 Special cases and examples

� 241

for all μ ∈ Π. We have K = {ΩΠu | u ∈ U}. Let C0 be a collection of subsets ΩΓw of 𝒞V (X, E) ⊗ F ˚ , such that Γ ⊂ Π and w ⊂ 𝒞V (X, E) is 𝒞 (X)-convex and U-bounded. The Choquet cone C is the cone of subsets of 𝒞V (X, E) ⊗ F ˚ generated by K and C0 . We note that all operators 𝒯 ∈ LK (𝒞V (X, E), F) are weakly compact and positive in this setting. Hence the values of their representation measures are positive operators in L(E, F) (see Corollary 2.6.2). Indeed, if pΩΠu (𝒯 ) < +∞ for ΩΠu ∈ K, then by our assumption u contains all functions f ∈ 𝒞V (X, E− ), hence 𝒯 (λf ) ∈ Π˝ for all λ ≥ 0. That is, (𝒯 (f ))(μ) ≤ 0 for all μ ∈ Π and indeed for all μ in F+˚ since all elements of F+˚ are absorbed by Π. Thus 𝒯 (f ) ≤ 0 as claimed. We continue to formulate our observations for the case that both E and F are ordered topological vector spaces over ℂ. The real case is similar. We further specify our settings for C0 , hence for C, as we did in the sample settings for the general case. ρ Let ℋ be a subcone of 𝒞V (X, E), and let 𝒢 and the sets uV be as stated in the general {μ} {μ} settings. That is, C0 consists of all sets Ω{f } for f ∈ ℋ together with all Ω ρ for uV ∩G

G ∈ 𝒢 and all μ ∈ Π.

3.3.17 Proposition. Suppose that 𝒮 ≺C 𝒯 for 𝒮 , 𝒯 ∈ LK (𝒞V (X, E), F). (a) Then 𝒮 (f ) ≤ 𝒯 (f ) for all f ∈ ℋ. ρ ρ (b) If the conic hull of Π is F ˚ , then 𝒮 (uV ∩G ) ⊂ 𝒯 (uV ∩G ) + F− for all G ∈ 𝒢 . Proof. Let 𝒮 ≺C 𝒯 for 𝒮 , 𝒯 ∈ LK (𝒞V (X, E), F). (a) For f ∈ ℋ, we have Re μ(𝒮 (f )) ≤ Re μ(𝒯 (f )) for all μ ∈ Π, that is, Re μ(𝒮 (f )) ≤ Re μ(𝒯 (f )) for all μ ∈ F+˚ and, thereρ

ρ

ρ

fore, 𝒮 (f ) ≤ 𝒯 (f ). (b) If 𝒮 (uV ∩G ) ⊄ 𝒯 (uV ∩G ) + F− for G ∈ 𝒢 , then there are f ∈ uV ∩G and ν ∈ F ˚ such that ρ

Re ν(𝒮 (f )) > sup{Re ν(𝒯 (g) + c) | g ∈ uV ∩G , c ∈ F− }. The latter implies that Re ν(c) ≤ 0 for all c ∈ F− , that is, ν ∈ F+˚ . Thus ρ

sup{Re ν(𝒯 (g) + c) | g ∈ uV ∩G , c ∈ F− }

ρ

= sup{Re ν(𝒯 (g)) | g ∈ uV ∩G }

= pΩ{ν} (𝒯 ). u

ρ V ∩G

By our assumption, there are μ ∈ Π and λ ≥ 0 such that ν = λμ. Then pΩ{μ} (𝒮 ) ≤ pΩ{μ} (𝒯 ) u

ρ V ∩G

u

ρ V ∩G

implies that pΩ{ν} (𝒮 ) ≤ pΩ{ν} (𝒯 ), u

ρ V ∩G

u

ρ V ∩G

contradicting the above. 3.3.18 Proposition. Let φ1 , . . . , φn be continuous real-valued functions on X and let a ∈ E+ such that the functions φi ⋅ a are contained in ℋ. Then for all μ ∈ Π the set {μ} Ω{φ⋅a} , where φ = ∧ni=1 φi , is C-superharmonic.

242 � 3 Choquet theory Proof. Let 𝒯 , 𝒟xT ∈ LK (𝒞V (X, E), F) such that 𝒯 ≺C 𝒟xT and 𝒟xT is a K-extremal point evaluation. Then both 𝒯 and 𝒟xT are positive operators, hence, likewise is T ∈ L(E, F). Let μ ∈ Π. We have Re μ(𝒯 (φi ⋅ a)) ≤ Re μ(𝒟xT (φi ⋅ a)) = φi (x)Re μ(T(a)) for all i = 1, . . . , n. Since φ ⋅ a ≤ φi ⋅ a, both 𝒯 , and 𝒟xT are positive and since μ ∈ F+˚ , this yields n

Re μ(𝒯 (φ ⋅ a)) ≤ ⋀ Re μ(𝒯 (φi ⋅ a)) i=1 n

≤ ⋀ φi (x)Re μ(T(a)) i=1

= φ(x)Re μ(T(a)), that is, pΩ{μ} (𝒯 ) ≤ pΩ{μ} (𝒟xT ). {φ⋅a}

{φ⋅a}

The case that both E and F are topological vector lattices In addition to the assumptions of the preceding setting, we presume that both E and F are topological vector lattices over ℝ and that all elements of the subset Π of F+˚ are lattice homomorphisms. The statement of Proposition 3.3.18 then can be strengthened in the following way. 3.3.19 Proposition. Let f1 , . . . , fn ∈ ℋ. Then for all μ ∈ Π the set Ω{f } , where f = ∧ni=1 fi , is C-superharmonic. {μ}

Proof. Let 𝒯 , 𝒟xT ∈ LK (𝒞V (X, E), F) such that 𝒯 ≺C 𝒟xT and 𝒟xT is a K-extremal point evaluation. Both 𝒯 and 𝒟xT are positive operators, so is T ∈ L(E, F). Let μ ∈ Π. We have μ(𝒯 (fi )) ≤ μ(𝒟xT (fi )) = μ(T(fi (x))) for all i = 1, . . . , n. Since f ≤ fi and μ is a lattice homomorphism, this renders n

n

i=1

i=1

μ(𝒯 (f )) ≤ ⋀ μ(𝒯 (fi )) ≤ ⋀ μ(T(fi (x))) = φ(x)Re μ(T(a)), that is, pΩ{μ} (𝒯 ) ≤ pΩ{μ} {φ⋅a}

{φ⋅a}

(𝒟xT ).

List of Symbols (C)Vε 129 (Π1) 179 (Π2) 179 (a)V 30 A˝ 2 E˚ 2 f 2 E E+ 15 F ˚ 19 F+˚ 156 F ˚˚ 19 F+˚˚ 156 F ˚‚ 39 T ˝ θ ˝ S 67 T ˚ 67 V (a) 30 V s (a) 30 Vεs (C) 129 Vε (C) 129 Xω 230 ˝

Y 5 A 4 𝒞(X) 5 𝒞(X, E) 5 𝒞𝒦 (X) 5 𝒞𝒦 (X, E) 5 𝒞V (X) 10 𝒞V (X, E) 10 𝒞V (X, E) ⊗ F ˚ 169 Conv(E) 6 𝒟xT 163 ΔC 197 ΔΩC 197 Ẽ 20 Ẽ ⊗ F ˚ 21 ℱ (X) 5 ℱ (X, Conv(E)) 45 ℱ(θ,B) (X, Conv(E)) 103 ℱ (X, E) 5 ℱ(θ,B) (X, E) 103 ω

Im f 234 ω Im 𝒯 236 LK (𝒞V (X, E), F) 190 L(𝒞V (X, E), F) 47 L(E, F) 19 L(E, F)+ 156 L(E, F ˚‚ ) 48

https://doi.org/10.1515/9783111315478-004

𝒩v 7 O 5 ΩΠu 178 𝒫E 36 𝒬F 37 R 4 ℝ 5 ℝ+ 5 ω Re f 234 ω Re 𝒯 236 ℝ 5 ℝ 5 ℱ𝒮 (X, Conv(E)) 68 ℱ𝒮 (X, E) 68 𝒯 ↓C 195 𝒯(θ,B) 139 𝒱 1 𝒱c 6 ‖θ‖(v,Π) 53 𝒲 19 β(E, F) 4 [|θ|(V )(μ)] 99 B↘ 130 δxT 62 =B 46 = 61 a.e. B

C 189 HΩ 170 K 189 K0 205 P 170 Q 170 V 8 V˚ 8, 14 Vθ 139 Vc 142 ⩾B 46

∫B f dθ 88 ⩽B 46 ⩽ 61 (v)

a. e. B

𝒯φ 161 LSU (E, F) 26

LΠu (𝒞V (X, E), F) 179 Lb (E, F) 20 S 3 ↗ 130 ˝

244 � List of Symbols

ωf 230 Y 5 pΩ 170 pΩΠ 178 u 𝜕C 197 𝜕CΩ 197 ≺C 190 ≼V 78 qΩ 170 → 130 B ↘ 130 σ(E, F) 4 τ(E, F) 4 bal(A) 1 bconv(A) 1 bcore(A) 26 b𝒞X (u) 185 cone(A) 1 conv(A) 1 supp(θ) 67 supp(f ) 5 θ‖G 162

θ|B 60 u|Y 233 ρ uU 206 ↗B 130 v⩽u 8 vW 89 v(θ,W ) 139 φ⋅f 5 ∨ 15, 157 |θ| 50 ∧ 15, 157 q(𝒯 ) 191 p ̂ (𝒯 ) 191 q ̃ 204 C f ⩽ g + v 45 f ⩽ g + v 61 a.e. B

fn B↘ gn 130 fn a.e. B ↘ f 131 fn a.e. B ↘ gn 131 fn a.e. 󳨀→ f 131 B fn ↗a.e. B f 131 pΠ 178

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Index A absorbs B 2 E-valued function 5 Fσ -set 203 Gδ -set 203 Lp -norm 14 Lp (μ)-spaces 14 Conv(E)-valued function 6 L(𝒫E , F)-valued measure 57 A-measurable function 71, 80 𝒞(X)-convex functional 173 𝒞(X)-convex subset of 𝒞V (X, E) 180 𝒞(X)-convex subset of 𝒞V (X, E) ⊗ F ˚ 172 L(E, F ˚‚ )-valued measure 48 L(𝒫E , 𝒬F )-valued measure 48 L(𝒫E , F ˚˚ )-valued measure 57 Ω-bounded 189 Ω-bounded above 198 Ω-bounded below 198 Ω-extremal point evaluation 175, 197 C-superharmonic set 204 K-bounded 189 K-extremal point evaluation 197 ω-imaginary part of a function 234 ω-imaginary part of a functional 236 ω-real part of a function 234 ω-real part of a functional 236 σ-algebra (see also σ-field) 4 σ-field 4 σ-ring 4 θ-almost everywhere 61 θ-dense 61, 204, 216 absolute value of a complex measure 59, 229 absolute value of a linear functional 229 absorbing set 2 abstract neighborhood system 30 adjoint operator 67 algebra homomorphism 158 algebraic dual space 2 balanced convex hull 1, 26 balanced core 26 balanced hull 1 balanced set 1 Banach space 47 Banach–Alaoglu theorem 29 barrel 26 https://doi.org/10.1515/9783111315478-006

barreled space 4, 26 bidual (see also second dual) 19 bipolar (see also second polar) 4 Bochner integrable 115 Borel measure V, 47 Borel set 4 bornological space 28 bounded below or above function 46, 69, 124 bounded below or above set of functions 46, 69, 124 bounded below or above subset of a locally convex cone 31 bounded linear operator 20 bounded measure 51 bounded semivariation 51 bounded set 2 bounded set of functions 46 bounded variation 56 cancellation law 30 characteristic function 5 Choquet boundary 197 Choquet cone 189 Choquet core 189 Choquet ordering 190 Choquet–Bishop–DeLeeuw theorem 203 closure of a set 2 coarser function space neighborhood system 8, 141 compact measure 60 compact operator 23 compactness in the strong operator topology 25 compactness in the weak operator topology 25 comparison of function space neighborhoods 7, 141 completion of a Choquet cone 204 composition measure 67, 117 composition of an operator with a scalar-valued function 161 composition of measurable functions 85 conditional convergence 49 cone 30 cone extension 36 conic hull 1 convex hull 1 convex set 1 countably additive 48

250 � Index

downward directed 8 dual cone 31 dual pair of vector spaces 3 dual space 2 element of 𝒬F representing a convex set 42 elementary function 11 elementary operator 185, 225 equicontinuous 25 equivalent function space neighborhood systems 8 extended neighborhood system 7, 83, 101, 106 extended superlinear functional 32 extension of ‖θ‖(v,Π) 55 extension of an operator 44 family of weights 13 finer function space neighborhood system 8, 141 finest function space neighborhood system 14 finite rank operator 185 full cone extension 36 full locally convex cone 30 function 46 function space 10 function space neighborhood 7 function space neighborhood system 8 function subspaces 19 function supported by a set 5 function vanishing at infinity 8 functional representing a convex set 42 general convergence theorem 133 Hahn–Banach theorem 32 Hilbert space 160, 209 inductive limit topology 14 inner regular measure 62 integral of a measurable function 88 integral of a set-valued function 101 integral of a step function 87 integral of a vector-valued function 111 integral over a set in A 103 integral over a set in R 101 integral representation 149 integral with respect to a composition measure 117 interior of a set 2 involution 15, 158 Krein–Milman theorem 179

lattice cone extension 37 lattice homomorphism 157, 242 linear functional 2 linear operator 2, 19, 31 locally convex cone 30 locally convex topological vector space 1, 19 lower semicontinuous function 6, 9, 13, 206, 210, 217 lower semicontinuous functional 41, 57, 170, 180 Mackey topology 4, 35, 98 maximal representation measure 168 measurable real-valued function 71 measurable set-valued function 73 measurable vector-valued function 70 measure 48 measure supported by a set 61 minimal elements 196 modulus of a composition measure 67 modulus of a measure 50 monotone operator 43, 156 multiples of a function space neighborhood 7 mutually singular measures 61 natural topology 40, 56 neighborhood basis for a topological vector space 1 neighborhood function 7 nondegenerate bilinear form 3 nonnegative function 5 normal operator 160 nuclear operator 187 operator-valued measure 48 order for function space neighborhoods 7 ordered cone 30 ordered topological vector space 14, 156 Orlicz–Pettis theorem 49 outer regular measure 62 partition of the unity 5, 11 point evaluation measure 62 point evaluation operator 163 pointwise convergence of sequences of functions 130 polar 2, 3, 31 polar topology 3 positive cone 15 positive function 5 positive operator 156 properties that hold almost everywhere 61

Index

quotient space 3, 20 quotient space topology 20 reflexive space 4, 24, 25, 159 regular measure 63 relative neighborhoods in Conv(E) 129 relative topologies 31, 73, 130 relatively compact 5 relatively weakly compact 23 representation measure 150, 160, 161, 163, 182, 197, 203 restriction of a measure 60, 66 Riesz representation theorem 47, 59, 150 sample settings in the general case 215 Sandwich theorem for locally convex cones 194 second dual 19, 40 second polar 4 semi reflexive space 4, 23, 29, 60, 149, 187 semivariation of a measure (see also modulus of a measure) 50 sequence spaces 14 sequences of convex sets 130 sequences of functions 133 sequences of set-valued functions 130 set of measure zero 60, 66 set-valued step function 68 spectral measure 160 spectral theorem 160 spectrum 160 standard representation of a step function 69 step function 11 strong operator topology 19 strong topology 4, 19 strongly θ-measurable 83 strongly additive at an element 49 strongly additive integral 128 strongly additive measure 49, 53 subcone of a locally convex cone 30 subcone of a vector space 1 sublinear functional 32, 37, 41, 57, 170 sums of function space neighborhoods 7 superlinear functional 32, 170 support of a function 5 support of a measure 67

� 251

tensor product 21 topological algebra 15, 158 topological boundary 5 topological closure 5 topological dual space 2 topological interior 5 topological vector lattice 15, 157 topology consistent with a duality 4 topology of compact convergence 12 topology of pointwise convergence 13, 39 topology of uniform convergence 12, 18, 164, 229 uniform operator topology 19 unimodular function 230 upper, lower and symmetric neighborhoods 30 upper, lower and symmetric topologies 30, 40, 130 upper and lower envelopes 191 upper semicontinuous function 6, 13, 192 upper semicontinuous functional 170 Urysohn’s lemma 10 variation of a measure 53, 182 weak operator topology 19, 49 weak subseries convergence 49 weak topology 3, 4 weak* operator topology 19, 49 weak* topology 3 weakly θ-measurable 83 weakly bounded set 3 weakly compact measure 60 weakly compact operator 23, 150, 176 weakly continuous operator 19, 158 weakly lower continuous 9, 14, 149, 154, 182 weakly lower continuous function space neighborhood 9 weakly lower continuous function space neighborhood system 9 weakly lower continuous neighborhood function 8, 113 weighted space topologies 13

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