The dual of L_infty(X,L,lambda), finitely additive measures and weak convergence 9783030347314, 9783030347321

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SPRINGER BRIEFS IN MATHEMATICS

John Toland

), The Dual of L∞(X, L, λ), Finitely Additive Measures and Weak Convergence A Primer

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John Toland

The Dual of L1ðX; L; ‚Þ, Finitely Additive Measures and Weak Convergence A Primer

123

John Toland Department of Mathematical Sciences University of Bath Bath, UK

ISSN 2191-8198 ISSN 2191-8201 (electronic) SpringerBriefs in Mathematics ISBN 978-3-030-34731-4 ISBN 978-3-030-34732-1 (eBook) https://doi.org/10.1007/978-3-030-34732-1 Mathematics Subject Classification (2010): 46E30, 28C15, 46T99, 26A39, 28A25, 46B04 © The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Preface

Assuming some familiarity with Lebesgue measure, integration and related functional analysis summarised in Chap. 2, this is an exposition of topics that arise when identifying elements of the dual space of L1 ðX; L; ‚Þ with finitely additive measures on a r-algebra L when the measure ‚ is complete and r-finite. Such a representation has its origins in the independent work of Fichtenholz and Kantorovitch [14] and Hildebrandt [20] and culminated in a general abstract theory due to Yosida and Hewitt [35] in 1952. However, even now it is not unusual ([16] is an exception) for books on measure theory to give a detailed account of the dual space of Lp ðX; L; ‚Þ, 1 6 p\1, while relegating the case p ¼ 1 to references, e.g. [12, p. 296] which invokes the theory of finitely additive measures on algebras such as is developed in [35]. An explanation may be that a study of finitely additive measures on algebras necessitates a possibly unwelcome diversion from the mainstream theory of countably additive measures that suffices for p 2 ½1; 1Þ. Whatever the reason, a consequence is that L1 ðX; L; ‚Þ has acquired an aura of mystery, to the extent that it is often not very clear beyond the mere definition what is meant by saying that a bounded sequence in L1 ðX; L; ‚Þ is weakly convergent. The aim here is to take Yosida and Hewitt theory on r-algebras beyond the representation theorem for L1 ðX; L; ‚Þ , pointing out some of its consequences for measurable functions generally and in particular for weak convergence of sequences in L1 ðX; L; ‚Þ. The target audience is anyone who feels nervous about representing the dual of L1 ðX; L; ‚Þ by finitely additive measures in the knowledge that there exist uncountably many, linearly independent, finitely additive measures m > 0 defined on the Lebesgue r-algebra of ð0; 1Þ with the property that Z 0

11k

Z u dm ¼ 0 for all u 2 L1 ð0; 1Þ and k 2 N; but

1

1 dm ¼ 1:

ð†Þ

0

An essential goal will be to come to terms with observations such as this one.

v

vi

Preface

In their seminal work, Yosida and Hewitt [35] studied general Banach spaces L1 ðX; M; N Þ of essentially bounded measurable functions, where measurability is determined by an algebra M (closed under complementation and finite unions) and essential boundedness is defined in terms of a family N  M (closed under countable unions with the added property that A  B 2 N implies A 2 N ) that mimics null sets. Obviously, L1 ðX; L; ‚Þ is a special case of L1 ðX; M; N Þ but in general no measure of any kind is involved in the definition of L1 ðX; M; N Þ. However, although [35] shows that the dual of L1 ðX; M; N Þ can be expressed in terms of finitely additive measures, the exposition here is restricted to L1 ðX; L; ‚Þ because properties of finitely additive measures on r-algebras are less circumscribed by hypotheses than on algebras, and replacing the algebra M by a r-algebra L and N by the family of null sets fE 2 L : ‚ðEÞ ¼ 0g, where ‚ is complete and r-finite, yields a theory which is relevant in applications, including when X is a Lebesgue measurable subset of Rn or a differentiable manifold, or when X ¼ N with counting measure.

For a r-finite measure space the ultimate aim is to develop theory sufficient to characterise weakly convergent sequences in L1 ðX; L; ‚Þ in terms of their ‚almost-everywhere pointwise behaviour. However, in the process, when ðX; ‰Þ is a locally compact Hausdorff topological space and ðX; B; ‚Þ is a corresponding Borel measure space, there emerges a natural way to localise weak convergence. A sequence is weakly convergent in L1 ðX; B; ‚Þ if and only if it is weakly convergent at every point x0 in the one-point compactification of ðX; ‰Þ. Here, weak convergence at x0 is defined in terms of functionals which are zero outside every neighbourhood of x0 ; for an example of such, see (†). The essential range RðuÞðx0 Þ1 of a Borel measurable function u at x0 is similarly defined in terms of those elements of L1 ðX; B; ‚Þ which are localised at x0 . Since it need not be a singleton, RðuÞðx0 Þ can be interpreted as a multivalued representation of the fine structure at x0 of u 2 L1 ðX; B; ‚Þ which is intimately related to weak convergence. The Literature In her Foreword to the monograph by Bhaskara Rao and Bhaskara Rao [6], Dorothy Maharam Stone cites Salomon Bochner as having said that “contrary to popular mathematical opinion finitely additive measures were more interesting, more difficult to handle, and perhaps more important than countably additive measures”. Oxtoby [25] described [6] as a comprehensive account of finitely additive measures which effectively organises a large body of material that is widely scattered in the literature and deserves to be better known, and in their preface the authors themselves described it as a reference book as well as a textbook.

RðuÞðx0 Þ is sometimes referred to as the cluster set of u at x0 .

1

Preface

vii

The origins of this theory are to be found in the early days of modern integration theory when there were many contributors: see [12, §III.15, p. 233 and §IV.16, p. 3882] and the comprehensive bibliography with notes in [6]. However, presumably because they could not match the versatility of Lebesgue’s theory of integration and the power of its convergence theorems, finitely additive measures seem to have fallen out of fashion. Nevertheless, they continue to have significant roles in, for example, mathematical economics, probability, statistics, optimization, control theory and analysis [7, 9, 25, 35]. In a series of three papers on additive set functions on abstract topological spaces, A. D. Alexandroff [2] studied bounded regular finitely additive measures that represent linear functionals on spaces of continuous functions. On a similar theme, but in a more general setting, a much-cited reference for the dual of L1 ðX; L; ‚Þ is Dunford and Schwartz [12, p. 296] which covers the theory of finitely additive set functions on algebras and includes extensive historical notes. For a recent account, see Fonseca and Leoni [16, Theorem 2.24], and Aliprantis and Border [3] for the abstract theory in which it is embedded. It will soon be apparent that key results, including (†), rely on the axiom of choice. For a discussion of the role of the axiom of choice, geometrical and paradoxical aspects of finitely additive measures, and their invariance under group actions on X, see Tao [32]. Oxtoby’s commentary [25] is of independent interest. A key role is played throughout by the set G of finitely additive measures that take only the values 0 and 1 and the observation that L1 ðX; L; ‚Þ is isometrically isomorphic to a space of real-valued continuous functions on ðG; ¿Þ with the maximum norm, where ¿ is a compact Hausdorff topology. Further analysis of G in a Borel setting then leads to localization, and to other developments mentioned above and outlined in the Introduction. What follows is in large part an extension of a simplified version of Yosida and Hewitt [35], set out in the notation and terminology of Chap. 2. Bath, UK

John Toland

Acknowledgements I am indebted to Charles Stuart (Lausanne) for many things including his encouragement of this project. I am grateful to Anthony Wickstead (Belfast) who obtained for me a copy of [33] and drew my attention to [34], and to Geoffrey Burton (Bath) and Eugene Shargorodsky (King’s London) for their interest and many comments. In addition, I would like to thank Eugene Shargorodsky who contributed Sect. 9.4 and Mauricio Fernández (Stuttgart) who on a visit to Cambridge asked a question that the account that follows attempts to answer.

2

The reference to Theorem 8.15 on p. 388 is a misprint; 8.15 is a Definition and obviously Theorem 8.16 was intended.

Contents

1

1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2

Notation and Preliminaries . . . . . . . . . . . . . . . . 2.1 Partial Ordering and Zorn’s Lemma . . . . . . 2.2 Algebras and r-Algebras . . . . . . . . . . . . . . 2.3 Measurable Sets and Measurable Functions 2.4 Measures and Real Measures . . . . . . . . . . . 2.5 Splitting Families of Measurable Sets . . . . . 2.6 Integration . . . . . . . . . . . . . . . . . . . . . . . . 2.7 Function Spaces . . . . . . . . . . . . . . . . . . . . 2.8 Dual Spaces and Measures . . . . . . . . . . . . 2.9 Functional Analysis . . . . . . . . . . . . . . . . . 2.10 Point Set Topology . . . . . . . . . . . . . . . . . .

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7 7 8 9 10 14 15 17 19 20 23

3

L1 and Its Dual . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

27

4

Finitely Additive Measures . . . . . . . . . . . . . . . . . . . . . 4.1 Definition, Notation and Basic Properties . . . . . . . 4.2 Purely Finitely Additive Measures . . . . . . . . . . . . 4.3 Canonical Decomposition: baðLÞ ¼ RðLÞ  PðLÞ 4.4 L1 ðX; L; ‚Þ . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . .

31 31 36 38 39

5

G: 0–1 Finitely Additive Measures . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 G and Ultrafilters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 G and the ‚-Finite Intersection Property . . . . . . . . . . . . . . . . .

41 42 44

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ix

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Contents

6

Integration and Finitely Additive Measures . . . . . . . . . . . 6.1 The Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Yosida–Hewitt Representation: Proof of Theorem 3.1 . 6.3 Integration with Respect to ! 2 G . . . . . . . . . . . . . . . 6.4 Essential Range of u 2 L1 ðX; L; ‚Þ . . . . . . . . . . . . . . 6.5 Integrating u 2 ‘1 ðNÞ with Respect to G . . . . . . . . . 6.6 The Valadier–Hensgen Example . . . . . . . . . . . . . . . .

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47 47 50 51 52 52 54

7

Topology on G . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 The Space ðG; ¿Þ . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 L1 ðX; L; ‚Þ and CðG; ¿Þ Isometrically Isomorphic . 7.3 Properties of G and ¿ . . . . . . . . . . . . . . . . . . . . . . 7.4 G and the Weak* Topology on L1 ðX; L; ‚Þ . . . . . 7.5 G as Extreme Points . . . . . . . . . . . . . . . . . . . . . . .

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57 57 58 59 63 64

8

Weak Convergence in L1 (X, L; ‚) . 8.1 Weakly Convergent Sequences 8.2 Pointwise Characterisation . . . . 8.3 Applications of Theorem 8.7 . .

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67 67 68 72

9

L1 When X is a Topological Space . . . . . . . . . . . . . . . . 9.1 Localising G . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Localising Weak Convergence . . . . . . . . . . . . . . . . 9.3 Fine Structure at x0 of u 2 L1 ðX; B; ‚Þ . . . . . . . . . 9.4 A Localised Range from Complex Function Theory

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77 77 80 81 84

10 Reconciling Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1 ðG; ¿Þ Versus L1 ðX; L; ‚Þ . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 Restriction to C0 ðX; .Þ of Elements of L1 ðX; B; ‚Þ . . . . . . . . . .

87 87 89

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

95

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Chapter 1

Introduction

Overview In a normed linear space V , a sequence {vk } converges weakly to v (vk  v) if v ∗ (vk ) → v ∗ (v) for all v ∗ ∈ V ∗ , the dual space of V and, from the uniform boundedness principle, weakly convergent sequences are bounded in norm. However, it has been known since the work of Banach that when V is a complete normed linear space it may not be necessary to use all elements of V ∗ when testing for weak convergence. Indeed, when C(Z ) denotes the space of real-valued continuous functions on a compact metric space Z with the maximum norm, he showed that vk  v in C(Z ) if and only if vk (z) → v(z) for all z ∈ Z and {vk } is bounded. To do so he observed [5, Annexe, Thm. 7] that Dirac δ-functions satisfy conditions for a set W ∗ in the dual space of a Banach space to have the property that {vk } bounded and w ∗ (vk ) → 0 for all w ∗ ∈ W ∗ imply vk  0.

(W)

When (X, ) is a locally compact Hausdorff space and (C0 (X, ),  · ∞ ) is the Banach space of real-valued continuous functions on X that vanish at infinity (see (2.9)), weakly convergent sequences are pointwise convergent because δ-functions belongs to the dual space of C0 (X, ), and bounded by the uniform boundedness principle. Conversely, from Theorem 2.37 (Riesz) and Lebesgue’s Dominated Convergence Theorem [15, Thm. 2.24], sequences that are norm-bounded and pointwise convergent on X are weakly convergent. In particular, when Z is a compact Hausdorff space, for {vk } ⊂ C(Z) (the space of real-valued continuous functions on Z with the maximum norm) vk  v0 in C(Z) ⇔ sup vk  < ∞ and vk (z) → v0 (z) for all z ∈ Z.

(V)

k

The possibility of usefully extending these observations to L ∞ (X, L, λ) (the real Banach space of essentially bounded real-valued functions defined by (2.8)) at first appears limited because, for example, in an open set  ⊂ Rn with Lebesgue measure, © The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 J. Toland, The Dual of L ∞ (X, L, λ), Finitely Additive Measures and Weak Convergence, SpringerBriefs in Mathematics, https://doi.org/10.1007/978-3-030-34732-1_1

1

2

1 Introduction

u k  u in L ∞ () implies that u k (x) → u(x) almost everywhere and {u k ∞ } is bounded, but the converse is false (Lemma 8.4 and Example 8.6). Nevertheless, for bounded sequences it is shown in Theorem 8.7 that weak convergence is equivalent to strong convergence in L ∞ (X, L, λ) of elements of a family of related sequences and the resultant test is illustrated by several examples in Sect. 8.3. The proof, which inter alia is developed from first principles in the following pages, depends on the construction a compact Hausdorff topological space (G, τ ) with the property that L ∞ (X, L, λ) and C(G, τ ) are isometrically isomorphic (Theorem 7.4). Here G (Gothic G) is the set of the finitely additive measures that take only values 0 and 1 on L, and are 0 on null sets in a general measure space (X, L, λ). When the essential range of u ∈ L ∞ (X, L, λ) is defined as   R(u) = α ∈ R : λ({x : |u(x) − α| < }) > 0 for all  > 0 , it is shown that  u dω : ω ∈ G

 R(u) = X

and its relation to the isometric isomorphism is given by (7.2) and (7.3). If the setting is a Borel measure space (X, B, λ) corresponding to a locally compact Hausdorff spaces (X, ), these considerations can be localised to points x0 in the one-point compactification X ∞ of X . This in turn leads to the definition of the essential range of u ∈ L ∞ (X, B, λ) localised at x0 ∈ X ∞ and the possibility of regarding u as multivalued at every point while at the same time being single-valued almost everywhere. These ideas are closely related to the dual of L ∞ (X, B, λ) and weak convergence. The fact that L ∞ (X, L, λ) and C(G, τ ) are isometrically isomorphic appears to be at variance with L ∞ (X, L, λ)∗ being represented by finitely additive measures while C(G, τ )∗ is represented by σ-additive measures. Moreover, when (X, ) is a locally compact Hausdorff space, elements of L ∞ (X, B, λ)∗ are represented by finitely additive measures but their restrictions to C0 (X, ) are also represented by Borel measures. Both these issues will be addressed. As explained in the Preface, finitely additive measures will be considered only on σ-algebras. The classical text [12] and the exhaustive monograph [6] both consider more general situations from the outset, as does Yosida and Hewitt [35] which is the motivation and main source for this account of the simplified theory. The approach here is referred to as Yosida–Hewitt theory. Layout Chapter 2 is a brief survey of background material, collected with references from different sources and organised in a consistent notation that will be used in the chapters that follow. Although most of the material is entirely standard and does not need to be digested unless a need arises in later chapters, some of it may be less familiar. For example, Sect. 2.5 is a slight extension of Kirk’s [22] application of Baire’s

1 Introduction

3

category theorem to show that if (X, L, λ) has no atoms and G is a denumerable family of sets of positive measure, there exists an infinite family of disjoint sets A ∈ L  such that λ(G A) > 0 and λ(G \ A) > 0 for all G ∈ G. This construction is used in Example 9.11 when considering possibilities for the localised essential range of u ∈ L ∞ (X, L, λ) at x0 defined in (9.4). Chapter 3 begins by stating Theorem 3.1, which is the Yosida–Hewitt representation of the dual space L ∞ (X, L, λ)∗ as the space of finitely additive measures on L which are zero on sets of zero λ-measure. These finitely additive measures will be denoted by L ∗∞ (X, L, λ), which should therefore be identified with, but not confused with, L ∞ (X, L, λ)∗ , the bounded, real-valued, linear functionals on L ∞ (X, L, λ). From Theorem 3.1, it follows intuitively that if {Ak } ⊂ L is any sequence of mutually disjoint sets, then χ Ak  0 in L ∞ (X, L, λ) as k → ∞, where χ Ak denotes the characteristic function of Ak . Moreover, there are finitely additive measures that represent non-zero bounded linear functionals f on L ∞ (0, 1) with f (v) = 0 when v is continuous. Since no such countably additive measure can exist because of the dominated convergence theorem and Theorem 2.35 (Lusin), observations such as these reflect the delicacy of Theorem 3.1 and the differences between σ-additive and finitely additive measures. Chapter 4 introduces basic notation and definitions, for example, of partial ordering, lattice operations, absolute continuity and singularity, for finitely additive measures. The important notion of pure finite additivity of measures follows, and it is shown that every finitely additive measure is uniquely the sum of a σ-additive and a purely finitely additive measure. Chapter 5 introduces the set G of elements in L ∗∞ (X, L, λ) which take only the values 0 or 1 on L and explains the sense in which every u ∈ L ∞ (X, L, λ) is constant ω-almost everywhere when ω ∈ G. Elements ω ∈ G give rise to families of sets Uω = {E ∈ L : ω(E) = 1} which are ultrafilters (Definition 5.3). The collection of ultrafilters is denoted by U (Gothic U), and there is a one-to-one correspondence (Theorem 5.4) between G and U. The existence of ultrafilters, and consequently of elements of G with certain properties, follow from Zorn’s lemma. G will dominate subsequent developments. Chapter 6 defines the integral of essentially bounded measurable functions with respect to the finitely additive measures that featured in Theorem 3.1 (the Yosida– Hewitt representation theorem). In Theorem 6.2, it is noted that for all ω ∈ G and u ∈  L ∞ (X, L, λ), u(x) = X u dω, ω-almost everywhere in the sense of finitely additive measures (Remark 5.2). Observation (†) in the Preface is justified in Remark 6.1. The chapter ends with an account of the Valadier–Hensgen example [19, 33] of purely finitely additive measures on [0, 1] that are definitely not σ-additive but which coincide with σ-additive Lebesgue measure when integrating continuous functions. Chapter 7 introduces a compact Hausdorff topology τ on G and from theory already developed, derives the existence, Theorem 7.4, of an isometric isomorphism between the Banach algebra L ∞ (X, L, λ) and the space of real-valued continuous functions C(G, τ ). It is immediate that the functionals corresponding to G have property (W) in the Introduction.

4

1 Introduction

Independently, it is shown that τ coincides with the restriction to G of the weak* topology on L ∗∞ (X, L, λ) and that G is a closed subset of L ∗∞ (X, L, λ) with the weak* topology. Also, it is shown that ±G coincide with the extreme points of the closed unit ball in L ∗∞ (X, L, λ) and consequently Theorem 2.55 (Rainwater) yields an alternative proof that G has property (W) in the Preface. Chapter 8 opens with the observation, based of the duality between Dirac measures acting on C(G, τ ) and elements of G acting on L ∞ (X, L, λ), that u k  u 0 in L ∞ (X, L, λ) if and only if 



u k ∞  M and

u k dω → X

u 0 dω as k → ∞ for all ω ∈ G. X

It follows that when F : R → R is continuous, u → F(u) is sequentially weakly continuous [4] on L ∞ (X, L, λ). Necessary pointwise conditions for a sequence to be weakly convergent in L ∞ (X, L, λ) are given but examples show that they are not sufficient. However, a necessary and sufficient pointwise condition for a sequence to be weakly convergent in L ∞ (X, L, λ), Theorem 8.7, follows from Theorem 5.6 and the fact that any u ∈ L ∞ (X, L, λ) is a constant ω-almost everywhere in the sense of finitely additive measures when ω ∈ G. In Sect. 8.3, some quite subtle questions about the weak convergence of specific sequences are settled using Theorem 8.7. Chapter 9 deals with refinements of the theory to measure spaces (X, B, λ) where (X, ) is a locally compact Hausdorff space with measure λ on its Borel σ-algebra B. Prototypical examples of this setting are when X is an open subset of Rn with the Euclidian metric and Lebesgue measure and X = N with the discrete topology and counting measure. The key observation is that G = x0 ∈X ∞ G(x0 ) where, for distinct points x0 ∈ X ∞ (the one-point compactification of X ), the sets G(x0 ) are closed in (G, τ ) and disjoint, and elements of G(x0 ) are zero outside every open neighbourhood of x0 ∈ X ∞ . In Sect. 9.2, this localisation result leads to a characterization of weakly convergent sequences in terms of the pointwise behaviour of related sequences of functions in neighbourhoods of points of X ∞ . (Here “pointwise” has its usual λ-almost everywhere meaning which is familiar in any measure space, whereas “localisation” refers to the behaviour of Borel measures and functions restricted to open neighbourhoods of points in a topological space.) Writing R(u), u ∈ L ∞ (X, B, λ), as a union of disjoint compact sets R(u) =

x0 ∈X ∞





R(u)(x0 ) where R(u)(x0 ) =

u dω, ω ∈ G(x0 ) X

localises the essential range and reflects the fine structure of u at x0 . As a consequence of Theorem 2.25, if (X, ) is completely separable and (X, B, λ) has no atoms, there exist u ∈ L ∞ (X, B, λ) such that  u(x) ∈ Q (the rational numbers) for all x ∈ X yet R(u)(x) is the closed interval 0, u for all x. The chapter ends

1 Introduction

5

with an example due to Shargorodsky of a similar phenomenon occurring naturally in the theory of complex Hardy spaces, see Sect. 9.4. Chapter 10 first reconciles the general fact that L ∞ (X, L, λ) and C(G, τ ) are isometrically isomorphic while the dual of L ∞ (X, L, λ) is represented by finitely additive measures, whereas the dual of C(G, τ ) is represented by regular, real Borel measures which are σ-additive. It goes on to consider the special case L ∞ (X, B, λ) where (X, ) is a locally compact Hausdorff space and B denotes the Borel subsets of X . In that case, elements of L ∞ (X, B, λ)∗ are represented by finitely additive measures but when restricted to C0 (X, ) they are also represented by Borel measures. As seen in Chap. 6, Valadier and Hensgen independently noted that Riemann sums have Banach limits (Definition 2.41) which on L ∞ [0, 1] are represented by purely finitely additive measures which are not σ-additive, but which coincide with the Lebesgue integral for continuous functions. That observation is the motivation for Sect. 10.2 which considers the relation between the finitely additive measures ν that yield elements of L ∞ (X, B, λ)∗ (Theorem 3.1) and the Borel measures νˆ that by Theorem 2.37 (Riesz) represent their restrictions to C0 (X, ). In particular, those ν for which νˆ is singular with respect to λ are characterised in Corollary 10.10 and a minimax formula for νˆ in terms of ν is given in Theorem 10.11. It follows that νˆ may be zero when ν  0 if (X, ) is not compact. When ω ∈ G, either ωˆ ∈ D (a Dirac measures on X ) or ωˆ may be zero if (X, ) is not compact; if (X, ) is compact ωˆ ∈ D. Note from Remark 6.7 that an arbitrary Hahn–Banach extension to L ∞ (X, B, λ) of a Dirac δ-function acting on C0 (X, ) need not be in G and from Chap. 9 there may be infinitely many extensions that are in G. Starting from (and referring to Chap. 2 only when necessary), a self-contained approach to the pointwise characterisation of weakly convergent sequences is presented by • • • • • •

Chapter 3 Section 4.1 Chapter 5, up to Corollary 5.7(a) Chapter 6 up to Sect. 6.3 Chapter 7 up to Sect. 7.2 Chapter 8

To begin, Chap. 2 reviews background material, cites references and fixes notation.

Chapter 2

Notation and Preliminaries

 The set of natural numbers {1, 2, · · · , } is denoted by N, N0 = N {0}, R is the real numbers, C  is the complex numbers, Q is the rational numbers, and extended real numbers, R {+∞, −∞}, are denoted by R. The empty set is denoted by ∅, and a set with only one element is called a singleton. For an arbitrary set S, ℘ (S) denotes the collection of all its subsets, including the empty set ∅ and S itself. A set S is said to be denumerable or countable if there is an injection from  S into N, and uncountable otherwise. If Sk is denumerable for all k ∈ N, then k Sk is denumerable. If there is an injection from S into the set {1, 2, · · · , K } for some K ∈ N, S is said to be finite, and infinite otherwise. If S is an infinite denumerable set, there is a bijection from S onto N.

2.1 Partial Ordering and Zorn’s Lemma For a non-empty set S and R ⊂ S × S, a relation  on S is defined by writing x  y if (x, y) ∈ R. Definition 2.1 (Partial and Total Ordering) A partial ordering on S is a relation which satisfies the following axioms [12, Sect. I.2]: (i) x  x for all x ∈ S, (ii) x  y and y  z implies that x  z for all x, y, z ∈ S, (iii) x  y and y  x implies x = y for all x, y ∈ S. If in addition at least one of x  y or y  x holds for every (x, y) ∈ S × S, the partial ordering  is said to be a total ordering on S. If x  y but x = y, write x ≺ y.  Example 2.2 The usual ordering ≤ on R is a partial ordering which is a total ordering on every subset S of R. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 J. Toland, The Dual of L ∞ (X, L, λ), Finitely Additive Measures and Weak Convergence, SpringerBriefs in Mathematics, https://doi.org/10.1007/978-3-030-34732-1_2

7

8

2 Notation and Preliminaries

Any collection S of subsets of a set is partially ordered by set inclusion, i.e. A  B if and only if A ⊂ B, but if S is not a singleton it may not be totally ordered.  Definition 2.3 (Upper Bounds and Maximal Elements) If  is a partial ordering on S and A ⊂ S, (i) u ∈ S is an upper bound of A if a  u for all a ∈ A, (ii) m ∈ A is a maximal element of A if m ⊀ a for all a ∈ A.



Lemma 2.4 (Zorn) Suppose  is a partial ordering on S and every totally ordered subset A ⊂ S has an upper bound. The S has a maximal element. The proof depends on an assumption that for any collection T of subsets of S there is a set which contains a point from each of the sets in T . More precisely: The Axiom of Choice Let S be non-empty and T a non-empty subset of ℘ (S) \ {∅}. Then there exists a function f : T → S such that f (T ) ∈ T for all T ∈ T . Conversely, the Axiom of Choice can be proved using Zorn’s lemma. So these two seemingly different statements are in fact equivalent formulations of a fundamental axiom of set theory. Definition 2.5 (Equivalence Relation) A relation ∼ on a set S is called an equivalence relation if for all x, y, z ∈ S, (i) x ∼ x; (ii)x ∼ y ⇔ y ∼ x; (iii) x ∼ y and y ∼ z ⇒ x ∼ z. The set [x] = y ∈ S : x ∼ y is the equivalence class which contains x. 

2.2 Algebras and σ-Algebras For an arbitrary set X , a collection M ⊂ ℘ (X ) is an algebra if ∅ and X are in M, A ∈ M if and only X \ A ∈ M,  K k=1 Ak ∈ M when Ak ∈ M, k = 1, 2, · · · , K , K ∈ N.  If M is an algebra and A, B ∈ M, it follows that A B ∈ M, A \ B ∈ M and M is closed under finite unions and intersections. A σ-algebra, denoted by L, is an algebra with the additional property that  Ak ∈ L, k ∈ N, implies k∈N Ak ∈ L. Hence, a σ-algebra is closed under complementation, and countable unions and intersections. Examples. For any set X , {∅, X } and ℘ (X ) are σ-algebras. With X = [0, 1), the collection of unions of finitely many intervals of the form [a, b), 0 ≤ a ≤ b ≤ 1, is an algebra, but not a σ-algebra.

2.2 Algebras and σ-Algebras

9

M ⊂ ℘ (N), defined by A ∈ M if and only if either A or N \ A is finite, is an algebra. Clearly, the singletons {2n + 1} ∈ M for all n ∈ N0 . So M is not a σ-algebra  / M. because n∈N0 {2n + 1} ∈ On the other hand, saying A ∈ L if and only if A or X \ A is denumerable defines a σ-algebra L on any set X. (When X is uncountable, L = ℘ (X ).)  It is obvious that if Lκ , κ ∈ K, where K is an arbitrary set, are σ-algebras, L := κ∈K Lκ is also a σ-algebra. Therefore, since ℘ (X ) is a σ-algebra, for any V ⊂ ℘ (X ) the intersection of all σ-algebras which contain V is the smallest σ-algebra containing V, called the σ-algebra generated by V. In particular, an algebra M generates a σ-algebra, denoted by M in [35]. Another special case arises when (X, ) is a topological space,  ⊂ ℘ (X ) being the collection of open sets in X . Then the σ-algebra generated by  is called the Borel σ-algebra of (X, ), often denoted by B and elements of B are called Borel subsets of X .

2.3 Measurable Sets and Measurable Functions A measurable space is a pair (X, L) where X = ∅ is arbitrary and L ⊂ ℘ (X ) is a σ-algebra. Elements of L are referred to as measurable sets and a function u : X → R is said to be measurable if   x : u(x) > α ∈ L for all α ∈ R. The set of measurable functions has the following properties. (i) If u n , n ∈ N, is measurable and u n (x) → u(x) for all x ∈ X , u is measurable. (ii) If {u n } is a sequence of extended real-valued measurable functions, u(x) := inf u n (x) and u(x) := sup u n (x), x ∈ X, are measurable. n

n

(iii) If u is measurable, u ± are measurable where u ± (x) := sup{±u(x), 0}, so that u ± (x) ≥ 0, x ∈ X , and u = u + − u − and |u| = u + + u − .

(2.1)

(iv) If u, v are real-valued and measurable, c ∈ R and g : R → R is continuous, then cu, g(u) = g ◦ u, u ± , uv, u + v are measurable. (v) If u is non-negative and measurable, there is a sequence {ϕn } of non-negative measurable functions on X with (a) 0 ≤ ϕn (x) ≤ ϕn+1 (x) ≤ u(x), x ∈ X, n ∈ N, (b) u(x) = limn ϕn (x), x ∈ X , (c) ϕn is real-valued with only finitely many values for each n ∈ N.

10

2 Notation and Preliminaries

The characteristic function χ A of a set A ∈ L, defined by  χ A (x) =

1 if x ∈ A , 0 otherwise

is a measurable function. Finite linear combinations of characteristic functions ϕ(x) =

K

ak χ Ak , ak ∈ R, Ak ∈ L, K ∈ N,

(2.2)

k=1

are called simple functions and are measurable. The functions ϕn in (v) above are non-negative simple functions.

2.4 Measures and Real Measures In the measure theory literature, subtly different meanings are often assigned by different authors to the same terminology. For example, in [15, Chap. 3, p. 85] a signed measure may have infinite values, whereas in [28, Sect. 6.6, p. 119] it may not. See also Remark 2.38. For this reason, the terminology chosen for subsequent chapters is described in some detail below. Definition 2.6 (Measures) In a measurable space (X, L), a measure λ is an extended real-valued function on L with  λ(X ) > 0, λ(∅) = 0, λ(A) ≥ 0 for all A ∈ L, and when E k ∈ L, k ∈ N, and E i E j = ∅, i = j, λ

 k∈N

Ek = λ(E k ).

(2.3)

k∈N

When L is a σ-algebra this identity is referred to as the σ-additivity of λ. Some texts use the term “positive measure” for what here is referred to as a measure.  The triple (X, L, λ) is called a  measure space. A measure is said to be finite if λ(X ) < ∞, and σ-finite if X = n∈N X n where X n ∈ L and λ(X n ) < ∞ for all n. In a measure space a set E ∈ L is null, written as E ∈ N , if λ(E) = 0, and a property is said to hold λ-almost everywhere if the exceptional set where it does not hold is null. A measure space is said to be complete if A ⊂ B ∈ N implies that A ∈ N . Definition 2.7 In a measure space (X, L, λ), a sequence {u k } of measurable functions is said to converge in measure to a measurable function u if for all α > 0

λ {x : |u k (x) − u(x)| > α} → 0 as k → ∞.



A sequence of measurable functions which converges in measure has a subsequence which converges pointwise λ-almost everywhere [15, Thm. 2.30].

2.4 Measures and Real Measures

11

Definition 2.8 (Atom) A ∈ L is an atom if A ∈ / N and A ⊃ B ∈ L implies that either λ(B) = 0 or λ(B) = λ(A).  Remark 2.9 An atom A in a σ-finite measure space has finite measure. If λ(A) < ∞ and u is a bounded measurable function let   a = inf b ∈ R : λ({x ∈ A : u(x) ≤ b}) = λ(A) .



 Then λ {x ∈ A : u(x) ≤ a} = λ(A) and λ {x ∈ A : u(x) < a} = 0, since λ

is σ-additive and λ {x ∈ A : u(x) ≤ a − 1/k} = 0 for all k. In other words u = a, a constant, λ-almost everywhere on A. The next result is used in Sect. 2.5.  Lemma 2.10 If (X, L, λ) has no atoms, for all G ∈ L with λ(G) > 0 and > 0 there exists a subset E ∈ L of G with λ(E) ∈ (0, ]. Proof Since there are no atoms, the given G can be replaced by a subset, also denoted by G ∈ L, with λ(G) ∈ (0, ∞). Let > 0. Now since G is not an atom, there exists E 1 ⊂ G, E 1 ∈ L, with λ(E 1 ) ∈ (0, λ(G)). If λ(E 1 ) > for all such E 1 it follows that λ(E 1 ) > and λ(G \ E 1 ) > . Since there are no atoms,  by the same argument there exists E 2 ⊂ G \ E 1 with λ(E 2 ) > and λ(G \ (E 1 E 2 )) > . By induction there is a sequence {E k } ⊂ L of disjoint subsets of G with λ(E k ) > . Since λ(G) < ∞ this is false. Hence, there exists E ⊂ G, E ∈ L with λ(E) ∈ (0, ].  Remark 2.11 Lemma 2.10 implies the Darboux property of atomless measures [8, Cor. 1.12.10] which is apparently stronger: If (X, L, λ) has no atoms and λ(F) > 0 for some F ∈ L, then for any a ∈ (0, λ(F)) there exists E ∈ L with E ⊂ F and λ(E) = a. (The prototype is due independently to Fichtenholz [13] and Sierpi´nski [30].)  Definition 2.12 (Regular Borel Measure) When B is a Borel σ-algebra on a topological space (X, ), a measure λ on B is called a Borel measure. For B ∈ B, a Borel measure is said to be   outer regular on B if λ(B) = inf λ(U ) : B ⊂ U ∈  ,   inner regular on B if λ(B) = sup λ(K ) : K ⊂ B, K compact . A Borel measure which is both outer and inner regular on every B ∈ B is called regular.  Example 2.13 (Dirac Measures) For any measurable space (X, L) and x ∈ X , let δx be defined on L by δx (E) = 1 if x ∈ E and δx (E) = 0 otherwise. Then δx is a measure, called a Dirac measure at x, and (X, L, δx ) is a finite, complete measure space in which {x} is an atom.  Example 2.14 (Counting Measure on N) Let X = N, L = ℘ (N) and define λ(E) as the number of elements in E ⊂ N if E is finite, and λ(E) = +∞ if E is infinite. Then (N, ℘ (N), λ) is a σ-finite, complete measure space, λ is called counting measure and every singleton {n}, n ∈ N, is an atom. 

12

2 Notation and Preliminaries

Example 2.15 (Lebesgue Measure on Rn ) Let B denote the Borel subsets of X = Rn with the standard metric and (Rn , B) the corresponding measurable space. Then although there is a unique σ-finite measure λ on B which coincides with the n-dimensional volume of balls in Rn , it is not complete. However, a complete, σ-finite measure space (Rn , L, λ) is defined as follows:    L= B A : B ∈ B, A ⊂ B  ∈ B, λ (B  ) = 0 ,

  λ B A = λ (B), B A ∈ L. This is the classical Lebesgue measure space and λ is Lebesgue measure on Rn . Thus every Borel measurable function is Lebesgue measurable, and every Lebesgue measurable function is equal almost everywhere to a Borel measurable function. To see that Rn with Lebesgue measure has no atoms, suppose

E ∈ L and λ(E) > 0. Now define a function f : [0, ∞) → R by f (r ) = λ E Br where Br denotes the ball of radius r centred at 0 in Rn . From the σ-additivity of λ, it follows that f is continuous with f (0) = 0 and f (r ) → λ(E) > 0 as r → ∞. So, by the intermediate value theorem, there exists rˆ with

  λ E Brˆ = f (ˆr ) = λ(E)/2 ∈ (0, λ(E)), which shows that E is not an atom. n If Ω ∈ L, a typical case  being when Ω is an open subset of R , the restriction of λ to the σ-algebra {Ω E : E ∈ L} creates a complete measure space which is denoted by (Ω, L, λ). 

Definition 2.16 (Real Measures) In a measurable space (X, L), a real measure [28, Sects. 1.8 and 6.6] is a real-valued function μ (not necessarily one-signed) on L with μ(∅) = 0 and

  E j = ∅, i = j ∈ N. Ei = μ(E i ) when E i ∈ L and E i μ i∈N

i∈N

Since the left side is independent of the ordering of i ∈ N, the sum of the series on the right is the same if i is replaced by σ(i), where σ is a permutation of N. Hence, the series is absolutely convergent.  For a real measure μ and E ∈ L let |μ|(E) = sup

∞ i=1

|μ(E i )| : E =

∞ 

E i , E i ∈ L, E j



 E j = ∅, i = j ∈ N .

i=1

Then |μ(E)| ≤ |μ|(E), E ∈ L, and |μ| is a measure in the sense of Definition 2.6 [28, Thm. 6.2] with |μ|(X ) < ∞ [28, Thm. 6.4]. Hence μ± are real measures where

2.4 Measures and Real Measures

0 ≤ μ+ := 21 (|μ| + μ) and 0 ≤ μ− := 21 (|μ| − μ),

13

(2.4)

whence μ = μ+ − μ− and |μ| = μ+ + μ− , 0 ≤ μ± (E) ≤ |μ|(X ) < ∞, E ∈ L. |μ| is called the total variation of μ, μ± are the positive and negative parts of μ, and μ = μ+ − μ− is the Jordan decomposition of μ. Theorem 2.17 (Hahn Decomposition [28, Thm 6.14])  Let μ be a real measure on a σ-algebra L. Then there exist A± ∈ L such that A+ A− = X , A+ A− = ∅ and for E ∈ L,   E), μ− (E) = −μ(A− E) μ+ (E) = μ(A+ where μ± are defined in (2.4). Definition 2.18 (Absolute Continuity and Singularity) A real measure μ is absolutely continuous with respect to a measure λ, written as μ  λ if and only if λ(E) = 0 implies |μ|(E) = 0 and singular with respect to λ, written as λ ⊥ μ if and only if λ(E) + |μ|(X \ E) = 0 for some E ∈ L.



This notation will be generalised in Definition 4.6 and Remark 4.7 to accommodate finitely additive measures that are not σ-additive. In Chap. 4, real measures on L are seen as examples of finitely additive measures, in which context they are denoted by Σ(L). Remark 2.19 Note that in [15, Chap. 7] the term “signed measure” allows μ to take one, but not both, of the values ±∞. Thus in that terminology a real measure would be a signed measure but a signed measure might not be a real measure.  Example 2.20 Any finite linear combination of finite measures is a real measure. A class of real measures that are absolutely continuous with respect to λ will be defined in Remark 2.27.  Definition 2.21 (Regular Real Borel Measures) Let B denote the Borel σ-algebra of a locally compact Hausdorff space (X, ). Then a real Borel measure μ is said to be regular if both the measures μ± are regular in the sense of Definition 2.12: for all B∈B     μ± (B) = inf μ± (U ) : B ⊂ U ∈  = sup μ± (K ) : K ⊂ B, K compact , where μ± are defined by (2.4).



14

2 Notation and Preliminaries

2.5 Splitting Families of Measurable Sets This section concerns measurable sets which in an arbitrary measure space (X, L, λ) split families of measurable sets in the following sense.  Definition 2.22 (Splitting Measurable Sets) A ∈ L splits G ⊂ L \ N if λ(G A) > 0 and λ(G \ A) > 0 for all G ∈ G.  Remark 2.23 If A splits G, then X \ A also splits G. Moreover, if G1 , G2 ⊂ L \ N are such that for each G 1 ∈ G1 there exists G 2 ∈ G2 with G 2 ⊂ G 1 , then A splits G1 if A splits G2 .  The main result, Theorem 2.25, depends on Baire’s category theorem. To set the scene define an equivalence relation ∼ on L by writing  E ∼ F if an only if EΔF is null, i.e. the symmetric difference EΔF = (E \ F) (F \ E) ∈ N . For E, F ∈ L let  

 −1 −1 λ(EΔF) = tan |χ E − χ F | dλ , (2.5) ∂(E, F) = tan X

where χ A is the characteristic function of A ∈ L and tan−1 maps R onto [−π/2, π/2]. Since L 1 (X, L, λ) is a Banach space and since a convergent sequence in L 1 (X, L, λ) has a subsequence that converges pointwise λ-almost everywhere, it is immediate   that the space (L, ∂) is a complete metric space. Let Lˆ = G ∈ L : λ(G) ∈ (0, ∞) and for G ∈ Lˆ let    F(G) = E ∈ L : λ(E G) ∈ {0, λ(G)} .

The following simple adaption of Kirk’s argument [22] yields Theorem 2.25, Lemma 7.9 and Example 9.11. Lemma 2.24 For any G ∈ Lˆ the set F(G) is closed with empty interior in (L, ∂). In other words F(G) is nowhere dense in (L, ∂).  Proof For G ∈ Lˆ let g : L → R be given by g(E) = λ(G E). Then g is continuous because for E 1 , E 2 ∈ L,     |g(E 1 ) − g(E 2 )| =  χG  E1 − χG  E2 dλ  X

 ≤ |χ E1 − χ E2 | dλ = tan ∂(E 1 , E 2 ) . X

So F(G) = g −1 {0, λ(G)} is closed in (L, ∂). To see that F(G) is nowhere dense let E ∈ F(G) and > 0 be arbitrary. By Lemma 2.10 there exists E ⊂ G with E ∈ L and 0 < λ(E    ) < min{tan , λ(G)}. 0 n(n + 1)/2 (the sum of the first n natural numbers). Let u k = χ Ik . Then u k → 0 in L p (0, 1) for all p ∈ [1, ∞), from which it follows that u k → 0 in measure, and hence there exists a subsequence with u k j (x) → 0 for λ-almost all x ∈ (0, 1). However, it is obvious from the construction that

18

2 Notation and Preliminaries

lim sup u k (x) = 1 for λ-almost all x ∈ (0, 1).



k→∞

A measurable function u is said to be essentially bounded if |u(x)| ≤ c, a constant, λ-almost everywhere. With norm   u∞ = inf c > 0 : |u(x)| ≤ c λ-almost everywhere

(2.8a)

the space L ∞ (X, L, λ) of (equivalence classes of) essentially bounded measurable functions is a real Banach space. Let   [u]+ ∞ = inf c : u(x) ≤ cλ-almost everywhere ,   [u]− = sup c : u(x) ≥ cλ-almost everywhere . ∞

(2.8b) (2.8c)

Then [u]− ≤ u(x) ≤ [u]+ for λ-almost all x ∈ X . Definition 2.30 (Local Compactness [28, Defn. 2.3]) A topological space (X, ) is locally compact if for every x ∈ X there is an open set G with x ∈ G and the closure G of G is compact.  Definition 2.31 (σ-Compactness [28, Defn. 2.16]) A topological space (X, )  is σ-compact if X = k∈N K k for a denumerable family of compact sets K k .  Definition 2.32 (One-point Compactification [21, p. 150]) The one-point compactification of a topological space (X, ) is the compact space (X ∞ , ∞ ) defined by  / X (x∞ is referred to as the point at infinity) putting X ∞ = X {x∞ } where x∞ ∈ is open, i.e. G ∈ ∞ , if either G ⊂ X and G ∈ , or x∞ ∈ G and saying G ⊂ X  ∞  and G = {x∞ } (X \ K ) for a compact set K in (X, ). Remark 2.33 (X ∞ , ∞ ) is Hausdorff if and only if (X, ) is locally compact and Hausdorff, and (X, ) is compact if and only if {x∞ } is isolated (both open and  closed) in (X ∞ , ∞ ) [21, loc. cit.]. Continuous Function Spaces For a locally compact Hausdorff space (X, ), let C0 (X, ) denote the linear space of real-valued continuous functions v with the property that for all > 0 there exists a compact set K ⊂ X such that |v(x)| < for all x ∈ X \ K . With the maximum norm (2.9) v∞ = max |v(x)|, v ∈ C0 (X, ), x∈X

C0 (X, ) is a real Banach space. When X is compact C0 (X, ) consists of all the realvalued continuous functions on X . When (X, ) is not compact C0 (X, ) consists of the restrictions to X of real-valued functions that are continuous on the compact space (X ∞ , ∞ ) and zero at x∞ .

2.7 Function Spaces

19

Remark 2.34 In a Hausdorff topological space (X, ) with Borel σ-algebra B every singleton {a} is closed, and hence χ{a} , a ∈ X , is Borel measurable.

Now suppose that all Borel measurable functions are continuous. Then {a} = χ−1 {a} (0, 2) is open, because χ{a} is continuous. Therefore, all subsets of X are open, and hence  = ℘ (X ) ( is the discrete topology) when all measurable functions are continuous. Conversely, if  is the discrete topology every function is continuous. An important case is given by Example 2.14.  The following observation is a corollary of Lusin’s theorem. Theorem 2.35 ([28, Cor. 2.24]) Suppose (X, ) is a locally compact Hausdorff space, λ(X ) < ∞ and λ is a complete regular real Borel measure. Then for u ∈ L ∞ (X, B, λ) there exists a sequence {gk } of continuous functions on (X, ) such that each gk has compact support (the closure of {x : gk (x) = 0} is compact), gk ∞ ≤ u∞ and u(x) = limk→∞ gk (x) for λ-almost all x ∈ X .

2.8 Dual Spaces and Measures Given a normed linear space (V,  · ), a metric is defined on V by ρ(x, y) = x − y, x, y ∈ V , and V is a Banach space if (V, ρ) is complete (i.e. Cauchy sequences converge). A linear function f : V → R is continuous if and only if there is a constant M such that | f (x)| ≤ Mx for all x ∈ V . The set of all such continuous linear functions, which are usually called bounded linear functionals, forms a linear space V ∗ over R, and V ∗ , the dual space of V , is a Banach space (whether or not V is a Banach space) when endowed with the norm    f V ∗ = sup | f (x)| : x ≤ 1, x ∈ V . In the special case when V is a space of real-valued functions v on X there has historically been a great deal of interest in describing the action of f ∈ V ∗ on v ∈ V as an integral of v over X . Indeed much of the subsequent chapters concern the consequences of such a description of L ∞ (X, L, λ)∗ . The classical result for the dual space L p (X, L, λ)∗ , p ∈ [1, ∞), a Banach space with norm    f  L p (X,L,λ)∗ = sup | f (u)| : u p ≤ 1, u ∈ L p (X, L, λ) ,

(2.10)

is the following. Theorem 2.36 ([15, Thm. 6.15]) For 1 ≤ p < ∞ let f ∈ L p (X, L, λ)∗ . Then for p > 1 there exists a unique g ∈ L q (X, L, λ) such that for all u ∈ L p (X, L, λ)  f (u) =

ug dλ where q = X

p and  f  L p (X,L,λ)∗ = gq . p−1

20

2 Notation and Preliminaries

If (X, L, λ) is σ-finite, for p = 1 there exists g ∈ L ∞ (X, L, λ) such that  f (u) = X

ug dλ for all u ∈ L 1 (X, L, λ) and  f  L ∞ (X,L,λ)∗ = g∞ .

Therefore, L p (X, L, λ)∗ and L q (X, L, λ) are isometrically isomorphic when p ∈ (1, ∞) and p −1 + q −1 = 1, and also in the limiting case p = 1 and q = ∞ if (X, L, λ) is σ-finite. For a necessary and sufficient for L ∞ (X, L, λ) to be the dual space of L 1 (X, L, λ), see [16, Cor. 2.41]. The dual space C0 (X, )∗ is a Banach space with norm    f C0 (X,)∗ = sup | f (v)| : v∞ ≤ 1, v ∈ C0 (X, ) .

(2.11)

Theorem 2.37 (Riesz [28, Thm. 6.19]) For f ∈ C0 (X, )∗ there exists a unique, regular, real Borel measure μ on X (Definition 2.21) with  f (v) =

v dμ for all v ∈ C0 (X, ).

(2.12)

X

Moreover, |μ|(X ) =  f C0 (X,)∗ < ∞ and f (v) ≥ 0 when 0 ≤ v ∈ C0 (X, ) implies that μ ≥ 0 on Borel sets. The space of all regular, real Borel measures on B is denoted by C0∗ (X, τ ) and for every μ ∈ C0∗ (X, ), (2.12) defines f ∈ C0 (X, )∗ . Remark 2.38 Although the statements are different because the nomenclature is different, the Riesz representation theorems in [15, Thm. 7.17] and [28, Thm. 6.19] are equivalent to Theorem 2.37. 

2.9 Functional Analysis Theorem 2.39 (Hahn–Banach [12, p. 62, II.3.10]) On a real linear space V let p : V → R be sublinear: p(x + y) ≤ p(x) + p(y), p(αx) = α p(x) for all x, y ∈ V, α ∈ [0, ∞), and suppose f : Y → R is linear with f (y) ≤ p(y) for all y ∈ Y , where Y is a linear subspace of V . Then there is a linear function f˜ : V → R with f˜(y) = f (y) for all y ∈ Y and f˜(x) ≤ p(x) for all x ∈ V. It follows that − p(−x) ≤ f˜(x) ≤ p(x) for all x ∈ V . f˜ is known as as a Hahn–Banach extension of f . The following is immediate from the fact that p(x) = x is sublinear on a normed linear space.

2.9 Functional Analysis

21

Corollary 2.40 Let Y be a closed linear subspace of a normed linear space V and let v0 ∈ V \ Y . Then there exists f ∈ V ∗ such that f (v0 ) = 1 and f (y) = 0 for all y ∈ Y. Proof Let Z = {y + αv0 : y ∈ Y, α ∈ R}, a linear subspace of V . Since v0 = 0, z ∈ Z determines a unique α ∈ R and y ∈ Y such that z = y + αv0 and since Y is closed, a bounded linear functional g is defined on Z by g(z) = g(y + αv0 ) = α, z ∈ Z . By Theorem 2.39 (Hahn–Banach), there exists f ∈ V ∗ which coincides with g on  Z . Since g(v0 ) = 1, f has the required properties. Definition 2.41 (Banach Limits [5, p. 34, Chap. II]) As in Example 2.14 and Remark 2.34, let ∞ (N) denote the space of bounded, real sequences indexed by N. Now let c(N) be the subspace of convergent sequences and let l(u) denote the limit of {u(k)} ∈ c(N). Since l : c(N) → R is positive, bounded and linear, there exists [11, Thm. III.7.1] a positive bounded linear functional L on ∞ (N) such that for u ∈ ∞ (N), L(u) = l(u), u ∈ c(N), L(u) = L(u n ) where u n (k) = u(k + n), n ∈ N, and L∞ (N) = lc(N) = 1. L : ∞ (N) → R is called a Banach limit; see [31] for development of the theory. For an account based on finitely additive measures, see Remark 6.5.  Let C(Z) with the maximum norm denote the Banach space of real-valued continuous functions on a compact Hausdorff space (Z, ). Theorem 2.42 (Stone–Weierstrass [11, Thm. V.8.1], [15, Thm. 4.45]) Suppose A ⊂ C(Z) is a linear space in which (i) f, g ∈ A implies the product f g ∈ A; (ii) the constant function 1 ∈ A; and (iii) for z 1 = z 1 ∈ Z there exists f ∈ A such that f (z 1 ) = f (z 2 ). Then A = C(Z), where A is the closure of A in C(Z). Theorem 2.43 (Weierstrass) For a continuous real-valued function f on [a, b] and > 0 there exists a polynomial p on [a, b] with | f (x) − p(x)| < for all x ∈ [a, b]. Definition 2.44 (Weak Convergence) In a normed linear space V a sequence {yk } converges weakly to y0 in V , written as yk  y0 , if v ∗ (yk ) → v ∗ (y0 ) for all  v∗ ∈ V ∗. By the uniform boundedness principle (also known as the Banach–Steinhaus theorem) [28, Thm. 5.8], yk  y0 in V implies {yk } is bounded. Theorem 2.45 (Mazur [23, Chap. 10, Thm. 6]) If {yk } ⊂ K , where K is closed and convex in a normed linear space, and yk  y0 , then y0 ∈ K .

22

2 Notation and Preliminaries

Corollary 2.46 If yk  y0 in a normed linear space V and {k j } ⊂ N is strictly increasing, there exists {y i } with y i − y0  → 0 as i → ∞, where yi =

mi

γ ij yk j , γ ij ∈ [0, 1] and

j=1

mi

γ ij = 1, for some m i ∈ N.

j=1

Proof For the sequence {k j } ⊂ N let K be the closure in V of the set of convex combinations of elements of {yk j : j ∈ N}. Since yk j  y0 and yk j ∈ K , which is  closed and convex, y0 ∈ K follows from Theorem 2.45. Definition 2.47 (Weak* Topology on V ∗ [11, Chap. IV, Example 1.8]) For a normed linear space V a set U ⊂ V ∗ is open in the weak* topology if and only if for every u ∗ ∈ U there exists v1 , · · · , vn ∈ V and > 0 such that n    v ∗ ∈ V ∗ : |(v ∗ − u ∗ )(v j )| < ⊂ U. j=1



The collection   V ,v,u ∗ = v ∗ ∈ V ∗ : |(v ∗ − u ∗ )(v)| < , > 0, v ∈ V, u ∗ ∈ V ∗ is said to form a sub-base for the weak* topology on V ∗ . Theorem 2.48 (Alaoglu [11, Thm. V.3.1]) For a normed linear space V , B ∗ = {v ∗ ∈ V ∗ : v ∗ V ∗ ≤ 1}, the closed unit ball in V ∗ , is weak* compact. Definition 2.49 (Weak* Convergence in V ∗ ) For a normed linear space V a sequence ∗ {vk∗ } ⊂ V ∗ converges weak* to v0∗ ∈ V ∗ , written as vk∗  v0∗ , if vk∗ (v) → v0∗ (v) for all v ∈ V .  Note that when V is a Banach space, weak* convergent sequences in V ∗ are bounded, but not necessarily if V is not a Banach space. Definition 2.50 (Weak* Sequential Compactness in V ∗ ) When V is a normed linear space K ⊂ V ∗ is weak* sequentially compact if every sequence {vk∗ } ⊂ K has a  subsequence which converges weak* to a point v0∗ ∈ K . Remark 2.51 If V is separable (has a denumerable dense subset) the closed unit ball in V ∗ with the weak* topology is metrizable [11, Thm. V.5.1], and hence [23, Chap. 10, Thm. 12] weak* sequentially compact.  Definition 2.52 (Extreme Points) If C is a convex subset of a linear space, c ∈ C is an extreme point of C if c = αc1 + (1 − α)c2 for c1 , c2 ∈ C and α ∈ (0, 1) implies  that c = c1 = c2 .

2.9 Functional Analysis

23

For example, the four vertices of a closed square in the plane are its extreme points, and there are no others. Every point of the unit circle is an extreme point of the closed unit disc in the plane. Theorem 2.53 (Krein–Milman [11, Thm. V.7.4]) If K is non-empty, compact and convex in a locally convex space, K is the closed convex hull of its extreme points. In other words, K is the closure of the intersection of all convex sets which contain the extreme points of K . (Equivalently, K is the intersection of all closed convex sets which contain the extreme points of K .) The next result is a consequence of Theorem 2.48 (Alaoglu) and Theorem 2.53 (Krein–Milman), since V ∗ with the weak* topology is a locally convex topological space when V is a normed linear space. Corollary 2.54 When V is a normed linear space the closed unit ball B ∗ in V ∗ is the weak* closed convex hull of its extreme points (i.e. B ∗ is the intersection of all weak* closed convex set which contains the extreme points of B ∗ ). In particular, the set of extreme points of B ∗ is non-empty. The final theorem of this section says that when V is a Banach space the extreme points of B ∗ satisfy property (W) in the Introduction. Theorem 2.55 (Rainwater1 [26]) In a Banach space, xk  x if and only if {xk } is bounded and x ∗ (xk ) → x ∗ (x) for all extreme points x ∗ in the closed unit ball of the dual space.

2.10 Point Set Topology Definition 2.56 (Base and Sub-base for a Topology) In a topological space (X, ) a collection G of open sets such that every non-empty open set is a union of sets in G is called a base for the topology . A sub-base for  is a collection G of open sets with the property that the finite intersections of elements of G form a base for .  Remark 2.57 Note that for any collection T ⊂ ℘ (X ) with the property that X =  G, there is a unique topology τ for which T is a sub-base.  G∈T Definition 2.58 (Separable) A topological space is separable if it has a denumerable dense subset.  Definition 2.59 (First Axiom of Countability) A topological space (X, ) satisfies the first axiom of countability if for every x ∈ X there is a denumerable collection Gx of open sets G with x ∈ G such that x ∈ U ∈  implies that x ∈ G ⊂ U for some  G ∈ Gx . The collection Gx is called a local base at x for the topology . 1 Many

internationally renowned mathematicians have published under the name John Rainwater, see https://en.wikipedia.org/wiki/John_Rainwater.

24

2 Notation and Preliminaries

Definition 2.60 (Complete Separability or Second Axiom of Countability) A space is said to be completely separable (or to satisfy the second axiom of countability) if its topology has a denumerable base.  A completely separable space satisfies the first axiom of countability and is separable. In metric spaces separability and completely separability are equivalent. If, for any pair of disjoint closed sets Fi , i = 1, 2, in a topological space there exists a pair of disjoint open sets G i with Fi ⊂ G i , the space is normal; if this holds for any closed F1 when F2 is a singleton the space is regular, and if it holds when both Fi are singletons the space is Hausdorff. Thus normal implies regular implies Hausdorff and a compact Hausdorff space is normal. Metric spaces are normal because, with {i, j} = {1, 2}, Fi ⊂ G i = {x : dist(x, Fi ) < dist(x, F j )}. A famous characterisation of normality is the following. Lemma 2.61 (Urysohn [21, Chap. 4, Lemma 4]) A topological space is normal if and only if for every pair of disjoint closed sets F1 and F2 there is a continuous function from the space into [0, 1] with f (F1 ) = 0 and f (F2 ) = 1. This is not to say that f −1 {0} = F1 and f −1 {1} = F2 . To obtain that level of precision, see Lemma 2.62(b). A set is a G δ -set if it is the intersection of a denumerable family of open sets. Closed G δ -sets feature in Corollary 7.11, Theorems 9.4, 10.2, 10.4, and the following lemma. Lemma 2.62 (a) For A = ∅ in a normal topological space X there exists a continuous function f : X → [0, 1/2] with f (x) = 0 for x ∈ A and f (x) > 0 otherwise, if and only if A is a closed G δ -set. (b) For every disjoint pair, F1 and F2 , of closed G δ -sets in a normal space X there is a continuous function f : X → [0, 1] such that F1 = f −1 {0} and F2 = f −1 {1}. Proof (a) For the “only if” part suppose such an f exists. Then A = f −1 {0} is closed. Also  f −1 [0, 1/k), A = f −1 {0} = k∈N

 and so A is a G δ -set. Conversely, suppose A = ∅ is closed and A = k∈N Uk where Uk is open. Since X \ Uk and A are disjoint closed sets, by Lemma 2.61 (Urysohn), there exists a continuous function f k : X → [0, 1] suchthat f k (A) = {0} and f k (X \ Uk ) = {1}. Now define f : X → [0, 1/2] by f (x) = k∈N 2−(k+1) f k (x). Since the series is uniformly convergent f is continuous, f (x) = 0 on A and f (x) ∈ (0, 1/2] elsewhere. (b) By Urysohn’s lemma there exists a continuous h 0 : X → [0, 1] with h 0 (F1 ) = 0 and h 0 (F2 ) = 1. Since X is a G δ -space, by part (a) there exists continuous h 1 : [0, 1] → [0, 1/2] with h −1 1 {0} = F1 and h 2 : [0, 1] → [1/2, 1] with {1} = F . Now f : X → [0, 1] defined by h −1 2 2

2.10 Point Set Topology

25

   f (x) = inf h 2 (x), sup h 0 (x), h 1 (x) X

X



is continuous and has the required property.

A topological space is a G δ -space if all its closed  sets are G δ -sets. Metric spaces are G δ -spaces because, for every closed set, F = k G k where G k = {x : dist {x, F} < 1/k}, k ∈ N. However, a G δ -space need not be normal and a normal space need not be a G δ -space. In this context, the following characterisation of normality is useful. Theorem 2.63 A topological space is normal if and only if for all closed F and open G with F ⊂ G there exist open sets Wn , n ∈ N, such that F ⊂ n Wn and Wn ⊂ G for all n ∈ N. Proof By definition of normality, for given F (closed) ⊂ G (open) in a normal space there is an open set, W1 say, with F ⊂ W1 ⊂ W1 ⊂ G. Conversely, let A and B be disjoint X . Then, by hypothesis, there  closed sets in B = ∅. Similarly, there exist A ⊂ W and W exist open sets Wn such that n n n   open sets Vn such that B ⊂ n Vn and Vn A = ∅. Now define open sets G n , Hm , W and V by G n = Wn \



Vi , Hm = Vm \

i≤n



Wj, W =



Gn, V =

n

j≤m



Hm .

m

Then A ⊂ W and B ⊂ V , and W and V are open.  To see they are disjoint suppose x ∈ W V . Then for some m ∗ and n ∗ , x ∈ G n ∗ Hm ∗ . But then



  x ∈ Wn ∗ \ Vi ) Vm ∗ \ W j ) = ∅. i≤n ∗

j≤m ∗

Thus V and W are disjoint, and hence the space is normal.



Lemma 2.64 A completely separable, regular topological space X is a normal G δ space. Proof Let F (closed) ⊂ G (open) ⊂ X . By regularity, for every x ∈ F there exist disjoint open sets G x and G x with x ∈ G x , X \ G ⊂ G x and G x G x = ∅. Therefore x ∈ G x ⊂ X \ G x ⊂ G which implies that x ∈ G x ⊂ G x ⊂ G for all x ∈ F. Since there is no loss in assuming that G x ∈ G, the denumerable base for the topology, the normality of X follows from Theorem 2.63. Then for every x ∈ / F there exists To see that X is a G δ -space, let F be closed.        and G with F ⊂ G , x ∈ G , G = ∅ and G ∈ G. Since F = G open sets G x x x x x x x    x ∈F / G x and at most countably many G x are needed, F is a G δ -set and so X is a normal G δ -space. 

26

2 Notation and Preliminaries

Lemma 2.65 A locally compact Hausdorff space is regular. A completely separable, locally compact Hausdorff space is a normal G δ -space. Proof For F closed and x ∈ / F,let x ∈ G where G is open and G is compact. Since X is Hausdorff and x∈ / F G which is compact, there are disjoint open sets V, W with x ∈ V and  F G ⊂W . Since G is compact in a Hausdorff space it is closed, and so V  areopen.Therefore, X is regular since  G and W (X \ G) x ∈ V G, F ⊂ W (X \ G) and (V G) (W (X \ G)) = ∅. Lemma 2.64 now implies that a completely separable, locally compact Hausdorff space is a normal  G δ -space.

Chapter 3

L ∞ and Its Dual

Henceforth, (X, L, λ) is a complete σ-finite measure space and L ∞ (X, L, λ) is defined by (2.8); similarly for (X, B, λ) and L ∞ (X, B, λ). The general terminology will be that of Chap. 2. In notation for finitely additive measures that will be considered in detail in Chap. 4, the representation theorem for the dual space L ∞ (X, L, λ)∗ , which is analogous to the Riesz Theorem 2.37 for continuous functions, is the following special case of [35, Thm. 2.3], see also [12, Thm. IV.8.16] and the references therein. This result will be referred to as the Yosida–Hewitt representation of L ∗∞ (X, L, λ). Theorem 3.1 For every bounded linear functional on L ∞ (X, L, λ) there exists a finitely additive measure (Definition 4.1) ν on L such that  f (u) =

ν(N ) = 0 for all N ∈ N ,

(3.1a)

u dν for all u ∈ L ∞ (X, L, λ),

(3.1b)

|ν|(X ) =  f ∞ < ∞.

(3.1c)

X

Conversely if ν is a finitely additive measure on X with ν(N ) = 0 for all N ∈ N , then f in (3.1b) is a well-defined bounded linear functional on L ∞ (X, L, λ). The set of finitely additive measure on X with ν(N ) = 0 for N ∈ N will be denoted by L ∗∞ (X, L, λ). (Note that |ν|(N ) = 0 if N ∈ N and ν ∈ L ∗∞ (X, L, λ).) Once the integral with respect to finitely additive measures in Theorem 3.1 has been defined, the proof in Sect. 6.2 of the theorem is straightforward (indeed, an interested reader might like to have a quick look now). However, there are differences from standard theory, see Remark 2.26, and the development needs to be handled carefully. In the next few chapters, notation and terminology for finitely additive measures, © The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 J. Toland, The Dual of L ∞ (X, L, λ), Finitely Additive Measures and Weak Convergence, SpringerBriefs in Mathematics, https://doi.org/10.1007/978-3-030-34732-1_3

27

28

3 L ∞ and Its Dual

their existence and basic properties on σ-algebras will be established. However, the following implications of Theorem 3.1 are more or less immediate.    Corollary 3.2 If {Ak } ⊂ L is a sequence in L with λ Ai A j = 0, i = j, then χ Ak  0 as k → ∞ in L ∞ (X, L, λ), where χ Ak denotes the characteristic function of Ak . Proof By Theorem 3.1, for any f ∈ L ∞ (X, L, λ)∗ there is a finitely additive measure ν with  χ Ak dν = ν(Ak ) = ν + (Ak ) − ν − (Ak ), f (χ Ak ) = X

where ν ± (the positive and negative parts of ν, see (4.1c)) are non-negative finitely additive measures. Since, by finite additivity and (6.2d), 0

K 

ν ± (Ak ) = ν ±

k=1

K 

Ak  ν ± (X ) < ∞, for all K ∈ N,

k=1

ν ± (Ak ) → 0 and the result follows.



When combined with the Hahn–Banach theorem, Theorem 3.1 yields a wide variety of finitely additive measures. Corollary 3.3 If (X, ) is a Hausdorff topological space and  is not the discrete topology, there exists a finitely additive measure ν = 0 which is zero on N and

v dν = 0 for all v ∈ L ∞ (X, B, λ) which are continuous on X . X Proof From the hypothesis and Remark 2.34, elements of L ∞ (X, B, λ) which are continuous on X form a proper closed linear subspace of L ∞ (X, B, λ). The result is then immediate from Theorem 3.1 and Corollary 2.40 of the Hahn–Banach theorem.  The existence of finitely additive measures with particular properties will be considered again in Chap. 5. Corollary 3.4 Suppose B is the Borel σ-algebra of a locally compact Hausdorff space (X, ), and (X, B, λ) is a corresponding measure space. Then for every ν ∈ L ∗∞ (X, B, λ) there exists a unique, regular, real Borel measure ν˜ such that 

 v dν = X

v d ν˜ for all v ∈ C0 (X, ), |ν|(X ˜ )  |ν|(X ),

(3.2)

X

and ν˜  0 on Borel sets when ν  0 on L. Proof For ν ∈ L ∗∞ (X, B, λ), define f ∈ L ∞ (X, B, λ)∗ by (3.1b). The restriction f˜ of f to C0 (X, ) has  f˜C0 (X,)∗   f  L ∞ (X,B,λ)∗ = |ν|(X ) < ∞, and the result follows from Theorem 2.37 (Riesz). 

3 L ∞ and Its Dual

29

It follows from Corollary 3.3 that the inequality in (3.2) may be strict and, from the Valadier–Hensgen example in Sect. 6.6, that for a given regular real Borel measure ν˜ there may be uncountably many finitely additive measures ν which satisfy (3.2).

Chapter 4

Finitely Additive Measures

Finitely additive measures are naturally defined on algebras (collections of sets which are closed under complementation and finite unions), but here they are considered on σ-algebras (closed under complementation and countable unions) because L in Theorem 3.1 is a σ-algebra. Although the terminology “finitely additive measures” may suggest only a slight modification of familiar measure theory, intuition based on experience with σ-additivity can seriously mislead, see (†) in the Preface which is justified in Remark 6.1.

4.1 Definition, Notation and Basic Properties Definition 4.1 (Finitely Additive Measures [35, Sects. 1.2–1.7]) A finitely additive measure ν on L is a mapping from L into R with ν(∅) = 0 and sup |ν(A)| < ∞; A∈L

ν(A



B) = ν(A) + ν(B) for all A, B ∈ L with A



B = ∅.

A finitely additive measure ν is σ-additive if and only if ν

 k∈N

   E k = ∅, j = k. Ek = ν(E k ) for all {E k } ⊂ L with E j k∈N

In the literature, it is usual for ba(L) (bounded and additive on L) to denote the set of finitely additive measures on L. Let Σ(L) ⊂ ba(L) denote the σ-additive measures.  © The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 J. Toland, The Dual of L ∞ (X, L, λ), Finitely Additive Measures and Weak Convergence, SpringerBriefs in Mathematics, https://doi.org/10.1007/978-3-030-34732-1_4

31

32

4 Finitely Additive Measures

Obviously, the set Σ(L) coincides with the set of real measures in Sect. 2.4. However λ ∈ ba(L) if and only if λ(X ) < ∞. Hence λ ∈ / Σ(L) when λ(X ) = ∞ although it is σ-additive on L. However, because by hypothesis λ is σ-finite on L, there exists [28, Lemma 6.9] an integrable function f which is positive everywhere on X . Hence γ(E) := E f dλ defines γ ∈ Σ(L) which is positive on L \ N and can be used as a surrogate for λ in arguments involving finitely additive measures, for example, in the proof of Theorem 10.4. There follows a review of standard theory sufficient for our purposes. For a comprehensive account see [35], [12, Chap. III] or [6, Chap. 4]. Lemma 4.2 Suppose ν ∈ ba(L). (a) Then ν ∈ Σ(L) if and  only if ν(Ak ) → 0 as k → ∞ for any nested sequence Ak+1 ⊂ Ak , Ak ∈ L with k Ak = ∅. (b) If 0  ν  γ for some γ ∈ Σ(L), then ν ∈ Σ(L).  Proof (a) Suppose ν ∈ Σ(L) and, for {Ak } ⊂ L nested with k Ak =∅, let E k = Ak \ Ak+1 ∈ L. Then {E k } is a sequence of mutually disjoint sets with k E k = A1 . Since ν(A1 ) is finite and ν is σ-additive ν(A1 ) =

∞ 

ν(E k ) =

k=1

∞ 

 ν(Ak \ Ak+1 ) = lim ν(A1 ) − ν(Ak ) , k→∞

k=1

whence limk→∞ ν(Ak ) = 0. For the  converse suppose ν ∈ ba(L) and limk→∞ ν(Ak ) = 0 for all nested {Ak } ⊂ sets {E j } in L let L with k Ak = ∅. Then for any sequence of mutually disjoint  Ak = ∞ k E j . By hypothesis lim k→∞ ν(Ak ) = 0 because k Ak = ∅ and by finite additivity, k−1 

ν(E j ) = ν

j=1

∞ 

∞    E j − ν(Ak ) → ν E j as k → ∞.

j=1

j=1

Hence ν ∈ Σ(L). Part (b) is immediate because its hypothesis implies that ν satisfies the criterion for σ-additivity in (a).  The next result, which is a simple case of Nikodym’s convergence theorem [8, Thm. 4.6.3(i)], depends on the following trivial observation. Suppose a jk ∈ R, j, k ∈ N, 0  a jk  a j  k   M when j  j  , k  k  . Then lim lim a jk = sup sup a jk = sup sup a jk = lim lim a jk .

k→∞ j→∞

k

j

j

k

j→∞ k→∞

4.1 Definition, Notation and Basic Properties

33

Lemma 4.3 Suppose {γk } ⊂ Σ(L) and 0  γk (E)  γk+1 (E)  M < ∞ for all E ∈ L, k ∈ N. Then γ ∈ Σ(L) where γ(E) = limk→∞ γk (E), E ∈ L. Proof It is obvious that γ is finitely additive. To show j γ ∈ Σ(L), let {E  } ⊂ L be a sequence of mutually disjoint sets and put a jk = =1 γk (E  ) so that 0  a jk  a j  k   M if j  j  , k  k  . Now since γk ∈ Σ(L), γ

∞ 

j ∞     E  = lim γk E  = lim lim γk (E  ) k→∞

1

k→∞ j→∞

1

 j

= sup sup k

j

γk (E  ) = sup sup j

=1

= lim lim

j→∞ k→∞

j 

k

γk (E  ) = lim

j→∞

=1

=1 j 

γk (E  )

=1 j 

γ(E  ) =

=1

∞ 

γ(E  ),

=1

which shows γ ∈ Σ(L).



Finitely additive measures form a real linear space with, for E ∈ L, (α1 ν1 + α2 ν2 )(E) = α1 ν1 (E) + α2 ν2 (E), νi ∈ ba(L), αi ∈ R, i = 1, 2, and a partial ordering ν1  ν2 is defined by ν1 (E)  ν2 (E) for all E ∈ L. For ν1 , ν2 ∈ ba(L) and E ∈ L let (ν1 ∨ ν2 )(E) = (ν1 ∧ ν2 )(E) =

 ν1 (F) + ν2 (E \ F) ,

 ν1 (F) + ν2 (E \ F) ,

sup

{F∈L:F⊂E}

inf

{F∈L:F⊂E}

from which it is obvious that     ν1 ∧ ν2 = − (−ν1 ) ∨ (−ν2 ) and ν1 ∨ ν2 = − (−ν1 ) ∧ (−ν2 ) .

Theorem 4.4 (a) ν1 , ν2 ∈ ba(L) ⇒ ν1 ∧ ν2 , ν1 ∨ ν2 ∈ ba(L), ν1 , ν2 ∈ Σ(L) ⇒ ν1 ∧ ν2 , ν1 ∨ ν2 ∈ Σ(L).

(4.1a)

(4.1b)

34

4 Finitely Additive Measures

(b) For ν ∈ ba(L), let ν + = ν ∨ 0, ν − = (−ν) ∨ 0. Then ν ±  0, ν = ν + − ν − and ν + ∧ ν − = 0.

(4.1c)

(c) If ν ∈ ba(L) and γ1 , γ2 ∈ Σ(L) with γ1  ν  γ2 , then ν ∈ Σ(L). (d) If ν ∈ ba(L) and ν(E) = 0 for all E ∈ N , then ν ± (E) = 0 for all E ∈ N . Remark 4.5 When ν ∈ Σ(L) (equivalently when ν is a real measure) (4.1c) is equivalent to (2.4) in Sect. 2.4 because of the Hahn Decomposition Theorem 2.17. For ν ∈ ba(L), ν ± are referred to as the positive and negative parts of ν, and  |ν| := ν + + ν − as the total variation of ν (see Theorem 3.1).   Proof  (a) Let E 1 , E 2 ∈ L where E 1 E 2 = ∅ and for F ⊂ E 1 E 2 let Fi = E i F, i = 1, 2. Then by finite additivity, (ν1 ∨ ν2 )(E 1 = =



E2 ) =

sup {F∈L:F⊂E 1



E2 }

sup

{F1 ∈L:F1 ⊂E 1 }

+

sup {F∈L:F⊂E 1



E2 }

ν1 (F1 ) + ν1 (F2 ) + ν2 (E 1 \ F1 ) + ν2 (E 2 \ F2 )

ν1 (F1 ) + ν2 (E 1 \ F1 )

sup

{F2 ∈L:F2 ⊂E 2 }



 ν1 (F) + ν2 (E 1 E2 ) \ F 



 ν1 (F2 ) + ν2 (E 2 \ F2 ) = (ν1 ∨ ν2 )(E 1 ) + (ν1 ∨ ν2 )(E 2 ).

  So ν1 ∨ ν2 ∈ ba(L), and hence ν1 ∧ ν2 = − (−ν1 ) ∨ (−ν2 ) ∈ ba(L) by (4.1b). Now suppose ν1 , ν2 ∈ Σ(L) and {E k } ⊂ L is a sequence of mutually disjoint sets. Then, for  ∈ (0, 1) and k ∈ N, there exists Fk ⊂ E k , Fk ∈ L, such that ν1 (Fk ) + ν2 (E k \ Fk )  (ν1 ∧ ν2 )(E k ) + k . Then, by countable additivity, with F :=

 k

Fk ⊂

 k

E k =: E,

∞    ν1 (Fk ) + ν2 (E k \ Fk ) (ν1 ∧ ν2 )(E)  ν1 (F) + ν2 (E \ F) = k=1



∞  

∞   (ν1 ∧ ν2 )(E k ) + k = (ν1 ∧ ν2 )(E k ) +

k=1

k=1

for all  ∈ (0, 1). Hence (ν1 ∧ ν2 )

∞  k=1

∞   E k = (ν1 ∧ ν2 )(E)  (ν1 ∧ ν2 )(E k ). k=1

 , 1−

4.1 Definition, Notation and Basic Properties

35

For the opposite inequality note that, with F ∈ L and F ⊂ E = ν1 (F) + ν2 (E \ F) =

∞  

ν1 (F



(ν1 ∧ ν2 )

∞ 

k

Ek ,

∞   E k ) + ν2 (E k \ F)  (ν1 ∧ ν2 )(E k ),

k=1

whence



k=1

∞   E k = (ν1 ∧ ν2 )(E)  (ν1 ∧ ν2 )(E k ).

k=1

k=1

Hence ν1 ∧ ν2 ∈ Σ(L), and it follows that ν1 ∨ ν2 = −((−ν1 ) ∧ (−ν2 )) ∈ Σ(L). (b) By definition ν ±  0 and inf

{F∈L:F⊂E}

ν(F) = −ν − (E),

sup

{F∈L:F⊂E}

ν(F) = ν + (E).

(4.2)

Since ν(E) = ν(F) + ν(E \ F) for E, F ∈ L with F ⊂ E, it follows that ν(F) − ν − (E)  ν(E)  ν(F) + ν + (E),

(4.3)

and ν = ν + − ν − follows by taking the supremum on the left and the infimum on the right of (4.3), over F ∈ L with F ⊂ E, using (4.2). Finally suppose ν + ∧ ν − = 0. Then, since ν ±  0 are finitely additive by part (a) there exists E ∈ L and α > 0 such that for any F ⊂ E, F ∈ L, 0 < α  ν + (E \ F) + ν − (F) = ν + (E) − ν + (F) + ν − (F) = ν + (E) − ν(F). Hence for all F ⊂ E, F ∈ L, ν(F)  ν + (E) − α. Since α > 0 this contradicts (4.2) and it follows that ν + ∧ ν − = 0. (c) Since 0  ν − γ1  γ2 − γ1 and γ2 − γ1 ∈ Σ(L), Lemma 4.2 implies ν − γ1 ∈ Σ(L), and hence ν ∈ Σ(L) as required. (d) The result follows from the hypothesis and (4.2) since F ∈ N when F ∈ L and F ⊂ E ∈ N , because (X, L, λ) is complete.  Definition 4.6 (Absolute Continuity and Singularity [6, Chap. 6]) For ν1 , ν2 ∈ ba(L) write ν1 ν2 (ν1 is absolutely continuous with respect to ν2 ), if for all  > 0 there exists δ such |ν1 (E)| <  when |ν2 |(E) < δ, and write ν1 ⊥ ν2 (ν1 and ν2 are mutually singular) if for every  > 0 there exists E ∈ L such that  |ν1 |(E) + |ν2 |(X \ E) < . Remark 4.7 In the special case when ν1 , ν2 ∈ Σ(L) the above definitions are equivalent to simpler statements [6, Thms. 6.1.6 & 6.1.17]:

36

4 Finitely Additive Measures

ν1 ν2 if and only if |ν2 |(E) = 0 implies ν1 (E) = 0; ν1 ⊥ ν2 if and only if |ν1 |(E) + |ν2 |(X \ E) = 0 for some E ∈ L. However, it is important to remember that a finitely additive measure ν which vanishes on N need not satisfy ν λ in Definition 4.6 if ν ∈ / Σ(L). 

4.2 Purely Finitely Additive Measures Definition 4.8 (Purely Finitely Additive Measures) μ ∈ ba(L) is purely finitely additive if

   γ ∈ Σ(L) : 0  γ  μ+ = 0 = γ ∈ Σ(L) : 0  γ  μ− .  +

−

Clearly, μ is purelyfinitely additive if and only if μ and μ are purely finitely additive and Π (L) Σ(L) = {0}, where Π (L) denotes the set of purely finitely additive measures. Remark 4.9 When μ1 , μ2 ∈ Π (L) and μ1  ν  μ2 for some ν ∈ ba(L), it is imme− − diate that ν ∈ Π (L) since ν +  μ+ 2 , ν  μ1 . Similarly, |ν|  μ ∈ Π (L) implies ν ∈ Π (L).  Theorem 4.10 (a) ν ∈ ba(L) is purely finitely additive if and only if ν ± ∧ γ = 0 for all 0  γ ∈ Σ(L). (b) For μ, μ1 , μ2 ∈ Π (L) and α ∈ R, αμ, μ1 + μ2 , |μ|, μ1 ∧ μ2 , μ1 ∨ μ2 ∈ Π (L). Proof (a) Suppose ν ∈ Π (L). Then 0  ν ± ∧ γ  γ for any 0  γ ∈ Σ(L), which implies, by Theorems 4.4(a) and 4.2(b), that ν ± ∧ γ ∈ Σ(L) and hence by Definition 4.8 that ν ± ∧ γ = 0. Conversely, for ν ∈ ba(L) suppose that ν ± ∧ γ = 0 for all 0  γ ∈ Σ(L). Then 0  γ  ν ± , γ ∈ Σ(L), implies that γ = ν ± ∧ γ = 0. Hence ν ∈ Π (L) by Definition 4.8. (b) That αμ ∈ Π (L), α ∈ R, is immediate from Definition 4.8 since (−μ)± = ∓ μ . To show that μ1 + μ2 ∈ Π (L) consider first the case when μ1 and μ2 are nonnegative. By part (a) it suffices to show that (μ1 + μ2 ) ∧ γ = 0 for all 0  γ ∈ Σ(L). In other words, it suffices to show, for all E ∈ L and all 0  γ ∈ Σ(L), that inf

{F∈L:F⊂E}

{μ1 (F) + μ2 (F) + γ(E \ F)} = 0.

Since μi ∧ γ = 0, for any  > 0 there exist E ⊃ Fi ∈ L, with

4.2 Purely Finitely Additive Measures

37

μi (Fi ) + γ(E \ Fi )  , i = 1, 2. Therefore since μi  0, μ1 (F1



F2 ) + μ2 (F1



F2 ) + γ(E \ (F1



F2 ))

 μ1 (F1 ) + μ2 (F2 ) + γ(E \ F1 ) + γ(E \ F2 )  2.

(4.4)

This proves μ1 + μ2 ∈ Π (L) for non-negative μi ∈ Π (L) and hence |μ| = μ+ + μ− ∈ Π (L) for any μ ∈ Π (L). More generally, + − − |μ1 + μ2 |  μ+ 1 + μ2 + μ1 + μ2 ∈ Π (L) when μi ∈ Π (L), i = 1, 2,

and μ1 + μ2 ∈ Π (L) follows by Remark 4.9. Similarly |μ1 ∧ μ2 | + |μ1 ∨ μ2 |  2(|μ1 | + |μ2 |), μ1 , μ2 ∈ Π (L), implies that μ1 ∧ μ2 and μ1 ∨ μ2 are in Π (L).



By (4.1a) and Theorem 4.10(a), a non-negative ν ∈ ba(L) is purely finitely additive if and only if for every 0  γ ∈ Σ(L), E ∈ L and  > 0 there exists F ∈ L, F ⊂ E with ν(E \ F) + γ(F) < .

(4.5)

Since L is a σ-algebra this remark can be significantly refined. Lemma 4.11 For 0  γ ∈ Σ(L), 0  μ ∈ Π (L) and  > 0 there exists F ∈ L with γ(F)   and μ(F) = μ(X ). Proof If μ(X ) = 0 the result holds with F = ∅. So suppose μ(X ) > 0 and for given  > 0 let ek = 2−k . Then with E 0 = X , by (4.5) there exists F1 ∈ L with μ(E 0 \ F1 ) + γ(F1 ) < e1 . With E 1 = E 0 \ F1 , in L there exists F2 ⊂ E 1 , F2 ∈ L, with μ(E 1 \ F2 ) + γ(F2 ) < e2 . Then with E 2 = E 1 \ F2 , there exists F3 ⊂ E 2 , F3 ∈ L, with μ(E 2 \ F3 ) + γ(F3 ) < e3 . Proceeding by induction, for k ∈ N0 , μ(E k \ Fk+1 ) + γ(Fk+1 ) < ek+1 , E k ⊃ Fk+1 ∈ L, E k+1 = E k \ Fk+1 . It follows that for K ∈ N, K K K       Fk  e K and γ Fk  ek < . 0  μ E0 \ k=1

k=1

k=1

38

4 Finitely Additive Measures

 Now since L is a σ-algebra, F = ∞ k=1 Fk ∈ L with γ(F)  . Moreover, μ(X ) = μ(F) since μ(X \ F)   K for all K and μ is finitely additive. This completes the proof.  The sense in which a purely finitely additive measure on a σ-algebra is singular with respect to any σ-additive measure is captured by the following. Theorem 4.12 Let 0  ν ∈ ba(L). Then ν ∈ Π (L) if and only if for every nonnegative γ ∈ Σ(L) there exists a sequence {E k } ⊂ L with E k+1 ⊂ E k , ν(E k ) = ν(X ) for all k and γ(E k ) → 0 as k → ∞. Proof Suppose 0  ν ∈ Π (L) and 0  γ ∈ Σ(L). Then by the preceding lemma, Ek = for Let k n ∈ N there exists Fn ∈ L with ν(Fn ) = ν(X ) and 0  γ(Fn ) < 1/n. k F . Then ν(E ) = ν(X ) because, by finite additivity, ν(X \ E )  ν(X \ n k k n=1 n=1 Fn ) = 0, and 0  γ(E k )  ν(Fk )  1/k. Conversely, if for every non-negative γ ∈ Σ(L) such a sequence {E k } exists, then γ ∧ ν = 0. Hence ν ∈ Π (L) by Theorem 4.10. 

4.3 Canonical Decomposition: ba(L) = Σ(L) ⊕ Π(L) Theorem 4.13 Any ν ∈ ba(L) can be written uniquely as ν = μν + γν where μν ∈ Π (L), γν ∈ Σ(L). Moreover μν , γν  0 if ν  0. Proof To show that such a decomposition, if it exists, is unique let ν = μi + γi , μi ∈ Π(L), γi ∈ Σ(L), i = 1, 2. Then, by Theorem  4.10, μ1 − μ2 = γ2 − γ1 ∈ Π (L) Σ(L) and uniqueness follows since Π (L) Σ(L) = {0} as noted after Definition 4.8. Since, by (4.1c), ν = ν + − ν − and, by Theorem 4.10, Σ(L) and Π (L) are linear spaces, it suffices to show the decomposition exists for non-negative ν ∈ ba(L). Let

 ς := sup γ(X ) : 0  γ  ν, γ ∈ Σ(L) and let γk ∈ Σ(L) be such that 0  γk  ν and limk→∞ γk (X ) = ς. Replacing γk by ∨kj=1 γ j , by Theorem 4.4(a) there is no loss in assuming that γk  γk+1 for all k ∈ N. Consequently γν (E) := limk→∞ γk (E)  ν(E)  ν(X ) exists for all E ∈ L, and γν ∈ Σ(L) by Lemma 4.3. Clearly 0  μν := ν − γν ∈ ba(L) and γν (X ) = ς. To show that μν ∈ Π (L), suppose 0  γ  μν for some γ ∈ Σ(L). Then 0  γ + γν  ν which implies that γ(X ) + γν (X )  ς = γν (X ). Hence γ(X ) = 0, which, by Definition 4.8, implies 0  μν ∈ Π (L). Obviously from the construction, γν and μν are non-negative when ν  0. This completes the proof. 

4.4 L ∗∞ (X, L, λ)

39

4.4 L ∗∞ (X, L, λ) This chapter closes by specialising briefly to the set L ∗∞ (X, L, λ) of finitely additive measures that feature in Theorem 3.1 (Yosida–Hewitt). Definition 4.14 (L ∗∞ (X, L, λ)) ν ∈ L ∗∞ (X, L, λ) if ν ∈ ba(L) and ν(E) = 0 for all E ∈ N .  Theorem 4.15 Suppose ν ∈ L ∗∞ (X, L, λ).  Then ν ∈ Π (L) if and only if ν + ∧ γ = − ν ∧ γ = 0 for all non-negative γ ∈ Σ(L) L ∗∞ (X, L, λ).  Proof Suppose ν ∈ L ∗∞ (X, L, λ) and ν ± ∧ γ = 0 for all non-negative γ ∈ Σ(L) γˆ  ν ± . Since, by Theorem 4.4(d), L ∗∞ (X, L, λ). Let γˆ ∈ Σ(L) be arbitrary with 0  ± ∗ ν = 0 on N it follows that γˆ ∈ Σ(L) L ∞ (X, L, λ). Hence, by hypothesis, ν ± ∧ γˆ = 0 and, by Definition 4.8, ν ∈ Π (L).  Conversely, if ν ∈ Π (L), by Definition 4.8 ν ± ∧ γ = 0 for all 0  γ ∈ Σ(L)  L ∗∞ (X, L, λ). Theorem 4.16 Any ν ∈ L ∗∞ (X, L, λ) can be written uniquely as       Π (L) ⊕ L ∗∞ (X, L, λ) Σ(L) . ν = μν + γν ∈ L ∗∞ (X, L, λ)

(4.6)

Proof If 0  ν ∈ L ∗∞ (X, L, λ) in Theorem 4.13, 0  γν  ν implies that γν (E) = 0, E ∈ N . Hence γν ∈ L ∗∞ (X, L, λ) and since μν = ν − γν , so is μν . The result for general ν ∈ Π (L) follows since ν ∈ L ∗∞ (X, L, λ) implies ν ± ∈  L ∗∞ (X, L, λ), see Theorem 4.4(d).

Chapter 5

G: 0–1 Finitely Additive Measures

In the notation of Definition 4.14, this chapter concerns the set   G = ω ∈ L ∗∞ (X, L, λ) : ω(X ) = 1, ω(E) ∈ {0, 1}, E ∈ L

(5.1)

of finitely additive measures, each element of which would, by Theorem 3.1, correspond to an element f ∈ L ∞ (X, L, λ)∗ with the property that f (χ X ) = 1, f (χ E ) ∈ {0, 1}, E ∈ L;

f (χ E ) = 0, E ∈ N .

Much of the importance of G in the sequel can be traced back to the next theorem and subsequent remark which suggest that the action of elements of G on L ∞ (X, L, λ) is reminiscent of Dirac measures acting on continuous functions.  Theorem 5.1 For u ∈ L ∞ (X, L, λ) and ω ∈ G there is a unique α ∈ I := [u]− ∞,  [u]+ ∞ (see (2.8)) such that ω



 x ∈ X : |u(x) − α| <  = 1 for all  > 0.

(5.2)

This means (5.2) holds for a unique α ∈ R, independent of the function chosen to represent the equivalence class u ∈ L ∞ (X, L, λ). Proof For i = 1, 2 suppose u i is essentially bounded, αi ∈ R and u 1 (x) = u 2 (x) for λ-almost all x ∈ X . If (5.2) is satisfied by both (u i , αi ), i = 1, 2, and if α1 − α2 =: 20 > 0, then    x ∈ X : |u 2 (x) − α2 | < 0 ∈ N . x ∈ X : |u 1 (x) − α1 | < 0 Hence by finite additivity the ω-measure of their union would be 2, contradicting ω ∈ G. Thus (5.2) cannot be satisfied by both (u 1 , α1 ) and (u 2 , α2 ) simultaneously © The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 J. Toland, The Dual of L ∞ (X, L, λ), Finitely Additive Measures and Weak Convergence, SpringerBriefs in Mathematics, https://doi.org/10.1007/978-3-030-34732-1_5

41

5 G: 0–1 Finitely Additive Measures

42

if α1 = α2 when u 1 = u 2 λ-almost everywhere. On the other hand if α1 = α2 and u 1 = u 2 λ-almost everywhere, either they both satisfy (5.2) or neither satisfies (5.2). This shows that the value of α (if any) for which (5.2) holds is independent of the function used to represent the equivalence class u ∈ L ∞ (X, L, λ) and, for given u ∈ L ∞ (X, L, λ), (5.2) holds for at most one α. By a similar argument α ∈ I if (5.2) holds. Now suppose there exists for given u there is no such α. Then for all α ∈ I  K (αk − α > 0 with ω {x ∈ X : |u(x) − α| < α } = 0. By compactness, I ⊂ k=1 αk , αk + αk ) and consequently

ω(X ) = ω



x : u(x) ∈

K

(αk − αk , αk + αk )



k=1 K  

 ω x : u(x) ∈ (αk − αk , αk + αk ) = 0. k=1

Since ω(X ) = 1, (5.2) holds for a unique α.



Remark 5.2 If ω ∈ G but ω ∈ / Σ(L) (which often happens, see Theorem 5.9 below) it does not follow from (5.2) that ω ({x ∈ X : u(x) = α}) = 1. Nevertheless (5.2) is what is meant by saying that in the sense of finitely additive measures u ∈  L ∞ (X, L, λ) is a constant ω-almost everywhere. The important fact that elements of G proliferate is a consequence of Zorn’s lemma in the light of a well-known one-to-one correspondence between elements of G and families of sets in L \ N called ultrafilters in (X, L, λ). This correspondence leads to Theorem 5.6 on the existence of elements of G with certain properties and ultimately to (†) in the Preface.

5.1 G and Ultrafilters Definition 5.3 (Ultrafilter) Given (X, L, λ), F ⊂ L is a filter if (i) X ∈ F and N



F = ∅,  E 2 ∈ F, (ii) E 1 , E 2 ∈ F ⇒ E 1 (iii) E 2 ⊃ E 1 ∈ F ⇒ E 2 ∈ F. A filter U which is maximal with respect to set inclusion, i.e. for a filter F (iv) U ⊂ F ⇒ F = U,

5.1 G and Ultrafilters

43

is called an ultrafilter in (X, L, λ). The set of ultrafilters is denoted by U.



Theorem 5.4 (a) For U ∈ U, an element ωU ∈ G is defined by  ωU (E) :=

 1 if E ∈ U . 0 otherwise

(5.3a)

(b) For ω ∈ G, an element Uω ∈ U is defined by   Uω := E ∈ L : ω(E) = 1 .

(5.3b)

Note that ωUω = ω when ω ∈ G and UωU = U when U ∈ U. Proof (a) To show that ωU , U ∈ U, is well defined on L it suffices to show that exactly one of E and E = X \ E is in U for every E ∈ L. To begin  note that at most one of E and E is in U, for otherwise E, E ∈ U implies ∅ = E E ∈ U, a contradiction since ∅ ∈ N .  / U. Then suppose that E F ∈ /N To see that one of E, E is in U, suppose E ∈ for all F ∈ U and let    G∈L:E F ⊂ G for some F ∈ U . F =U Clearly F is a filter with U ⊂  F and E ∈ F \ U. Since this contradicts the maximality of U in (iv) it follows that E Fˆ ∈ N for some Fˆ ∈ U.  It now follows that E ∈ / N , since E Fˆ ∈ N and Fˆ ∈ / N by (i). Thus E ∈ /U 



ˆ / N , equivalently N ∈ N implies N ∈ U. In particular, E F ∈ N implies that E ∈     implies that E Fˆ ∈ U, and hence by (ii) E Fˆ = (E Fˆ ) Fˆ ∈ U. Finally  / U implies X \ E ∈ U, E Fˆ ⊂ E implies that E ∈ U by (iii). This shows that E ∈ is well defined by (5.3a). It remains to check that ω ∈ ba(L). and hence ω  U If A B = ∅, A, B ∈ L, it follows from (i) and (ii) that at most one of ωU (A)  then ωU (A B) = 1 by (iii). If both are zero, andωU (B) is 1, and if one of them is 1,  A B ∈ U by (ii), and hence ωU (A B) = 0. Hence ωU ∈ ba(L) and it follows that ωU ∈ G. (b) Now suppose Uω is given by (5.3b) where ω ∈ G. Then Uω satisfies (i) and (iii), by (5.1) and finite additivity. Also, for E 1 , E 2 ∈ Uω , 1 = ω(E 1



 E2 ) =

ω(E 1 \ E 2 ) + ω(E 2 ) = ω(E 1 \ E 2 ) + 1 . ω(E 2 \ E 1 ) + ω(E 1 ) = ω(E 2 \ E 1 ) + 1

  Hence ω(E 1 \ E 2 ) = 0 = ω(E 2 \ E 1 ) which implies ω(E 1 E 2 ) = 1. So E 1 E 2 ∈ Uω and Uω satisfies (ii). F denote thecollection To see that U ω satisfies (iv), suppose it does not and let  F is totally ordered by inclusion, F ∈F F ∈ of filters which contain Uω . If F ⊂   F is an upper bound for F. Hence, by Zorn’s lemma,  F has a maximal element,

5 G: 0–1 Finitely Additive Measures

44

 ∈ U say. Since, by assumption, Uω is not maximal, there exists F ∈ U  \ Uω with U ωU (F) = 1 and ω(F) = 0. Hence ω(F ) = 1 which implies that F ∈ Uω and hence  that ωU (F ) = 1 and ωU (F) = 0. This contradiction proves Uω satisfies (iv). The observation that G is determined entirely by the ultrafilter structure of L \ N is considered further in Lemma 9.1 and Corollary 9.2. For example, in a locally compact Hausdorff space X , each ω ∈ G may be identified with a unique point of X just as a Dirac measure is identified with a singleton (see Remark 9.5). However unlike Dirac measures, infinitely many elements of G may be identified with the same point and some elements of G are identified with the point at infinity. A simple consequence of Theorem 5.4 is that ω(A



B) = ω(A)ω(B) for all A, B ∈ L, ω ∈ G.

(5.4)

5.2 G and the λ-Finite Intersection Property A finitely additive measure on L belongs to L ∗∞ (X, L, λ) if ν(N ) = 0 for all N ∈ N and Theorem 3.1 combined with the Hahn–Banach theorem yields the existence of a wide variety of elements of L ∗∞ (X, L, λ). The following complementary approach, also based on Zorn’s lemma, concerns the existence of elements of G which are determined (non-uniquely) by families of sets with the following property. Definition 5.5 (λ-finite intersection property) L has the λ-finite intersection  E ⊂  K property if E k ∈ E, 1  k  K , implies λ  E k=1 k > 0. Theorem 5.6 If E has the λ-finite intersection property, there exists ω ∈ G with ω(E) = 1 for all E ∈ E. Proof Consider the collection F of filters F with E ⊂ F. Then F = ∅ since 

A∈L: A⊃

K 

 E k , E k ∈ E, K ∈ N ∈ F.

k=1

Now suppose  that T ⊂ F is a family of filters which is totally ordered by set inclusion and let F ∗ = F ∈T F. Then F ∗ ∈ F is an upper bound for T. So by Zorn’s lemma, F has a maximal element U which is an ultrafilter. By Theorem 5.4 (a), ωU ∈ G satisfies the theorem.  The next result shows that for given E there may be uncountably many ω ∈ G that satisfy Theorem 5.6. Corollary 5.7 (a) For A ∈ L \ N there exists ω ∈ G with ω(A) = 1. (b) Suppose, for j ∈ N, that F j ∈ L \ N and Fi F j ∈ N when i = j. Let

5.2 G and the λ-Finite Intersection Property

45

  E = E k : k ∈ N where E k = Fj . jk

Then for uncountably many distinct ω ∈ G \ Σ(L), ω(E k ) = 1 for all k. Proof (a) This follows from Theorem 5.6 with E = {A}. (b) For a ∈ (0, 1) let q j (a) ∈ Q, j ∈ N, be suchthat q j−1 (a) < q j (a)  a as j → ∞. Let Q a = {q j (a) : j ∈ N} and note that Q a Q b is at most finite if a = b. Now let Na = η(Q a ) where η : Q → N is a bijection. So {Na : a ∈ (0,1)} is an uncountable collection of infinite subsets of N with the property that Na Nb is at most finite if  a = b. In particular, for a = b ∈ (0, 1) there exists K (a, b)∈ N such that { j ∈ Na Nb : j  K (a, b)} = ∅. Consequently a = b implies Fi F j ∈ N for all i ∈ Na , j ∈ Nb with i, j  K (a,b). Now for a ∈ (0, 1) note that E ka := k j∈Na F j ∈ L, since L is a σ-algebra, and let Ea = {E ka : k ∈ N}. Then by Theorem 5.6there exists ωa ∈ G with ωa (E ka ) = 1 for all k ∈ N. However, by construction, E ka E kb ∈ N for all k  K (a, b), a = b. It follows that ωa = ωb since 1 = ωa (E ka ) = ωb (E ka ) = 0 for k sufficiently large. Moreover, for all a, E ka ⊂ E k , k ∈ N, which implies ωa (E k ) = 1 for all a ∈ (0, 1) and k ∈ N.  a / Σ(L) follows from Lemma 4.2(a) since k E ka = ∅ but E k+1 ⊂ Finally, that ωa ∈ a a  E k and ω(E k ) = 1 for all k. This completes the proof. Example 5.8 As in Example 2.14, the space of all bounded sequences of real numbers indexed by N, usually denoted by ∞ , coincides with the space of bounded functions u : N → R. With ∞ (N) = L ∞ (X, L, λ), where X = N, L = ℘ (N) and λ is counting measure, let E = {E k : k ∈ N}, where E k = k + N. Then E has the λ-finite intersection property and E k satisfies the hypotheses of Corollary 5.7 with F j = { j}. Hence, there exist uncountably many ω ∈ G with ω(E k ) = 1 for all k. This example underlies Sect. 6.5 on integration of sequences in ∞ (N), and leads to Remark 9.13.  Theorem 5.9 For ω ∈ G (a) either ω ∈ Π (L) or ω ∈ Σ(L). (b) ω ∈ Σ(L) if and only if for an atom E ω ∈ L (Definition 2.8)  λ(E E ω ) ω(E) = for all E ∈ L. λ(E ω ) Proof (a) If this is false, by Theorem 4.13 ω = μ + γ where μ ∈ Π (L) and γ ∈ Σ(L) are non-negative, ω(X ) = 1, μ(X ) ∈ (0, 1), and by Theorem 4.12 there exists {E k } ⊂ L with μ(E k ) = μ(X ) for all k and γ(E k ) → 0 as k → ∞. If ω(E k ) = 0 for some k then 0 = ω(E k )  μ(E k ) = μ(X ), whence μ(X ) = 0 which is false. Therefore, since ω ∈ G, ω(E k ) = 1 for all k and it follows that 1 = ω(E k ) = μ(E k ) + γ(E k ) = μ(X ) + γ(E k ) → μ(X ) as k → ∞.

5 G: 0–1 Finitely Additive Measures

46

Hence μ(X ) = 1 which is also false. This proves (a). (b) Suppose ω ∈ G Σ(L). Then ω is a real measure and both ω and λ are σ-additive with ω λ. Since by hypothesis λ is σ-finite, by Theorem 2.28(b) (Radon–Nikodym) there exists g ∈ L 1 (X, L, λ) with  g dλ for all E ∈ L.

ω(E) = E

So g is non-negative λ-almost everywhere on X , λ ({x ∈ X : g(x)  n}) → 0 as n → ∞, and hence  g dλ → 0 as n → ∞. {x∈X :g(x)n}

In particular, ω



x ∈ X : g(x)  n



 =

g dλ → 0 as n → ∞. {x∈X :g(x)n}

Now ω ∈ G implies that for some N ∈ N, ω ({x ∈ X : g(x)  N }) = 0 and it follows that λ ({x ∈ X : g(x)  N }) = 0. Thus g ∈ L ∞ (X, L, λ) and by (5.2), for a unique α ∈ R,   ω x ∈ X : |g(x) − α| <  = 1 for all  > 0.

Since ω ∈ Σ(L) it follows that ω E ω = 1 where E ω = {x ∈ X : g(x) = α}. Moreover λ(E ω ) > 0 since ω(E ω ) = 1 and ω ∈ G. Therefore    ω(E) = ω(E E ω ) =  α dλ = αλ(E E ω ) for all E ∈ L. E

Eω

With α = 1/λ(E ω ), E ω is an atom with the required  properties since ω ∈ G. Conversely, when A is an atom ω(E) := λ(E A)/λ(A), E ∈ L, defines ω ∈  G Σ(L) since λ is σ-additive.  Remark 5.10 This shows that all elements of G are purely finitely additive when L has no atoms. In any case if X has infinitely many mutually disjoint subsets with positive λ-measure, by Corollary 5.7(b) there are uncountably many distinct ω ∈ G which are purely finitely additive. For example, in ∞ (N) G has countably many σ-additive elements given by the atoms {n}, n ∈ N, and uncountably many ω ∈ G which are purely finitely additive are given by Corollary 5.7 and Theorem 5.9. Further statements can be made about G, e.g. Lemmas 7.7, or 9.1 in a locally compact Hausdorff setting. 

Chapter 6

Integration and Finitely Additive Measures

The integral of u ∈ L ∞ (X, L, λ) with respect to ν ∈ L ∗∞ (X, L, λ) will been defined and sufficient of its properties established to prove the representation Theorem 3.1 (Yosida–Hewitt). Then the special case of integration with respect to ω ∈ G will be examined and the essential range of u ∈ L ∞ (X, L, λ) introduced. The chapter ends with an account of the Valadier–Hensgen example. For a more comprehensive account of integration with respect to finitely additive measures, see [6, Chap. 4], [12, Chap. III] and [35].

6.1 The Integral To begin note that if u ∈ L ∞ (X, L, λ) and w(x) = u(x) for λ-almost all x ∈ X , it follows that |ν|({x : w(x) = u(x)} = 0 for all ν ∈ L ∗∞ (X, L, λ) since |ν|(N ) = 0 when λ(N ) = 0. Therefore, by finite additivity the following construction is independent of whether u or w is used to represent an equivalence class in L ∞ (X, L, λ). Since, by (4.1c), ν = ν + − ν − and by Theorem 4.4(d) ν ± ∈ L ∗∞ (X, L, λ) when ν ∈ L ∗∞ (X, L, λ), it suffices first to define integration with respect to non-negative ν ∈ L ∗∞ (X, L, λ) and to extend by linearity. For u ∈ L ∞ (X, L, λ) and p > u∞ , let P:

− p = y0 < y1 < · · · < y K < y K +1 = p, K ∈ N,

(6.1a)

be a partition of [− p, p] and let S P (u) :=

K  k=0

yk ν(E k (u)) 

K 

yk+1 ν(E k (u)) =: S P (u),

(6.1b)

k=0

where E k (u) = {x : yk  u(x) < yk+1 }, 0  k < K . As remarked above, S P (u) and S P (u) are independent of which function u is used to represent an element of © The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 J. Toland, The Dual of L ∞ (X, L, λ), Finitely Additive Measures and Weak Convergence, SpringerBriefs in Mathematics, https://doi.org/10.1007/978-3-030-34732-1_6

47

48

6 Integration and Finitely Additive Measures

L ∞ (X, L, λ). Since ν is non-negative, finitely additive and ν(X ) < ∞, 0  S P (u) − S P (u) < ρ(P)ν(X ), where ρ(P) = max{yk+1 − yk : 0  k  K }. Furthermore, if a partition P is augmented by adding an additional point then S P (u) is not decreased and S P (u) is not increased. It follows that over all partitions P, inf P S P (u) = supP S P (u). With these observations, the integral of u ∈ L ∞ (X, L, λ) with respect to ν  0 is defined as  inf S P (u) = u dν = sup S P (u), P P X  u dν = lim S P (u). (6.2a) or equivalently lim S P (u) = ρ(P)→0

ρ(P)→0

X

It follows that when ν  0 and u  v λ-almost everywhere on X , 

 u dν  X

v dν.

(6.2b)

X

Now the integral of u ∈ L ∞ (X, L, λ) with respect to general ν ∈ L ∗∞ (X, L, λ) can be defined unambiguously by 



u dν + −

u dν := X

X



u dν − ,

(6.2c)

X

from which it follows that  χ E dν = ν(E), E ∈ L, ν ∈ L ∗∞ (X, L, λ).

(6.2d)

X

For simplicity of notation write 

 u dν := E

χ E u dν, E ∈ L.

(6.3a)

X

Note that for E ∈ L,       u dν   u∞ |ν|(E) and u dν = 0 if E ∈ N .   E

Moreover, when E = when i = j,

(6.4)

E

N n=1

E n , a finite union of sets in L with E i



Ej ∈ N

6.1 The Integral

49

 u dν = E

N   n=1

u dν.

(6.5)

En

Since (−ν)± = ν ∓ , it follows from (6.2c) that   αu dν = α u dν, u ∈ L ∞ (X, L, λ), α ∈ R. X

(6.6)

X

Therefore, the integral is linear on L ∞ (X, L, λ) if it is shown that 





(u + w) dν =

u dν +

X

X

w dν, u, w ∈ L ∞ (X, L, λ),

(6.7)

X

and by (6.2c) it is enough to consider ν  0. Let p = 1 + u∞ + w∞ and for  > 0 let P be a partition of [− p, p] for which S P (u) − S P (u) < . Then, by (6.5), 

 (u + w) dν = X

K k=0

E k (u)

(u + w) dν =

K   k=0

E k (u)

(u + w) dν



K

K

(yk + w) dν = S P (u) + k=0 w dν  k=0 K Ek (u) K Ek (u)  k=0 Ek (u) (yk+1 + w) dν = S P (u) + k=0 E k (u) w dν







= S P (u) + X w dν  X u dν + X w dν −  . = S P (u) + X w dν  X u dν + X w dν + 

Since  > 0 was arbitrary, (6.7) follows when 0  ν ∈ L ∗∞ (X, L, λ). The general case is immediate from (6.2c).  Remark 6.1 When γ ∈ Σ(L) L ∗∞ (X, L, λ), it is obvious that integration of u ∈ L ∞ (X, L, λ) with respect to γ coincides with Lebesgue integration. However, if ν is not σ-additive, familiar results such as the Monotone convergence theorem do not hold for this integral. For example, by Corollary 5.7(b), on the interval (0, 1) there are uncountably many finitely additive measures ν ∈ G such that ν(0, 1 − 1/k] = 0 and ν(1 − 1/k, 1) = 1 for all k > 1, k ∈ N. It follows from (6.2a) that for all such ν 

1 0



1− k1

1 dν = 1 but

u dν = 0 for all k ∈ N and all u ∈ L ∞ (X, L, λ).

0

This establishes (†) in the Preface.



∗ By Theorem be written as ν = μ + γ where μ ∈ L ∗∞  4.16, ν ∈ L ∞ (X, L, λ) can  (X, L, λ) Π (L) and λ γ ∈ Σ(L) L ∗∞ (X, L, λ). Since λ is a σ-finite

50

6 Integration and Finitely Additive Measures

measure, by Theorem 2.28(b) (Radon–Nikodym) there exists g ∈ L 1 (X, L, λ) with 

 u dγ =

ug dλ for all u ∈ L ∞ (X, L, λ).

X

(6.8)

X

So (4.6) can be rewritten as ν = μ + gλ, μ ∈ Π (L)



L ∗∞ (X, L, λ), g ∈ L 1 (X, L, λ).

(6.9)

The relation of (6.9) to Theorem 2.28(a) when λ is Borel measure is examined in Chap. 10.

6.2 Yosida–Hewitt Representation: Proof of Theorem 3.1 Having prepared the ground in Sects. 4.1 and 6.1, the proof of Theorem 3.1, which identifies elements of L ∞ (X, L, λ)∗ with elements of L ∗∞ (X, L, λ), and vice versa, is straightforward. In Sect. 6.1, it was shown that when ν ∈ L ∗∞ (X, L, λ) a bounded linear functional f ∈ L ∞ (X, L, λ)∗ is well defined on L ∞ (X, L, λ) by  u dν and | f (u)|  u∞ |ν|(X ).

f (u) = X

Hence  f ∞ = sup{| f (u)| : u∞ = 1}  |ν|(X ). To see that equality holds note that, since ν + ∧ ν − = 0, by (4.1c) for any  > 0 there exists a set A ∈ L with ν + (A) + ν − (A ) < , where A = X \ A. Let u = χ A − χ A ∈ L ∞ (X, L, λ). Then u∞ = 1 and      u dν = u dν + + u dν + − u dν − − u dν − f (u) = X +

A +

A



−

A

A −



+



= −ν (A) + ν (A ) + ν (A) − ν (A )  ν (A ) + ν − (A) −  = ν + (X ) − ν + (A) + ν − (X ) − ν − (A ) −   (|ν|(X ) − 2)u∞ . Therefore  f ∞  |ν|(X ) and equality follows. For the converse, given f ∈ L ∞ (X, L, λ)∗ define ν on L by ν(E) = f (χ E ), E ∈ L. The goal is to show that ν ∈ L ∗∞ (X, L, λ) and (3.1b) holds. ν is finitely First note that ν(E) = 0 if E ∈ N because χ E ∞ = 0. Moreover,  additive because f is linear and χ E1  E2 = χ E1 + χ E2 when E 1 E 2 = ∅. It remains only to show that (3.1b) holds. For u ∈ L ∞ (X, L, λ) let P in (6.1a) be a partition into K + 1 equal intervals, and K let u K := k=0 yk χ Ek (u) ∈ L ∞ (X, L, λ). Since u K → u in L ∞ (X, L, λ) and f is a bounded linear functional

6.2 Yosida–Hewitt Representation: Proof of Theorem 3.1

f (u) = lim f (u K ) = K →∞

K 

51

K  

yk f χ Ek (u) = yk ν(E k (u)) →

k=0

k=0

 u dν X



by (6.2a) and (6.2c). Thus (3.1b) holds and the proof is complete.

6.3 Integration with Respect to ω ∈ G Theorem 6.2 (a) For u ∈ L ∞ (X, L, λ) and ω ∈ G   u dω = α and |u| dω = |α| X

(6.10a)

X

where, as in Theorem 5.1, α is such that   ω x ∈ X : |u(x) − α| <  = 1 for all  > 0. (b) For u, v ∈ L ∞ (X, L, λ) and ω ∈ G 



 

uv dω = X

 v dω .

u dω X

(6.10b)

X

(c) If F : R → R is continuous, u ∈ L ∞ (X, L, λ) and ω ∈ G, 





u dω .

F(u) dω = F

(6.10c)

X

Proof (a) With α given by Theorem 5.1, the first part of (6.10a) follows because            u dω − α =  (u − α) dω  =       X

X

{x∈X :|u(x)−α| 0. The second part follows because, for all  > 0 ω (b) If



     x ∈ X : |u(x)| − |α| <   ω x ∈ X : |u(x) − α| <  = 1. 

 u dω = α and X

v dω = β, X

it follows that for all  > 0,     ω x ∈ X : |u(x) − α| <  = 1 and ω x ∈ X : |v(x) − β| <  = 1

52

6 Integration and Finitely Additive Measures

and hence, by (5.4), ω



 x ∈ X : |u(x)v(x) − αβ| <  = 1, for all  > 0.

By (6.10a) the proof of (b) is complete. (c) It follows from (6.10b) that (6.10c) holds when F is a polynomial, and therefore for continuous F by approximation using Theorem 2.43 (Weierstrass). 

6.4 Essential Range of u ∈ L ∞ (X, L, λ) In a general measure space (X, L, λ), the essential range [28, Chap.3, Ex.19] of u ∈ L ∞ (X, L, λ) is defined as  

 R(u) = α ∈ R : λ {x : |u(x) − α| < } > 0 for all  > 0 .

(6.11a)

Lemma 6.3 For u ∈ L ∞ (X, L, λ),  R(u) =

u dω :

 ω∈G .

(6.11b)

X

Proof From Theorems 5.6 and 6.2, the right side contains the left. Since ω(E) = 1 implies λ(E) > 0, by Theorem 6.2 the left contains the right.  For another viewpoint on the essential range, see (7.3). In a Borel setting the essential range can be localised, see Sect. 9.3.

6.5 Integrating u ∈ ∞ (N) with Respect to G Recall from Example 5.8 that bounded real sequences {u j } ∈ ∞ can be identified with the essentially bounded functions u ∈ ∞ (N) defined by u( j) = u j , j ∈ N, and there are uncountably many ωˆ ∈ G, each with the property that ω(E ˆ k ) = 1 for all k, where E k = k + N. Theorem 6.4 A sequence {u j } ∈ ∞ has a subsequence that converges to α ∈ R if

and only if there exists ω ∈ G with ω(E k ) = 1 for all k ∈ N and N u dω = α.

Proof Let ωˆ ∈ G have the property that ω(E ˆ k ) = 1 for all k and suppose α = N u d ωˆ where u ∈ ∞ (N) is defined by u( j) = u j . Then by (5.4) and Theorem 6.2(a) ωˆ



 j ∈ E k : |u j − α| <  = 1 for any k ∈ N and any  > 0.

6.5 Integrating u ∈ ∞ (N) with Respect to G

53

Hence, for any k ∈ N and  > 0, there exists j  k such that |u j − α| < . In particular, there exists j1 ∈ N such that |u j1 − α| < 1 and j2 > j1 such that |u j2 − α| < 1/2. Proceeding by induction, for each n ∈ N there exists jn > jn−1 such that |u jn − α| < 1/n. This shows that N u d ωˆ = α, ωˆ ∈ G, implies a subsequence u jn → α as n → ∞. Now suppose {u j } has a subsequence u jn → α, let E = { E˜ k : k ∈ N} where E˜ k = ˜ E˜ k ) = 1 for all k ∈ N { jn : n  k} and let ω˜ ∈ G be a finitely additive measure with ω( given by Theorem 5.6. Then for  > 0 there exists k ∈ N such that

 ˜ E˜ k )  ω˜ { j : |u j − α| < } . E˜ k ⊂ { j : |u j − α| < }, and hence 1 = ω( By Theorem 6.2 this implies

N

u d ω˜ = α.



In Remark 9.13, the preceding observation is related to the essential range at infinity of u ∈ ∞ (N). Remark 6.5 For u ∈ ∞ it is immediate that  u j → α as j → ∞ ⇒

N

u dω = α

for all ω ∈ G with ω(E k ) = 1 for all k. Let S : ∞ (N) → ∞ (N) be the shift operator defined by (Su)( j) = u( j + 1). Then by (6.10b), for ω ∈ G, 

 N

u S(u) dω =

  u dω N

N

 S(u) dω , u ∈ ∞ (N).

In particular, when u( j) = (−1) j it follows that u S(u) ≡ −1 and hence that 

  u dω N

N

 S(u) dω = −1



In other words, u → N u dω, u ∈ ∞ (N), is not shift invariant, and therefore is not a Banach limit (Definition 2.41 and [31]) for any ω ∈ G. However, a Banach limit can be constructed as follows. For u ∈ ∞ (N) let T u ∈ ∞ (N) be defined by j 1 (T u)( j) = u(i), i, j ∈ N. j i=1

Then T : ∞ (N) → ∞ (N) is a bounded linear operator with T  = 1 and  T u( j) − T (Su)( j)| = 1 |u( j + 1) − u(1)| → 0 as j → ∞. j

54

6 Integration and Finitely Additive Measures

  It follows that if ω ∈ G and ω(E k )=1 for all k then  ω j : |T u( j) − α| <  =1 for all  > 0 if and only if ω j : |T (Su)( j) − α| <  = 1 for all  > 0. Hence, by Theorem 6.2(a),   T u dω = T (Su) dω if ω ∈ G and ω(E k ) = 1 for all k. N

N

Since u( j) → α implies T u( j) → α as j → ∞, it follows that u → u ∈ ∞ (N), is a Banach limit.

N

T u dω, 

6.6 The Valadier–Hensgen Example Independently, Valadier [33] and Hensgen [19] made the following observation which contradicts a claim in [35]. Theorem 6.6  When L and λ refer to Lebesgue measure on X = [0, 1], there exists μ ∈ Π (L) L ∗∞ (X, L, λ) such that 

1

 v dμ =

0

1

v dλ for all continuous v : [0, 1] → R.

(6.12)

0

Proof 1] let μx ∈ G be given by Theorem 5.6 with the property that

For x ∈ [0, x μ (x − , x + ) [0, 1]) = 1 for all  > 0, and note from Theorem 6.2 that 

1

v dμx = v(x) for all v ∈ C[0, 1].

0

For n ∈ N, let x nj = j/2n , 0  j  2n , j ∈ N0 , and for v ∈ C[0, 1] let 1  = n v(x nj ) = 2 + 1 j=0 2n

snv



1 0

1  x nj v dμn , where μn = n μ . 2 + 1 j=0 2n

(6.13) n

For all n ∈ N, μn ∈ L ∗∞ (X, L, λ) and μn (E) ∈ [0, 1], E ∈ L, since each μx j ∈ G. Moreover, snv is an n th Riemann sum of v ∈ C[0, 1] corresponding to partitioning [0, 1] with 2n + 1 equally spaced points. Hence s v := {snv } ∈ ∞ is convergent and lim s v n→∞ n



1

= 0

or, in the notation of Definition 2.41,

v dλ, v ∈ C[0, 1],

6.6 The Valadier–Hensgen Example

55

v



v

1

s ∈ c(N) and l(s ) =

v dλ for all v ∈ C[0, 1].

0

Now define μ on L by



μ(E) = L {μn (E)} , E ∈ L,

(6.14)

where L : ∞ (N) → R is a Banach limit, Definition 2.41. Clearly μ  0 is finitely additive. Also μ(E) = 0 for all E ∈ N since μn is zero on N . To see μ is purely finitely additive, for k ∈ N let      n 1 1  . x j − 2n , x nj + 2n E k = 0, 1 2 k 2 k { j,n: 0 j2n n∈N}

Note that since μn (E k ) = 1 for all n ∈ N, it follows that μ(E k ) = 1 = μ(X ) for all k. Moreover, since {E  [0, 1] with λ(E k )  2/k,

 of k } is a nested sequence of open subsets = 0. Hence γ(E = E ) → γ E it follows that λ k k k k k  0 if 0  γ ∈ Σ(L). Since μ  0 this shows that μ ∧ γ = 0 for any 0  γ ∈ Σ(L) L ∗∞ (X, L, λ) whence, by Theorem 4.15, μ is purely finitely additive. It remains M to prove (6.12). gm χG m , gm ∈ R, G m ∈ L) When g is a simple function on [0, 1] (i.e. g = m=1 it follows from (6.2d) and (6.14) that 

1

L





1

=

g dμn

0

g dμ.

(6.15)

0

Now for an arbitrary u ∈ L ∞ ([0, 1], L, λ) let u k → u uniformly on [0, 1] where {u k } is a sequence of simple functions. Then for every k ∈ N    1   1   1    − u dμn u dμ   L u dμn − u k dμn  0 0 0 0   1   1    1  1        + L − u k dμn u k dμ +  u k dμ − u dμ .

   L 

1

0

0

0

0

Now the first and third terms on the right converge to 0 as k → ∞ and by (6.15) the middle term is zero for all k. Hence, 



1

L

u dμn



1

=

0

u dμ for all u ∈ L ∞ ([0, 1], L, λ).

0

In particular, for v ∈ C[0, 1] 

1 0



1

v dμ = L 0

 v dμn

= lim snv = n→∞



1 0

v dλ.

56

6 Integration and Finitely Additive Measures



Hence (6.12) is proved. Remark 6.7 Yosida and Hewitt [35, Thm. 3.4] claimed that 

1

v dν = 0 for all v ∈ C[0, 1]

(6.16)

0

implies ν is purely finitely additive but omitted the proof because it was considered straightforward. However, Valadier [33], and later Hensgen [19], independently showed that this claim is wrong. In Theorem 6.6 μ ∈ Π (L), λ ∈ Σ(L) and so, by the uniqueness statement in Theorem 4.13, ν := μ − λ ∈ L ∗∞ (X, L, λ) is neither purely finitely additive nor σ-additive yet satisfies (6.16).  Different choices of the points x nj lead to different μ. So if {y nj : j ∈ N} {x nj : j ∈ N} = ∅, there are two measures μ1 , μ2 ∈ Π (L) which satisfy (6.12). Hence 0 = μ1 − μ2 = ν ∈ Π (L) also satisfies (6.16). For further discussion of these issues, see Sect. 10.2. 

Chapter 7

Topology on G

7.1 The Space (G, τ ) For A ∈ L, let Δ A = {ω ∈ G : ω(A) = 1} and note that Δ A ⊂ Δ B if A ⊂ B and G \ Δ A = Δ X \A . More generally, from finite additivity it follows that for finite families, K 

Δ Ak = Δ1K

1

Ak

and

K 

Δ Ak = Δ1K

Ak ,

Ak ∈ L,

(7.1a)

1

whereas for arbitrary families only inclusions hold:  α

Δ Aα ⊂ Δα Aα and Δα Aα ⊂



Δ Aα .

(7.1b)

α

For further developments, see Theorem 7.10 and Corollary 7.11.   Lemma 7.1 Suppose A, B ∈ L. Then Δ A Δ B = ∅ if and only if A B ∈ / N. Moreover, B \ A ∈ / N if and only if  Δ B ⊂ Δ A . Consequently, Δ A = Δ B if and only if the symmetric difference (A \ B) (B \ A) ∈ N . Proof The first part is immediate from (7.1a), since elements of G are 0 on N , and  from Theorem 5.6 since A B ∈ / N implies Δ A  B = ∅. It follows that B \ A ∈ /N if and only if Δ B \ Δ A = ∅ and the observation on the symmetric difference is immediate.  Definition 7.2 ( (G, τ ) ) Let T := {Δ A : A ∈ L} be a sub-base for the topology τ on G. (See Remark 2.57.) Equivalently, by (7.1a), T is a base for τ , see Definition 2.56.  Note that Δ A , A ∈ L, is open and closed because G \ Δ A = Δ X \A . Theorem 7.3 (G, τ ) is a compact Hausdorff space. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 J. Toland, The Dual of L ∞ (X, L, λ), Finitely Additive Measures and Weak Convergence, SpringerBriefs in Mathematics, https://doi.org/10.1007/978-3-030-34732-1_7

57

7 Topology on G

58

Proof Let τ be the topology defined in Definition 7.2. To see that (G, τ ) is Hausdorff suppose that ω1 = ω2 . Then there exists A ∈L such that ω1 (A) = 1 and ω2 (A) = 0. Hence ω1 ∈ Δ A and ω2 ∈ Δ X \A where Δ A Δ X \A =  ∅. Thus (G, τ ) is Hausdorff. To show that (G, τ ) is compact suppose that G = A∈A Δ A for some A ⊂ L and   of A. Since Δ X \A = G \ Δ A , it follows that  G = A∈A Δ A for any finite subset A that A∈A Δ X \A = ∅ but, by (7.1a), Δ(

 X \A) A∈A

=



 ⊂ A is finite. Δ X \A = ∅ when A

 A∈A

  ⊂ A. By Lemma 7.1 this implies that A∈A(X \ A) ∈ / N for any finite A Since E := {X \ A : A ∈ A} satisfies the hypotheses of Theorem 5.6, there exists ω ∈ G such thatω(X \ A) = 1, and consequently ω(A) = 0, for all  A ∈ A. Since ω ∈ G and ω ∈ / A∈A Δ A , this is a contradiction. It follows that G ⊂ A∈A Δ A for  ⊂ A and hence, since T is a base for τ , (G, τ ) is compact. some finite A 

7.2 L ∞ (X, L, λ) and C(G, τ ) Isometrically Isomorphic The next observation is that L ∞ (X, L, λ) and C(G, τ ) are isometrically isomorphic with the consequence that their duals are isometrically isomorphic, even though one is represented by finitely additive measures, Theorem 3.1, and the other by σ-additive measures, Theorem 2.37 (Riesz). For u ∈ L ∞ (X, L, λ) let L[u] : G → R be defined by  u dω for all ω ∈ G.

L[u](ω) =

(7.2)

X

Note from Lemma 6.3 that the essential range of u ∈ L ∞ (X, L, λ) is the range of L[u] on G: R(u) = L[u](G). (7.3) Theorem 7.4 L[u] is continuous on (G, τ ), u∞ = L[u]C(G,τ ) =: sup |L[u](ω)|, ω∈G

(7.4)

and the mapping u → L[u], from L ∞ (X, L, λ) to C(G, τ ) is linear. Moreover, for u, v ∈ L ∞ (X, L, λ), L[u](ω)L[v](ω) = L[uv](ω) for all ω ∈ G.

(7.5)

7.2 L ∞ (X, L, λ) and C(G, τ ) Isometrically Isomorphic

59

Conversely, for every real-valued continuous function u on (G, τ ) there exists u ∈ L ∞ (X, L, λ) with u = L[u]. In other words, L is an isometric isomorphism between the Banach algebras L ∞ (X, L, λ) and C(G, τ ). Proof For u 0 ∈ L ∞ (X, L, λ) and ω0 ∈ G let α0 = L[u 0 ](ω0 ) = X u 0 dω0 and for 0 > 0 put A0 = {x ∈ X : |u 0 (x) − α0 | < 0 /2}. Then ω0 ∈ Δ A0 by Theorem 6.2 ˆ < 0 /2} A0 ∈ N if |αˆ − α0 |  0 . Therefore ω({x ∈ and {x ∈ X : |u 0 (x) − α| X : |u 0 (x) − α| ˆ < 0 /2}) = 0, and consequently X u 0 dω = α, ˆ if ω ∈ Δ A0 and |αˆ − α0 |  0 . It follows that









](ω) − α u 0 dω − α0 < 0 if ω ∈ Δ A0 .

L[u 0 0 =

X

Hence, for any u 0 ∈ L ∞ (X, L, λ), L[u 0 ] : (G, τ ) → R is continuous at any ω0 ∈ G, as required. Identity (7.5) is a restatement of (6.10b) in the present notation. It follows from (6.4), (6.6) and (6.7) that u → L[u] is linear from L ∞ (X, L, λ) to C(G, τ ) and supω∈G |L[u](ω)|  u∞ . To prove equality, suppose u∞ = [u]+ (see (2.8), if not replace u with −u). Then for  > 0 let D = {x : u∞ − u(x) < } ∈ / N . By Theorem 5.6 there exists ω ∈ G such that ω(D ) = 1. Hence, for any >0   u∞ − L[u](ω) = u∞ − u dω = (u∞ − u) dω  . X

D

Therefore (7.4) holds.  Now let L = L[u] : u ∈ L ∞ (X, L, λ) ⊂ C(G, τ ). Then L[1] ∈ L is the constant function 1 and by (7.5) the product of two elements of L is in L. If ω1 = ω2 there is a set E ∈ L with ω1 (E) = 1 and ω2 (E) = 0 and, since X χ E dω1 = 1 = 0 = χ dω2 , L separates points in G. Moreover, by (7.4) L is closed since L ∞ (X, L, λ) E X is a Banach space. Therefore, by Theorem 2.42 (Stone–Weierstrass), L = C(G, τ ) since (G, τ ) is a compact Hausdorff space. This completes the proof. 

7.3 Properties of G and τ Lemma 7.5 H is open and closed in (G, τ ) if and only if H = Δ A , A ∈ L. Proof First recall that Δ A is open and closed because Δ A and its complement G \ Δ A = Δ X \A are open. Conversely, if H is closed in (G, τ ) it is compact. Hence, if it is also open, there exists Ak ∈ L, 1  k  K , such that, by (7.1a), H=

K  k=1

and the result follows.

Δ Ak = Δ A where A =

K 

Ak ∈ L,

k=1



7 Topology on G

60

Theorem 7.6 (G, τ ) is totally disconnected, meaning all its connected sets are singletons. Proof Suppose H ⊂ G and ω1 , ω2 ∈ H are distinct. Then, for some A ∈ L, ω1 (A)=1 and ω2 (A) = 0, whence ω1 ∈ Δ A and ω2 ∈ G \ Δ A = Δ X \A . Therefore, H is not connected since Δ X \A and Δ A are disjoint open sets the union of which is G. Clearly a singleton {ω} is connected.  Lemma 7.7 For A ∈ L, Δ A is a singleton if and only if A is an atom in (X, L, λ).   Proof If A  is not an atom, / N . Hence  A = A1 A2 where A1 A2 ∈ N , A1 , A2 ∈ Δ A = Δ A1 Δ A2 , Δ A1 Δ A2 = ∅ and Δ A1 , Δ A2 = ∅, and so Δ A is not a singleton. Conversely, ω1 , ω2 ∈ Δ A , ω1 = ω2 , implies there exist A1 , A2 ⊂ A with ω1 (A1 ) = 1 = ω2 (A2 ), ω1 (A2 ) = 0 = ω2 (A1 ). It follows from Lemma 7.1 that A1 , A2 , (A1 \ / N , and so A is not an atom if Δ A is not a singleton.  A2 ) (A2 \ A1 ) ∈ Corollary 7.8 When (X, L, λ) has no atoms every element of G is purely finitely additive and no element of G is isolated. Proof Since by definition ω is isolated in (G, τ ) if and only if {ω} = Δ A for some A ∈ L, by Lemma 7.7 no element of G is isolated if there are no atoms. The pure finite additivity of ω ∈ G follows by Theorem 5.9.  In a Borel measure space (X, B, λ), a further implication of the absence of atoms is the following. Recall from Sect. 2.10 that a topological space (X, ) is completely separable and it has a denumerable basis (a denumerable collection T = {Ti } of non-empty open sets such that every open set is a union of elements of T ). Lemma 7.9 If B has no atoms, (G, τ ) is not completely separable. Proof Let {Ti : i ∈ N} be an arbitrary denumerable family of non-empty sets in τ and let ∅ = ΔG i ⊂ Ti , G i ∈ L. Since λ(G i ) > 0 for all i, by Theorem 2.25 there exists / N it follows from Lemma 7.1 A ∈ L which splits G = {G i : i ∈ N}. Since G i \ A ∈ that ΔG i ⊂ Δ A , and hence that Ti ⊂ Δ A , for all i ∈ N. Since Δ A ∈ τ , it follows that  {Ti : i ∈ N} is not a base for τ . This completes the proof. This discussion of (G, τ ) is continued in Sect. 9.1, but for that the following refinement of (7.1b) is needed.  Theorem 7.10 (i) If Γ = k∈N Δ Ak , Ak ∈ L: (a) Γ ◦ = Δ A0 , where Γ ◦ is the interior of Γ in (G, τ ) and A0 = (b) Both Γ and Γ ◦ are closed. (c) The boundary ∂Γ = Γ \ Γ ◦ of Γ is characterised by

 k∈N

 ∂Γ = ω ∈ G : ω(Ak ) = 1 for all k ∈ N but ω(A0 ) = 0 .

Ak .

7.3 Properties of G and τ

61

(d) If in addition Ak+1 ⊂ Ak , k ∈ N, Ak \ Ak+1 ∈ / N for infinitely many k, then ∂Γ is uncountable. (ii) If Λ = k∈N Δ Bk , Bk ∈ L: (a) Λ = Δ B 0 , where Λ is the closure of Λ in (G, τ ) and B 0 = (b) Both Λ and Λ are open. (c) The boundary ∂Λ = Λ \ Λ◦ of Λ is characterised by

 k∈N

Bk .

 ∂Λ = ω : ω(Bk ) = 0 for all k ∈ N but ω(B 0 ) = 1 . / N for infinitely many k, then (d) If, in addition, Bk ⊂ Bk+1 , k ∈ N, and Bk+1 \ Bk ∈ ∂Λ is uncountable. Proof (i)(a) Since Δ A0 is open, it follows from (7.1b) that Δ A0 = Δk Ak ⊂



Δ Ak

◦

= Γ ◦.

k

Suppose ω ∈ Γ ◦ \ Δ A0 and let E ∈ L \ N be such that ω ∈ Δ E ⊂ Γ . It follows that / N for some k0 ∈ N, for otherwise E \ A k0 ∈ E\



  Ak = (E \ Ak ) ∈ N ,

k

k

whence 1 = ω(E) = ω(E \ A0 ) + ω(E



   A0 ) = 0, A0 ) = ω E \ Ak + ω(E k

a contradiction. Since E \ Ak0 ∈ / N , it follows from Theorem 5.6 that there exists / Γ. ω0 ∈ G with ω0 (E \ Ak0 ) = 1. This implies that ω0 (Ak0 ) = 0 and hence that ω0 ∈ But ω0 (E \ Ak0 ) = 1 also implies that ω0 ∈ Δ E ⊂ Γ . This contradiction implies that Γ ◦ \ Δk Ak is empty and hence that Δ A0 = Γ ◦ . (i)(b, c) Since Γ is an intersection of closed sets, Γ = Γ is closed, Γ ◦ is closed by Lemma 7.5 and the formula for the boundary ∂Γ = Γ \ Γ ◦ follows. / N for infinitely many (i)(d) Now suppose Ak+1 ⊂ Ak , k ∈ N, and Ak \ Ak+1 ∈ k. Then there is no loss since, by Lemma 7.1, Ak \ Ak+1 ∈ N implies Δ Ak = Δ Ak+1 , / N for all k. Since the Fk are non-null and in assuming that F k := Ak \ Ak+1 ∈ disjoint, and since jk F j ⊂ Ak , Corollary 5.7 implies the existence of a family

  F {ωa : a ∈ (0, 1)} ⊂ G of distinct finitely additive measures with ωa jk j = ωa (Ak ) = 1, k ∈ N. Hence ωa ∈ Γ for all a ∈ (0, 1). However A0

  jk

   Ak \ A0 = ∅. F j = A0

7 Topology on G

62

Therefore, ωa (A0 ) = 0 for all a ∈ R and hence ωa ∈ Γ \ Γ0 = ∂Γ for all a ∈ (0, 1), as required. (ii) For the statements about Λ, let Ak = X \ Bk and Γ = G \ Λ in part (i). This completes the proof.  The next result generalises the first part of Theorem 7.10 since Γ is a closed G δ -set (a closed intersection of countably many open sets).  Corollary 7.11If H ⊂ G is a closed G δ -set, there exists {Ak } ⊂ L with Ak+1 ⊂ Ak such that H = k∈N Δ Ak and H◦ = Δ A0 where A0 = k Ak . The boundary ∂H is uncountable if it is non-empty, equivalently if H ∈ / τ. Proof Since (G, τ ) is compact and Hausdorff, it is normal (see Sect. 2.10) and since H is a closed G δ -set, by Lemma 2.62 there exists a non-negative continuous function f : G → [0, 1/2] with H = f −1 ({0}). By Theorem 7.4, for some non-negative u ∈ L ∞ (X, L, λ),  f (ω) = u dω for all ω ∈ G. X

Let Ak = {x ∈ X : u(x)  1/k}, k ∈ N. Then Ak+1 ⊂ Ak and if ω ∈ H, 



0 = f (ω) =



u dω = X

u dω + Ak

X \Ak

u dω  k −1 ω(X \ Ak )  0,

and it follows that ω(Ak ) = 1 for all k ∈ N. If ω ∈ / H let k ∈ N be sufficiently large that 1/k < f (ω). Then 1 0 < f (ω) −  k



1 + u dω − k Ak

 X \Ak

u dω  u∞ ω(X \ Ak )

and since ω ∈ G it follows that ω(Ak ) = 0 for all k sufficiently large if ω ∈ / H. Hence   Δ Ak . H = ω ∈ G : ω(Ak ) = 1 for all k ∈ N =

(7.6)

k∈N

 Since H = k∈N Δ Ak , by Theorem 7.10(i) H◦ = Δk∈N Ak . Finally, note that since H is closed its boundary ∂H = H \ H◦ is non-empty if / N for infinitely and only if H is not open. If H is not open it follows that Ak \ Ak+1 ∈ many k, for otherwise, by Lemma 7.1, the collection of sets {Δ Ak } is finite and so H = k Δ Ak is open. Thus when ∂H is non-empty {Ak } satisfies the hypothesis of Theorem 7.10(i)(d) and ∂H is uncountable. 

7.4 G and the Weak* Topology on L ∗∞ (X, L, λ)

63

7.4 G and the Weak* Topology on L ∗∞ (X, L, λ) Recall from Definition 2.47 that a set U ⊂ L ∞ (X, L, λ)∗ is open in the weak* topology if and only if for every f 0 ∈ U there exists u 1 , · · · , u n ∈ L ∞ (X, L, λ) and  > 0 such that n   f ∈ L ∞ (X, L, λ)∗ : |( f − f 0 )(u j )| <  ⊂ U. j=1

By analogy say W ⊂ L ∗∞ (X, L, λ) is weak* open in L ∗∞ (X, L, λ) if and only if the corresponding set of elements of L ∞ (X, L, λ)∗ , defined by Theorem 3.1, is open in the weak* topology of L ∞ (X, L, λ)∗ . In other words  W := W0 ,u 0 ,ν0 : 0 > 0, u 0 ∈ L ∞ (X, L, λ), ν0 ∈ L ∗∞ (X, L, λ) ,

(7.7a)

where

   



u 0 dν0

< 0 , W0 ,u 0 ,ν0 = ν ∈ L ∗∞ (X, L, λ) :

u 0 dν − X

(7.7b)

X

is a sub-base for the weak* topology on L ∗∞ (X, L, λ). Lemma 7.12 The topology τ on G coincides with the restriction to G of the weak* topology on L ∗∞ (X, L, λ). Proof It suffices to show that when  ω0 ∈ Δ A0 ∈ T (Definition 7.2),  there exists Δ A0 , and when ω0 ∈ W0 G, W0 ∈ W, W0 ∈ W (see (7.7)) with ω0 ∈ W0 G ⊂ there exists A0 ∈ L with ω0 ∈ Δ A0 ⊂ W0 G. First suppose ω0 ∈ Δ A0 and let



  

1 ∈ W. W0 = ν ∈ L ∗∞ (X, L, λ) :

χ A0 dω0 − χ A0 dν

< 2 X X  1 Then ω ∈ W0 G implies that |ω 0 (A0 ) − ω(A0 )| < 2 . Hence ω(A0 ) = 1 since ω0 (A0 ) = 1, and therefore ω0 ∈ W0 G ⊂ Δ A0 . Now let ω0 ∈ G and, for any u 0 ∈ L ∞ (X, L, λ) and 0 > 0, let

   



u 0 dω0

< 0 ∈ W. W0 = ν ∈ L ∗∞ (X, L, λ) :

u 0 dν − X

X

Let α0 = X u 0 dω0 and put A0 = {x ∈ X : |u 0 (x) − α0 | < 0 /2}. Then ω0 ∈ ˆ < 0 /2} A0 ∈ N if |αˆ − Δ A0 by Theorem 6.2 and {x ∈ X : |u 0 (x) − α| α0 |  0 . Therefore ω({x ∈ X : |u 0 (x) − α| ˆ < 0 /2}) = 0, and consequently X u 0 dω  =

α0 |  0 . It follows that X u 0 dω − α0 < 0 if ω ∈ Δ A0 . α, ˆ if ω ∈ Δ A0 and |αˆ −  Hence ω0 ∈ Δ A0 ⊂ W0 G, as required.

7 Topology on G

64

Lemma 7.13 G is closed in the weak* topology of L ∗∞ (X, L, λ). Proof It suffices to show that L ∗∞ (X, L, λ) \ G is open. If ν0 ∈ L ∗∞ (X, L, λ) \ G / {0, 1}. Let 2 = min{|a|, |1 − a|} > 0 and supthere exists A ∈ L with ν0 (A) = a ∈ pose ν ∈ L ∗∞ (X, L, λ) is in the weak* open neighbourhood of ν0 given by

   



∗



χ A dν0 <  . V = ν ∈ L ∞ (X, L, λ) : χ A dν − X

X

Then (see (6.2d)) ν ∈ V implies that |ν(A) − ν0 (A)| < , which means ν(A) ∈ / {0, 1}, and hence ν ∈ / G. Therefore L ∗∞ (X, L, λ) \ G is weak* open, and G is weak* closed.  It follows from Theorem 2.48 (Alaoglu) that (G, τ ) is compact. For a direct proof, see Theorem 7.3.

7.5 G as Extreme Points By Definition 2.52 f is an extreme point of B ∗ , the closed unit ball in L ∞ (X, L, λ)∗ , if and only if for f 1 , f 2 ∈ B ∗ and α ∈ (0, 1) f (u) = α f 1 (u) + (1 − α) f 2 (u) for all u ∈ L ∞ (X, L, λ) implies that f = f 1 = f 2 . Let ν, ν1 , ν2 ∈ U ∗ = {ν ∈ L ∗∞ (X, L, λ) : |ν|(X )  1} be defined in terms of f, f 1 , f 2 ∈ B ∗ by Theorem 3.1. Then a necessary and sufficient condition for f to be an extreme point of B ∗ is that for α ∈ (0, 1) 





u dν = α X

u dν1 + (1 − α) X

u dν2 for all u ∈ L ∞ (X, L, λ)

(7.8a)

X

implies that ν = ν1 = ν2 . Since simple functions are dense in L ∞ (X, L, λ), it is immediate that (7.8a) holds if and only if ν(E) = αν1 (E) + (1 − α)ν2 (E) for all E ∈ L.

(7.8b)

It is obvious that f is on the boundary of B ∗ , and so  f  = |ν|(X ) = 1, if f is extreme. But more can be said. Lemma 7.14 Let f ∈ L ∞ (X, L, λ)∗ and ν ∈ L ∗∞ (X, L, λ) be related by (3.1b) in Theorem 3.1. Then f is an extreme point of B ∗ if and only if ν or −ν is in G.

7.5 G as Extreme Points

65

Proof First suppose f is an extreme point of B ∗ . Then  f  = |ν|(X ) = 1. If ν is not one-signed then |ν| = ν + + ν − where ν + ∧ ν − = 0 and ν ± (X ) ∈ (0, 1). Let 0 < 0 = 21 min{ν + (X ), ν − (X )} and, by (4.1c), choose A ∈ L such that ν + (X \ A) + ν − (A) =  < 0 . If ν(A) = 0 then ν + (X ) = ν + (X \ A) + ν + (A) = ν + (X \ A) + ν − (A) =  < 0 , which is false. So ν(A) = 0 and hence |ν|(A) > 0. If |ν|(A) = 1 then ν − (X ) = 1 − ν + (X ) = ν − (A) + ν + (A) − ν + (X ) = ν − (A) − ν + (X \ A)  ν − (A) + ν + (X \ A) =  < 0 which is false. So |ν|(A) ∈ (0, 1). Let   ν(A E) ν((X \ A) E) , ν2 (E) = for all E ∈ L. ν1 (E) = |ν|(A) |ν|(X \ A) Then ν1 , ν2 ∈ U ∗ and, for all E ∈ L, ν(E) = αν1 (E) + (1 − α)ν2 (E), where α = |ν|(A), (1 − α) = |ν|(X \ A). Since α ∈ (0, 1), ν1 (A) = ν(A)/|ν|(A) = 0 and ν2 (A) = 0, this shows that f is not an extreme element of B ∗ if ν is not one-signed. So suppose ν is one-signed, say 0  ν ∈ U ∗ (for ν  0 replace ν with −ν), but ν∈ / G. Then there exists A ∈ L with ν(A) ∈ (0, 1). Let   ν(A E) ν((X \ A) E) , ν2 (E) = for all E ∈ L. ν1 (E) = ν(A) ν(X \ A) Then ν1 , ν2 ∈ U ∗ , ν(E) = αν1 (E) + (1 − α)ν2 (E) for all E ∈ L, where α = ν(A), (1 − α) = ν(X \ A). Since ν1 (A) = 1 = 0 = ν2 (A), ν is not extreme in U ∗ . Hence ±ν ∈ G if f is an extreme point of B ∗ . Now to show that f is an extreme point of B ∗ if ν ∈ G, suppose that ν1 , ν2 ∈ U ∗ and for all E ∈ L, ν(E) = αν1 (E) + (1 − α)ν2 (E), α ∈ (0, 1), ν1 , ν2 ∈ U ∗ . Then ν  0 and if ν(E) = 1, 1 = ν(E) = αν1 (E) + (1 − α)ν2 (E)  α|ν1 |(X ) + (1 − α)|ν2 |(X )  1 which implies that ν1 (E) = ν2 (E) = ν(E) = 1. In particular ν1 (X ) = ν2 (X ) = 1. If ν(E) = 0 then ν(X \ E) = 1 and so ν1 (X \ E) = ν2 (X \ E) = 1, whence ν1 (E) = ν2 (E) = ν(E) = 0. Thus ν = ν1 = ν2 which shows that f is an extreme point of  B ∗ if ν ∈ G.

66

7 Topology on G

Remark 7.15 Because of Theorem 2.55 (Rainwater), Lemma 7.14 implies that G has property (W) in the Introduction. In Sect. 8, this conclusion is derived independently, from the isometric isomorphism in Theorem 7.4.  Let (X, ) be a Hausdorff space, B its Borel σ-algebra and let D denote the set of regular Borel measures (Definition 2.12) δ that take only values 0 and 1 on B with δ(X ) = 1. Lemma 7.16 For δ ∈ D there exists a unique x0 ∈ X such that δ = δx0 , the Dirac measure introduced in Sect. 2.4. Proof Since δ ∈ D is regular, there is a compact K ⊂ X with δ(K ) = 1. Let K 0 denote the intersection of all such compact K and note that K 0 = ∅ because these compact sets have the finite intersection property. If x0 , y0 ∈ K 0 are distinct, then at least one of {x0 }, {y0 } has zero measure. If δ({y0 }) = 0, by the regularity of δ there is an open set G y0 with y0 ∈ G y0 and δ(G y0 ) = 0. So δ(K \ G y0 ) = 1 for all compact K with δ(K ) = 1. This contradicts the fact that y0 ∈ K 0 . Hence K 0 is a singleton, / K 0 . So K 0 = {x0 } and {x0 } say. If δ({x0 }) = 0, the same argument implies that x0 ∈  δ = δx0 .

Chapter 8

Weak Convergence in L ∞ (X, L, λ)

In this chapter, theory so far developed yields a necessary and sufficient condition in terms of behaviour λ-almost everywhere for a bounded sequence that is pointwise convergent to be weakly convergent in L ∞ (X, L, λ). The resulting test is illustrated by examples.

8.1 Weakly Convergent Sequences Since, by Sect. 7.2, L ∞ (X, L, λ) and C(G, τ ) are isometrically isomorphic, their duals are isometrically isomorphic and u k  u 0 in L ∞ (X, L, λ) if and only if L[u k ]  L[u 0 ] in C(G, τ ). Since (G, τ ) is a compact Hausdorff space, by (V) in the Introduction L[u k ]  L[u 0 ] in C(G, τ ) if and only if {L[u k ]C(G,τ ) } is bounded and L[u k ] → L[u 0 ] pointwise on G. In other words, u k  u 0 in L ∞ (X, L, λ) if and only if for some M 



u k ∞  M and

u k dω → X

u 0 dω as k → ∞ for all ω ∈ G.

(8.1)

X

Thus the collection G of finitely additive measures plays a rôle for weak convergence in L ∞ (X, L, λ) analogous to that of σ-additive Dirac measures for weak convergence in spaces of continuous functions. The sequential weak continuity of composition operators is an immediate consequence, in contrast to what happens for weak convergence in L p (X, L, λ), 1  p < ∞. (When u k (x) = sin(kx) in L p (0, 2π), u k  0 but |u k |  0.) Theorem 8.1 If u k  u 0 in L ∞ (X, L, λ) as k → ∞ and F : R → R is continuous, then F(u k )  F(u 0 ) in L ∞ (X, L, λ). Proof This is immediate from Theorem 6.2(c) and (8.1).



© The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 J. Toland, The Dual of L ∞ (X, L, λ), Finitely Additive Measures and Weak Convergence, SpringerBriefs in Mathematics, https://doi.org/10.1007/978-3-030-34732-1_8

67

8 Weak Convergence in L ∞ (X, L, λ)

68

Corollary 8.2 In L ∞ (X, L, λ), u k  0 if and only if |u k |  0. Proof “Only if” follows from Theorem 8.1 and “if” from (6.10a) and (8.1).



Recall from Remark 2.9 that if A is an atom, λ(A) ∈ (0, ∞) and u ∈ L ∞ (X, L, λ) is constant on A. Recall also from Theorem 5.9 that ω ∈ G is  either purely finitely additive or, for all E ∈ L, ω(E) is a constant multiple of λ(E ω E) where E ω is an atom. Hence from (8.1) if {u k } is bounded in L ∞ (X, L, λ), u k  u in L ∞ (X, L, λ) if and only if u k (A) → u(A) for all atoms A ∈ L,

(8.2a)

and 

 u k dμ → X

u dμ for all purely finitely additive μ ∈ G.

(8.2b)

X

Of course many familiar measure spaces have no atoms (Example 2.15), but when they do (8.2a) is equivalent to 

 χ A u k dλ → X

χ A u dλ for all atoms A. X ∗

Hence for bounded sequences (8.2a) holds if u k  u, and in particular if u k (x) → u(x) for λ-almost all x ∈ X . Since (X, L, λ) is σ-finite, L ∞ (X, L, λ) is the dual of L 1 (X, L, λ) [8, Thm. 4.4.1] and hence by Theorem 2.48 (Alaoglu) the closed unit ball in L ∞ (X, L, λ) is compact in the weak* topology of L 1 (X, L, λ)∗ (Definition 2.47). If in addition L 1 (X, L, λ) is a separable metric space, by Remark 2.51 the closed unit ball in L ∞ (X, L, λ) with the weak* topology is metrisable, and hence weak* sequentially compact (Definition 2.49). Thus a bounded sequence in L ∞ (X, L, λ) has a weak* convergent subsequence which therefore satisfies (8.2a). (For example (X, L, λ) is σ-finite and L 1 (X, L, λ) is separable when X ⊂ Rn and λ is any Borel measure on Rn which is finite on bounded sets [8, Cor. 4.2.2].) In circumstances such as these, deciding whether a bounded sequence is weakly convergent or has a weakly convergent subsequence is mainly a question of deciding whether (8.2b) is satisfied. Theorem 8.7 settles this issue by characterising sequences that converge weakly to 0 in terms of their λ-almost everywhere pointwise behaviour. To set the scene, recall some necessary conditions for the weak convergence of sequences in L ∞ (X, L, λ).

8.2 Pointwise Characterisation Lemma 8.3 When u k  u 0 in L ∞ (X, L, λ), there is a subsequence with u k j (x) → u 0 (x) for λ-almost all x ∈ X .

8.2 Pointwise Characterisation

69

Proof Since (X, L, λ) is σ-finite there exists f ∈ L 1 (X, L, λ) which is positive λalmost everywhere and, by Corollary 8.2, |u k − u 0 | f → 0 in L 1 (X, L, λ). Hence a  subsequence u k j (x) → u 0 (x) for λ-almost all x. In a metric space, let B(x, r ) denote the ball with centre x and radius r . Lemma 8.4 Suppose (X, ρ) is a metric space and for u ∈ L ∞ (X, B, λ), there exists a set E(u) ∈ L such that λ(X \ E(u)) = 0, 1 u(x) := lim 0 α , u ∈ L ∞ (X, L, λ), α > 0. The proof is independent of Mazur’s theorem. Theorem 8.7 In L ∞ (X, L, λ) bounded sequence {u k } converges weakly to zero if and only if for every α > 0 and every strictly increasing sequence {k j } in N there exists J ∈ N with J 

λ Aα (u k j ) = 0. (8.5) j=1

This criterion is equivalent to saying that for all strictly increasing sequences {k j }, the sequence {v J } in (8.4) converges strongly to zero in L ∞ (X, L, λ). Remark 8.8 In the Borel setting of the next chapter, this criterion for weak convergence in L ∞ (X, B, λ) will be localised to arbitrary open neighbourhoods of points in the one-point compactification of X .  Proof Suppose, for a strictly increasing sequence {k j } and α > 0, that (8.5) is false for all J ∈ N. Then E = {Aα (u k j ) : j ∈ N} satisfies the hypothesis of Theorem 5.6 and hence there exists ω ∈ G such that ω(Aα (u k j )) = 1 for all j. It follows that 

 |u k j | dω  X

Aα (u k j )

|u k j | dω  α > 0 for all j.

Hence |u k |  0 by (8.1) and so, by Corollary 8.2, u k  0. Conversely, suppose u k  0. Then by (8.1) and Corollary 8.2 there exists α > 0, a strictly increasing sequence {k j } ⊂ N and ω ∈ G such that  |u k j | dω =: α j > α > 0 for all j ∈ N. X

(8.6)

8.2 Pointwise Characterisation

71

Now α j − α > 0 for all j and 

     x : |u k j (x)| − α j  < α j − α ⊂ x : |u k j (x)| > α = Aα (u k j ).

From (8.6) and Theorem 6.2 it follows that 1=ω







  x : |u k j (x)| − α j  < α j − α  ω Aα (u k j ) ,

J

and so ω(Aα (u k j )) = 1 for all j. Hence ω j=1 Aα (u k j ) = 1, for all J , since ω ∈ G. Since ω is zero on N because ω ∈ G, it follows that (8.5) is false for all J . Finally, note that for a strictly increasing sequence {k j } and α > 0, λ



x : v J (x) > α



=λ =λ



x : |u k j (x)| > α for all j ∈ {1, · · · , J }

J 





Aα (u k j ) .

j=1

Since v J (x)  v J +1 (x)  0 it follows that v J → 0 in L ∞ (X, L, λ) if and only if, for every α > 0, (8.5) holds for some J . This completes the proof.  Dini’s theorem [27, Thm. 7.13] says that on compact spaces monotone pointwise convergence of a sequence of continuous functions to a continuous function is uniform; equivalently, by (V) in the Introduction, weak and strong convergence coincide for bounded monotone sequences of continuous functions on compact spaces. An analogue in L ∞ (X, L, λ) now follows either from Dini’s theorem in the light of Theorem 7.4, or directly from Theorem 8.7. The sequence in Example 8.6 converges, pointwise everywhere and monotonically, to zero, but not weakly in L ∞ (X, L, λ). Corollary 8.9 Suppose {u k } is bounded in L ∞ (X, L, λ) and |u k (x)|  |u k+1 (x)|, k ∈ N, for λ-almost all x ∈ X . Then u k  0 if and only if u k → 0 in L ∞ (X, L, λ). Proof Suppose u k  0 in L ∞ (X, L, λ). Since the monotonicity in k of {|u k (x)|} implies that v J = |u J |, by Theorem 8.7, u J → 0 in L ∞ (X, L, λ) as J → ∞. The converse is obvious.   Remark 8.10 Let u k = χ Ak where {Ak } ⊂ L and λ Ai A j = 0, i = j. Then, defined in terms of any increasing sequence {k j }, v J = 0 for all J  2. Hence u k  0 as k → ∞ in L ∞ (X, L, λ) by Theorem 8.7. This is a new perspective on Corollary 3.2.  With ∞ (N) as in Example 5.8 and Sect. 6.5, for every j ∈ N a bounded linear functional δ j is defined on ∞ (N) by δ j (u) = u( j), u ∈ ∞ (N). Hence u k  0 in ∞ (N) implies that u k ( j) → 0 as k → ∞ for each j ∈ N. However, pointwise convergence does not imply weak convergence.

8 Weak Convergence in L ∞ (X, L, λ)

72

Corollary 8.11 Let {u k } be a sequence in ∞ (N) with u k (i) → 0 as k → ∞ for each i ∈ N. Then u k  0 if and only if there exist α > 0 and a strictly increasing sequence {k j } for which there exist strictly increasing sequences {i n }, {Jn } such that for all n ∈ N (8.7) |u k j (i n )| > α for all j ∈ {1, 2 · · · · · · , Jn }. Proof Suppose u k  0. Then by Theorem 8.7 there exists α > 0 and a strictly increasing sequence {k j } such that for all J ∈ N   v J ∞ > α where v J (i) = min |u k j (i)| : j ∈ {1, · · · , J } for i ∈ N.

(8.8)

Let J1 = 1 and let i 1 denote the smallest i such that |u k1 (i)| > α. Since u k j converges pointwise to 0 as j → ∞, there exists a smallest J2 > J1 such that |u k j (i 1 )|  α for all j  J2 . Therefore by (8.8) there exists a smallest i > i 1 , denoted by i 2 , such that   min |u k j (i 2 )| : j ∈ {1, · · · , J2 } > α. Again there exists a smallest J3 > J2 such that |u k j (i 2 )|  α for all j  J3 . Let i 3 denote the smallest i > i 2 such that   min |u k j (i 3 )| : j ∈ {1, · · · , J3 } > α. By induction, for the strictly increasing sequence {k j } satisfying (8.8), this process yields strictly increasing sequences {i n } and {Jn } with the required properties. Conversely, if (8.7) holds it is immediate that v J ∞ → 0 as J → ∞ and hence  u k  0 as k → ∞ by Theorem 8.7. This completes the proof. Example 8.12 A sequence {u j } in ∞ (N) with  u j (i) =

u i j → 0 as j → ∞, i < j  α > 0, i j

 , i ∈ N.

is pointwise convergent to 0, but not weakly convergent in ∞ (N) by Corollary 8.11  with i n = Jn = n.

8.3 Applications of Theorem 8.7 Example 8.13 In Example 2.29 u k → 0 strongly in L p (0, 1) for all finite p, and therefore {u k } has a subsequence (see Sect. 2.27) which converges almost everywhere to 0, yet lim supk→∞ u k (x) = 1 for all x ∈ (0, 1). To see that u k  0 in L ∞ (0, 1) let kj = 1 +

 j ( j + 1) 1

and put Ik j = 0, . 2 j +1

8.3 Applications of Theorem 8.7

Then

73

 u k j (x)  1, x ∈ 0,

1

when j ∈ {1, 2, · · · , J }, J +1

and the conclusion follows from Theorem 8.7. However u k j = χ Ik where Ik j = j

 2 j − 1 2 j+1 − 1

, , 2j 2 j+1

defines a subsequence for which u k j  0 by Remark 8.10, since {Ik j } is a sequence of disjoint intervals.  Example 8.14 (Step Functions) In this example X = (−1, 1) with Lebesgue measure and  1 1

u k = χ Ak where Ak = k+1 , k . 2 2 Let   1

1

+ x + = χ , (x) = u , where A = 0, u+ k k k A+ k 2(k+1) 2(k+1)  1 1

3

− u− , (k+1) . k (x) = u k x − (k+1) = χ A− , where Ak = k k 2 2 2 − − So u k  = u ± k  = 1, and u k  0 and u k  0 by Remark 8.10 since {Ak } and {Ak } are sequences of disjoint measurable sets. −(J +1) ) for all J ∈ N, and hence However v J , defined in (8.4) by {u + k }, is 1 on (0, 2 + u k  0 in L ∞ (X, L, λ) by Theorem 8.7. The key observation is that χ Ak  0 for any family {Ak } of disjoint sets in L. For another illustration, see Example 9.9.  ∞ Example 8.15 (Simple Functions) In L ∞ (X, L, λ) let u k = i=1 αi χ Ai , where k

∞ Σi=1 |αi | < ∞ and, for each i ∈ N, {Aik }k∈N is a family of mutually disjoint measurable sets. Then u k  0. To see this, note that for each x ∈ X and i ∈ N there exists at most one k ∈ N, denoted, if it exists, by κ(x, i), such that x ∈ Aik if and only if k = κ(x, i). Note |αi | < . Hence, for any given also that for > 0 there exists I ∈ N such that Σ I∞

+1 k ∈ N and x ∈ X ,

|u k (x)| 

I  i=1

|αi |χ Aik (x) + =



|αi | + .

i∈{1,··· ,I } κ(x,i)=k

Since {κ(x, i) : i ∈ {1, · · · , I }} has at most I elements, there exists k ∈ {1, · · · , I + 1} such that k = κ(x, i) for any i ∈ {1, · · · , I }. Consequently, inf{|u k (x)| : 1  k  I + 1}  , independent of x ∈ X . Since this argument can be repeated with k ∈ N replaced by any strictly increasing subsequence {k j }, {v J } defined in terms of that subsequence by (8.4) has v J ∞ → 0 in L ∞ (X, L, λ), and u k  0 follows. 

8 Weak Convergence in L ∞ (X, L, λ)

74

Example 8.16 (Translations) Let u : R → R be essentially bounded and measurable with |u(x)| → 0 as |x| → ∞ and let u k (x) = u(x + k). Then u k  0 in L ∞ (R, L, λ) where λ is Lebesgue measure on R. To see this, for > 0 suppose that |u(x)| < if |x| > K . Then for any strictly increasing {k j } ⊂ N, v J ∞ < for all J  2K where {v J } is defined in terms of {u k j } by (8.4), and the result follows. For u : R → R essentially bounded and measurable with u(x) → 0 as x → ∞ and u(x) → 1 as x → −∞, let u k (x) = u(x + k). Then u k (x) → 0 as k → ∞ for all x ∈ R, but u k is not weakly convergent to 0 because of Theorem 8.7. However, in the notation of Definition 9.6, u k  0 at every point of R but not, as will be seen in Chap. 9, at the point at infinity in its one-point compactification.  Example 8.17 (Oscillatory Functions) With X = (0, 2π) and Lebesgue measure, let u k (x) = sin(1/(kx)). Clearly |u k (x)| → 0 as k → ∞ uniformly on ( , 2π) for any

∈ (0, 2π). Therefore, if a subsequence {u k j } is weakly convergent, its weak limit is zero. To see that no subsequence of {u k } is weakly convergent to 0, consider first a strictly increasing sequence {k j } of natural numbers for which there exists a prime power p m which does not divide k j for all j. Then, for J ∈ N sufficiently large, let  xJ =

π lcm {k1 , · · · , k J } pm

−1

∈ (0, 2π),

where lcm denotes the least common multiple. Then, for j ∈ {1, · · · , J }, since p m  k j and p is prime, 1 lcm {k1 , · · · , k J } = π where k j xJ pm k j lcm {k1 , · · · , k J } = r mod p m , r ∈ {1, · · · , p m − 1}. kj It follows that |u k j (x J )|  | sin(π/ p m )| > 0, independent of j ∈ {1, · · · , J } and, since u k j is continuous at x J , that v J  L ∞ (X,L,λ)  | sin(π/ p m )| > 0 for all J sufficiently large. By Theorem 8.7, this shows that u k j  0 if {k j } has a subsequence {k j } for which p m  k j for all j ∈ N. Note that if {k j } has no such subsequence for any prime p and m ∈ N, then every K ∈ N is a divisor of k j for all j sufficiently large, how large depending on K . Consequently, if u k j  0, {k j } has subsequence {k j } such that 2k j

divides

k j+1

and

k j+1

2

j 

ki for all j ∈ N.

i=1

Now for J ∈ N let

2   −1 k . π i=1 i J

xJ =

8.3 Applications of Theorem 8.7

75

Then the properties of {k j } imply that for j ∈ {1, 2, · · · , J },  1 = k j xJ

Hence

J i=1 ki

k j



π

2



J i= j+1 ki

=



+ k j +



j−1 i=1 ki



π

k j 2

 π

1 where N ∈ N and |z| < . = 2N + 1 + z 2 2 

 π   1  π

 , sin = √ , 1 , j ∈ {1, · · · , J }. |u k j (x J )| ∈ sin 4 2 2

It follows from Theorem 8.7 that u k j  0 as j → ∞ since it has a subsequence {u k } j √ which generates a sequence {v J } with v J ∞  1/ 2 for all J . 

Chapter 9

L ∗∞ When X is a Topological Space

This chapter considers what more can be said when (X, ) is a locally compact Hausdorff space, with B the corresponding Borel σ-algebra and λ a measure on B as described in Chap. 3. In addition, here λ is assumed regular, finite on compact sets and positive on open sets. Recall from Lemma 7.16 that a regular Borel measure that takes only values 0 or 1 is a Dirac measure concentrated at a point x0 ∈ X . Now it will be shown that G can be partitioned into a union of disjoint closed subsets G(x0 ), x0 ∈ X ∞ , where X ∞ is the one-point compactification of X . Because of (8.1), this yields a notion of weak convergence at points of X ∞ and leads to a related idea, the essential range at x0 of u ∈ L ∞ (X, B, λ) (sometimes called the set of cluster values of u at x0 ) which is parameterized by elements of G(x0 ). The pointwise criterion for weak convergence in the previous chapter can then be localised and related to the pointwise essential range. The distinction between the essential range at x0 of u ∈ L ∞ (X, B, λ) and its actual range can be rather striking. When Ω ⊂ Rn with positive Lebesgue measure, there  exists u ∈ L ∞ (Ω, B, λ) for which the range is denumerable, {u(x) : x ∈ Ω} = Q [0, 1], while at every x0 ∈ Ω the essential range of u is the continuum [0, 1]. The essential range of u ∈ ∞ (N) at the point at infinity in N∞ is the set of limits of convergent subsequences {u(k j )} of {u(k)}.

9.1 Localising G When (X, ) is a locally compact Hausdorff space, by (5.1),   G = ω ∈ L ∗∞ (X, B, λ) : ω(X ) = 1, ω(A) ∈ {0, 1}, A ∈ B . Lemma 9.1 Let (X, ) be locally compact and Hausdorff. (a) For ω ∈ G there exists a compact set K with ω(K ) = 1 if and only if there exists x0 ∈ K such that © The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 J. Toland, The Dual of L ∞ (X, L, λ), Finitely Additive Measures and Weak Convergence, SpringerBriefs in Mathematics, https://doi.org/10.1007/978-3-030-34732-1_9

77

9 L ∗∞ When X is a Topological Space

78

ω(G) = 1 for all open sets G with x0 ∈ G.

(9.1a)

For all ω ∈ G there is at most one x0 ∈ X satisfying (9.1a) and when (X, ) is compact there is exactly one such x0 . (b) For x0 ∈ X at least one ω ∈ G satisfies (9.1a). If (X, ) is not compact, at least one ω ∈ G satisfies, ω(X \ K ) = 1 for all compact K ⊂ X.

(9.1b)

Proof (a) Suppose that ω(K ) = 1 for some compact K and (9.1a) is false. Then for each x∈ K there is an open set G x with x ∈ G x and ω(G x ) = 0. Since K is compact, K G xi . Since ω(G xi ) = 0, 1  i  K and ω is finitely additive, ω(K ) = 0 K ⊂ i=1 which is false. Hence if ω(K ) = 1 and K is compact, there exists x0 ∈ K satisfying (9.1a). Since X is Hausdorff, there (9.1a). Suppose there is another x1 ∈ X satisfying  = ∅. But exist open sets with x0 ∈ G x0 , x1 ∈ G x1 and G x0  G x1  this is impossible because, by finite additivity, it would imply that ω G x0 G x1 = 2. Now suppose ω ∈ G and ω(K ) = 0 for all compact sets K . By local compactness, for any x ∈ X there is an open set G x with x ∈ G x and its closure G x is compact. Hence, since ω(G x )  ω(G x ) = 0, there is no x ∈ X with the required property. Finally, the existence of x0 when X is compact follows because ω(X ) = 1. (b) For x0 ∈ X , let E(x0 ) = {G ∈  : x0 ∈ G}. Since by hypothesis λ(G) > 0 when G is non-empty and open, by Theorem 5.6 there exists ω ∈ G with ω(G) = 1 for all G ∈ E(x0 ). Finally if (X, ) is not compact let E∞ = {X \ K : K compact}. Since sets in E∞ are non-empty and open (because compact sets in Hausdorff spaces are closed), the existence of ω ∈ G follows from Theorem 5.6 as before. This completes the proof.  For ω ∈ G, let ω∞ be defined on Borel subsets E of X ∞ , the one-point compactification of X (Definition 2.32), by ω∞ (E) = ω(E X ). Then ω∞ is the unique finitely additive measure on X ∞ which takes only values 0 and 1 and which coincides with ω on Borel sets in X . Lemma 9.1 can then be re-stated as follows. Corollary 9.2 (a) For ω ∈ G there is a unique x0 ∈ X ∞ such that ω∞ (G) = 1 when x0 ∈ G and G is open in X ∞ . Moreover x0 = x∞ if and only if ω(K ) = 0 for all compact K ⊂ X . (b) For x0 ∈ X ∞ there exists ω ∈ G such that ω(G) = 1 when x0 ∈ G and G is open in X ∞ . Definition 9.3 For x0 ∈ X ∞ , let G(x0 ) ⊂ G denote the set of ω ∈ G for which the conclusion of Corollary 9.2(b) holds, and let U(x0 ) ⊂ U denote the corresponding family of ultrafilters (Definition 5.3).  Properties of G(x0 ), x0 ∈ X ∞ (a) G(x0 )  = ∅ by Corollary 9.2(b); (b) G(x1 ) G(x2 ) = ∅ if x1 = x2 ∈ X ∞ , because (X ∞ , ∞ ) is Hausdorff;

9.1 Localising G

79

(c) G(x0 ) is compact because (G, τ ) is compact by Theorem 7.3 and G(x0 ) =



ΔG , where ΔG is closed.

{G:x0 ∈G∈∞ }

Theorem 9.4 Let (X, ) be locally compact and Hausdorff. (a) Suppose x0 ∈ X and {G k }k∈N is a nested, denumerable local base for  at x0 (i.e. x0 ∈ G k+1 ⊂ G k ∈  for all k and x0 ∈ G ∈  implies G k ⊂ G for some k). Then (i) G(x0 ) is a compact G δ -set in (G, τ ); (ii) G(x0 )◦ = ∅ if and only if {x0 } is an atom and G(x0 )◦ = {δx0 }, a Dirac measure; (iii) G(x0 )◦ = ∅ and G(x0 ) is uncountable if {x0 } is not an atom.

 (b) Suppose (X, ) is not compact and {x∞ } (X \ K k ) k∈N is a nested, denumerable local base for ∞ at x∞ (i.e. K k is compact, K k ⊂ K k+1 and for every compact K there exists k ∈ N with K ⊂ K k ). Then (i) G(x∞ ) is a closed G δ -set in (G, τ ), (ii) G(x∞ )◦ = ∅ and G(x∞ ) is uncountable.  Proof (a) (i) For x0 ∈ X it is immediate that G(x0 ) = k ΔG k and since ΔG k is both open and closed, G(x0 ) is a closed G δ -set in (G, τ ), and (G, τ ) is compact. (ii) Since (X, ) is Hausdorff, ∩k∈N G k = {x0 } and by Theorem 7.10 the interior G(x0 )◦ = Δ{x0 } . Hence G(x0 )◦ = ∅ if and only if λ({x0 }) > 0, i.e. {x0 } is an atom and by Lemma 7.7 G(x0 )◦ is the singleton Δ{x0 } . (iii) Since x0 ∈ G k , by hypothesis λ(G k ) > 0 because G k is non-empty and open, and λ(G k ) → 0, since λ is σ-additive and λ({x0 }) = 0. That G(x0 ) is uncountable and G(x0 )◦ = ∅ now follows from Theorem 7.10.  (b) Since a singleton is compact, the hypothesis implies that X = k∈N K k , and  since G(x∞ ) = k Δ(X \Kk ) the results follow from Theorem 7.10 as in part (a).   The hypothesis of part (b) implies that X = k K k ; in other words, (X, ) is σ-compact (Definition 2.31). However, (X, ) being σ-compact does not imply that the hypotheses of part (b) are satisfied. Remark 9.5 Given the one-to-one correspondence between the set G of 0–1 finitely additive measures and ultrafilters U in (X, B, λ) in Theorem 5.4, there is an obvious similarity with Lemma 7.16, but there are important differences. By Lemma 7.16 there is a one-to-one correspondence between points x0 ∈ X and Dirac measures δx0 ∈ D, and no Dirac measure is concentrated at infinity. By contrast, there is a oneto-one correspondence between points x0 ∈ X ∞ and the disjoint sets G(x0 ) ⊂ G which by Theorem 9.4(iii) may be uncountably infinite. 

9 L ∗∞ When X is a Topological Space

80

9.2 Localising Weak Convergence In the notation of Definition 9.3  G(x0 ), G=

U=

x0 ∈X ∞



U(x0 ),

(9.2)

x0 ∈X ∞

which leads to the following definition of pointwise weak convergence. Definition 9.6 (Weak Convergence at a Point) A bounded sequence {u k } in L ∞ (X, B, λ) converges weakly to u at x0 ∈ X ∞ (written as u k  u at x0 ) if and only if u k dω → X

u dω as k → ∞ for all ω ∈ G(x0 ). X

Equivalently, for all open sets G with x0 ∈ G,

u k dω → G

u dω for all ω ∈ G(x0 ). G

 Theorem 9.7 A bounded sequence converges weakly to u in L ∞ (X, B, λ) if and only if it converges weakly to u at every x0 ∈ X ∞ . Proof This is immediate from (8.1), (9.2) and Definition 9.6.



For u ∈ L ∞ (X, B, λ), α > 0 and G open in X ∞ , let    X : |u(x)| > α . Aα (u|G ) = x ∈ G Theorem 9.8 For bounded {u k } in L ∞ (X, B, λ) the following are equivalent: (a) u k  0 at x0 ∈ X ∞ ; (b) there exists α0 > 0 and a strictly increasing sequence {k j } ⊂ N such that for all open G ⊂ X ∞ with x0 ∈ G and every J ∈ N λ

J



 Aα0 (u k j |G ) > 0;

(9.3)

j=1

(c) there exists a strictly increasing {k j } such that, for all open G ⊂ X ∞ with x0 ∈ G, the non-increasing sequence {v J } defined by (8.4) satisfies v J |G  X → 0 in L ∞ (G



X, B, λ) as J → ∞.

9.2 Localising Weak Convergence

81

Proof That (b) and (c) are equivalent is immediate from the definitions. If (a) holds there exists α0 > 0, ω0 ∈ G(x0 ) and a strictly increasing sequence {k j } ⊂ N, such that for any open G with x0 ∈ G



0 < α0 

|u k j | dω0 = X

|u k j | dω0 for all j. G

Now (b) follow as in the proof of Theorem 8.7. If (b) holds then (9.3) implies that   E0 := Aα (u k j |G ) : j ∈ N, G open, x0 ∈ G satisfies the hypothesis of Theorem 5.6. Hence, there exists ω0 ∈ G with ω0 (A) = 1 for all A ∈ E0 . It follows that ω0 ∈ G(x0 ) and from (9.3), |u k j | dω0 = |u k j | dω0  α0 > 0 for all j, G

X



and (a) follows.

Example 9.9 Suppose (X, B, λ) has no atoms and (X, ) is completely separable (i.e.  has a denumerable base, G say, see Sect. 2.10). By Theorem 2.22 there exists {Ak }, a sequence of mutually disjoints sets in B, each of which splits G. Hence, for all k ∈ N and for every open set G ⊂ X the function χ Ak takes the value 1 on a subset of G with positive measure. However, since the sets Ak are mutually disjoint it follows from Example 8.14 that χ Ak  0 in L ∞ (X, B, λ), and therefore χ Ak  0 at every point of X . Note that {χ Ak } does not satisfy (9.3) for any {k j } ⊂ N j  and x0 ∈ X .

9.3 Fine Structure at x0 of u ∈ L ∞ (X, B, λ) By analogy with (6.11), in a Borel measure space (X, B, λ) the essential range of u localised at x0 ∈ X ∞ is defined as 

     α:λ x ∈G X : |α − u(x)| < > 0 (9.4) R(u)(x0 ) = { >0, G: x0 ∈G∈}



 u dω : ω ∈ G(x0 ) .

= X

From the first line of (9.4), the multivalued mapping x0 → R(u)(x0 ) reflects the measure-theoretic fine structure at x0 of u ∈ L ∞ (X, B, λ) and, since   R(u)(x0 ) = L[u] G(x0 ) ,

(9.5)

9 L ∗∞ When X is a Topological Space

82

the second line relates that fine structure to the isometric isomorphism L : L ∞ (X, B, λ) → C(G, τ ) in Sect. 7.2. R(u)(x0 ) is closed because G(x0 ) is compact and L[u] is continuous.  A point α ∈ R(u)(x0 ) given by α = X u dω, ω ∈ G(x0 ), may be thought of as the directional limit of u at x0 , the “direction” being determined by the ultrafilter Uω ∈ U(x0 ) (see (5.3b). Thus, by Theorem 9.7, weak convergence in L ∞ (X, B, λ) is equivalent to convergence, for each U ∈ U(x0 ), of the directional limits of u k at x0 to corresponding directional limits of u at x0 , for each x0 ∈ X ∞ . In other words, u k  u implies that for every x0 ∈ X ∞ and every α ∈ R(u)(x0 ) there exist αk ∈ R(u k )(x0 ) such that αk → α as k → ∞. This is not equivalent to αk → α when αk ∈ R(u k )(x0 ) and α ∈ R(u)(x0 ) because of the possibility that



αk =

u k dωk and α = X

u dω, but ωk = ω. X

Thus the multivalued R(u)(x0 ) may be interpreted as representing the fine structure at x0 of u ∈ L ∞ (X, B, λ) which is intimately related to weak convergence. A sufficient condition for u k  u is that for every x0 ∈ X ∞   sup |α| : α ∈ R(u k − u)(x0 ) → 0 as k → ∞. As already noted, the necessary condition is not sufficient. The following example shows that the sufficient condition is not necessary. Example 9.10 With {Ak } in Example 9.9, u k = χ Ak has the property that R(u k )(x) = {0, 1} for all x ∈ X while u k  0 in L ∞ (X, B, λ) by Corollary 3.2. For a simpler example, let u k = χ Ak where {Ak } is a sequence of disjoint open segments cen2 tred on the origin 0 of the unit disc  in R . Then R(u k )(0) = {0, 1} but u k  0 by Corollary 3.2. In these examples X u k dω → 0, but not uniformly for ω ∈ G(0).  The next example illustrates the strong distinction between the pointwise value u(x) and the essential range R(u)(x) at a point x. Example 9.11 Suppose (X, B, λ) has no atoms and (X, ) is completely separable with denumerable base G. Let {Ai : i ∈ N} be the disjoint family of measurable sets,  each of which splits G according to Theorem 2.25. Let q : N → Q (0, 1) be a bijection and define a measurable function by u(x) =

∞  i=1

 q(i)χ Ai =

 q(i) if x ∈ Ai , i ∈ N . 0 otherwise

From Theorem 2.25 and (9.4) it follows that Q



(0, 1) ⊂ R(u)(x0 ) ⊂ [0, 1] for all x0 ∈ X.

9.3 Fine Structure at x0 of u ∈ L ∞ (X, B, λ)

83

Since R(u)(x0 ) is closed, R(u)(x0 ) = [0, 1], even though u(x0 ) ∈ Q, for all  x0 ∈ X . Lemma 9.12 Suppose (X, ) is completely separable and u ∈ L ∞ (X, B, λ). Then u(x) ∈ R(u)(x) for λ-almost all x ∈ X . K αk χ Ak where the αk Proof Consider first the case when u is simple, say u = k=1 are distinct and Ak = {x : u(x) = αk } ∈ B. Then let   / R(u)(x) , k ∈ {1, · · · , K }, Bk = x ∈ X : u(x) = αk ∈ K    so that x ∈ X : u(x) ∈ / R(u)(x) = Bk . k=1

 Let y ∈ Bk . Then  u(y) = αk and there exists an open set G with y ∈ G and λ {x ∈ G : u(x) = αk } = 0. Since there is no loss in assuming that G ∈ G where G is the denumerable base for , Bk is covered bya denumerable family of sets of zero / R(u)(x)} = 0. measure. Hence λ(Bk ) = 0 for all k, and so λ {x : u(x) ∈ For the general case, let u ∈ L ∞ (X, B, λ) be arbitrary, let {u k } be a sequence of simple functions with u k − u∞ → 0 in L ∞ (X, B, λ) and for fixed k let ∈ R(u k )(x) for all x ∈ E k where λ(X \ E k ) = 0. Then λ(X \ E) = 0 where u k (x) E = k∈N E k , u k (x) ∈ R(u k )(x) for all k ∈ N and x ∈ E, and without loss of generality suppose u k (x) → u(x) for all x ∈ E. Now for x ∈ E and k ∈ N let ωk ∈ G(x) be such that u k (x) = X u k dωk . Then for x ∈ E, u k dωk u(x) = lim u k (x) = lim k→∞ k→∞ X   = lim (u k − u) dωk + u dωk = lim u dωk ∈ R(u)(x), k→∞

since

 X

X

X

k→∞

X

u dωk ∈ R(u)(x) and R(u)(x) is closed. The proof is complete.



Remark 9.13 As in Example 5.8, ∞ (N) = L ∞ (X, B, λ) where X = N, B = ℘ (N) and λ is counting measure. Then G(k), k ∈ N, has only one element, the σ-additive Dirac measure δk (Example 2.13). Moreover, {E k = k + N : k ∈ N} is a local base of the open neighbourhoods of ∞ in N∞ . In this notation, the results of Sect. 6.5 on integrating u ∈ ∞ (N) are examples of the formula  R(u)(∞) =

N

 u dω : ω ∈ G(∞) , u ∈ (N).

9 L ∗∞ When X is a Topological Space

84

Thus when u( j) = u j ∈ ∞ , the essential range of u at k ∈ N is the singleton {u k }, while at infinity the essential range is a singleton {α} if and only if u j → α as j → ∞. Lorentz [24] proved (see [31] for an elementary proof) that a sequence {xk } ∈ ∞ has the property that all its Banach limits coincide and equal s if and only if as n→∞ n 1 yn → y0 in ∞ (N), where yn ( j) = xi+ j and y0 ( j) = s for all j. n i=1



This does not imply that limk→∞ xk exists.

9.4 A Localised Range from Complex Function Theory The description of a localised essential range in (9.4) is reminiscent of the cluster set C D ( f, z 0 ) [10] of an analytic functions f at a boundary point z 0 of its domain of definition. In Corollary 9.15, Shargorodsky [29] shows that the phenomenon illustrated in Example 9.11 arises naturally in the theory of complex Hardy spaces. Let T and D denote the unit circle and the unit disc, respectively, and let   Bρ (α) := z ∈ C : |z − α| < ρ , α ∈ C, ρ > 0. Let f ∈ H ∞ (D). We say that ζ ∈ T is a singularity of f if f cannot be extended analytically to ζ (see, e.g. [17, Chap. II, Sect. 6]). At almost every ζ ∈ T, f has a non-tangential limit, denoted by f (ζ), at ζ. The following notation differs from that of [17, Chap. II, Sect. 6]:  

   R( f )(ϑ0 ) := α ∈ C : λ ϑ ∈ R : |ϑ − ϑ0 | < δ,  f (eiϑ ) − α < ε > 0  for all ε, δ > 0 , ϑ0 ∈ (−π, π], where λ denotes the standard Lebesgue measure on R. Theorem 9.14 Let f ∈ H ∞ (D) be an inner function and let eiϑ0 ∈ T be its singularity. Then R( f )(ϑ0 ) = T. Proof Since | f | = 1 almost everywhere on T, one has R( f )(ϑ0 ) ⊆ T. Suppose α ∈ T does not belong to R( f )(ϑ0 ). Then there exist ε, δ ∈ (0, 1) with λ



     ϑ ∈ R : |ϑ − ϑ0 | < δ,  f eiϑ − α < ε = 0.

(9.6)

Let α±ε be the two points where the circle {ζ ∈ C : |ζ− α| = ε} intersects T and let T(α, ε) and D(α, ε) be the intersections of T and D T with the closed half-plane

9.4 A Localised Range from Complex Function Theory

85

containing 0 and bounded by the straight line through α±ε : 

  T(α, ε) := ζ ∈ T| Re αζ ≤ Re (αα±ε ) , 

   D(α, ε) := ζ ∈ D T| Re αζ ≤ Re (αα±ε ) . Let  > 0 be the distance from α to D(α, ε). (It is not difficult to see that  is the distance from α to the midpoint of the chord [α−ε , αε ] and  = 1 − cos(2 arcsin 2ε ).) For the Poisson kernel Pr (ϑ) :=

1 − r2 , 0 ≤ r < 1, ϑ ∈ R, 1 − 2r cos ϑ + r 2

there exists δ0 > 0 such that Pr (θ − ϑ) dϑ < /2, δ≤|ϑ−ϑ0 |≤π

for all z = r eiθ ∈ Bδ0 (eiϑ0 ).

(9.7)

  It follows from (9.6) that f eiϑ ∈ T(α, ε) for almost all ϑ with |ϑ − ϑ0 | < δ. Since Pr ≥ 0, |ϑ−ϑ0 | 0 there are sets F ⊂ E ⊂ G with F closed, G open and ν(G \ F) < . If X is compact and ν is regular in this sense, by a theorem of Alexandroff [2, Pt. I, p. 590], [12, III.5.13] ν is σ-additive and hence νˆ = ν. By Lemma 10.6, if ν(X ) = ν(X ˆ ) and F ⊂ E ⊂ G, where F is closed and G is open,  ν(E) ˆ  ν(G) ˆ  ν(G). ν(F)  ν(F) ˆ Hence, ν  0 regular implies ν = νˆ is σ-additive on B if ν(X ) = ν(X ˆ ).



Theorem 10.8 Suppose that (X, ) is a locally compact Hausdorff space, K ⊂ G where K is compact, G is open and ν ∈ L ∗∞ (X, B, λ) is non-negative. Then for n ∈ N there exists a compact set K n and an open set G n with K ⊂ G n ⊂ K n ⊂ G, G n ⊂ G n−1 , K n ⊂ K n−1 , ν(K ˆ )  ν(K n ), ν(G) ˆ  ν(G n ) and λ(K n ) < λ(K ) + 1/n. Proof Since λ is a regular Borel measure that is finite on compact sets there exist open sets G k with K ⊂ G k ⊂ G and λ(G k ) < λ(K ) + 1/k for k ∈ N. Since K ⊂ G k , for k ∈ N there exists, by Lemma 10.5, a continuous function f k : X → [0, 1] such that f k (K ) = 1 and {x : f k (x) > 0} is a compact subset of G k . For x ∈ X , let gn (x) = min{ f k (x) : k  n} so that gn  gn−1 , gn is continuous on X , gn (K ) = 1 and {x : gn (x) > 0} ⊂ G n is compact. Let G n = {x : gn (x) > 0} and K n = {x : gn (x) > 0}. Then K ⊂ G n ⊂ K n ⊂ G n ⊂ G and, by Lemma 10.6, ˆ  ν(K ˆ n )  ν(K n )  ν(G n ), ν(K ˆ )  ν(G ˆ n )  ν(G n )  ν(K n ), ν(G) and λ(K n ) < λ(K ) + 1/n because K n ⊂ G n . Now {G n } and {K n } are nested sequences of open and compact sets, respectively, because gn (x) is decreasing in n, with the required properties. This completes the proof.  Corollary 10.9 For G open, K compact and ν ∈ L ∗∞ (X, B, λ) non-negative, ν(G) ˆ = sup{ν(K ) : K ⊂ G, K compact}, ν(K ˆ ) = inf{ν(G) : K ⊂ G, G open}. Proof From Lemma 10.6, for any open set G and all compact K ⊂ G, ν(K )  ν(K ˆ )  ν(G). ˆ Since νˆ is a regular real Borel measure, for any > 0 there exists

92

10 Reconciling Representations

compact K ⊂ G with ν(K ˆ ) > ν(G) ˆ − . Now by Theorem 10.8 there exists compact ˆ ) > ν(G) ˆ − . This establishes the first K 1 with K ⊂ K 1 ⊂ G and ν(K 1 )  ν(K identity. For a given compact set K and any open set G with K ⊂ G, ν(K ˆ )  ν(G) ˆ  ν(G) and for > 0, there exists an open G with K ⊂ G and ν(G) ˆ < ν(K ˆ ) + . By ˆ )+ > Theorem 10.8, there exists an open set G 1 with K ⊂ G 1 ⊂ G with ν(K  ν(G) ˆ  ν(G 1 ), and the result follows. Say that νˆ has a singularity with respect to λ if ν(E) ˆ = 0 (equivalently ρ(E) = 0) for some E ∈ N . Corollary 10.10 Let (X, ) be a locally compact Hausdorff space and 0  ν ∈ L ∗∞ (X, B, λ). Then 0  νˆ ∈ (B) has a singularity if and only if there exists α > 0 and a sequence of compact sets with ν(K n )  α, K n+1 ⊂ K n for all n, and λ(K n ) → 0 as n → ∞. Proof If α > 0 and such a sequence exists, by Lemma 10.6, ν(K ˆ n )  α for all n. ˆ )  α where K = n K n . Since {K n } is nested and νˆ is σ-additive it follows that ν(K Since K ∈ N , because limn→∞ λ(K n ) = 0 and λ is σ-additive, νˆ has a singularity. Conversely, if νˆ  0 has a singularity there exists E ∈ N and α > 0 with ν(E) ˆ = 2α. Since νˆ is a real Borel measure which is regular (Definition 2.21), there exists a compact K ⊂ E with ν(K ˆ )  α > 0. Now since λ(K ) = 0 because K ⊂ E ∈ N , ˆ )  α, K n+1 ⊂ the existence of a sequence {K n } of compact sets with ν(K n )  ν(K  K n for all n, and λ(K n ) → 0 as n → ∞ follows from Theorem 10.8. Theorem 10.11 For (X, ) locally compact and 0  ν ∈ L ∗∞ (X, B, λ), ν(B) ˆ = inf

G open B⊂G

⎫ ⎪ ⎬

⎧ ⎪ ⎨

sup ν(K ) = ⎪ ⎪ ⎭ ⎩ K compact K ⊂G

⎫ ⎬

⎧ ⎨ sup K compact K ⊂B

inf ν(G) , ⎭ ⎩ G open K ⊂G

for all B ∈ B. Proof Since νˆ is a regular real Borel measure, ν(B) ˆ = inf{ν(G) ˆ : B ⊂ G, G open} = sup{ν(K ˆ ) : K ⊂ B, K compact}, and the formulae follow from Corollary 10.9.



Corollary 10.12 For a locally compact Hausdorff space (X, ) and ω ∈ G, (a) either ωˆ is zero or ωˆ is a Dirac measure δx0 ; (b) both possibilities may occur when (X, ) is not compact; (c) if ωˆ = δx0 ∈ D, then ω ∈ G(x0 ), i.e.(9.1a) holds . Proof (a) By Lemma 9.1, either ω(K ) = 0 for all compact K , in which case ωˆ = 0 by the first formula for ω(B), ˆ or ω(K ) = 1 for some compact K . In the latter case,

10.2 Restriction to C0 (X, ) of Elements of L ∗∞ (X, B, λ)

93

there is a unique x0 ∈ X for which ω(G) = 1 if x0 ∈ G and G is open. From the second formula for ω(B) ˆ it is immediate that ω(B) ˆ = 1 if and only if x0 ∈ B. Hence ωˆ = δx0 ∈ D. (b) For an example of both possibilities let X = (0, 1) with the standard (locally compact but not compact) topology and Lebesgue measure. Let ω ∈ G be defined by Theorem 5.6 with E  = (0, 1/),  ∈ N. Then ω(K ) = 0 for all compact K ⊂ (0, 1) and hence ωˆ = 0. On the other hand, if E  = (1/2 + 1/, 1/2) in Theorem 5.6, ω ∈ G with ω([1/2 + 1/, 1/2]) = 1 for all  and hence ωˆ = δ1/2 ∈ D. (c) If ωˆ = δx0 let G an open set with x0 ∈ G. Since {x0 } is compact, there exists v ∈ C0 (X, ) with v(X ) ⊂ [0, 1], v(x0 ) = 1, v(X \ G) = 0 and 



1  ω(G) 



v dω = G

v dω = X

Hence, ω(G) = 1 for every open set with x0 ∈ G.

v d ωˆ = v(x0 ) = 1, X



References

1. Y.A. Abramovich, A.W. Wickstead, Singular extensions and restrictions of order continuous functionals. Hokkaido Math. J. 21, 475–482 (1992) 2. A.D. Alexandroff, Additive set-functions in abstract spaces I, II, III. Rec. Math. (Mat. Sbornik) N.S. 8(50), 307–348 (1940); 9(51), 563–628 (1941); 13(55), 169–238 (1943) 3. C.D. Aliprantis, K.C. Border, Infinite Dimensional Analysis—A Hitchiker’s Guide (Springer, Berlin, 2006) 4. J.M. Ball, Weak continuity properties of mappings and semigroups. Proc. Roy. Soc. Edin. Sect. A 72(4), 275–280 (1975) 5. S. Banach, Théorie des Opérations Linéaires. Monografje Matematyczne, Tom I, Z Subwehcji Fundusku Kultryy Narodowej, Warszawa, 1932; reprinted with corrections and added material, Chelsea Publishing Co. New York, 1955; Translated from the French by F. Jellett, with comments by A. Pelczy´nski and Cz. Bessaga. North-Holland Mathematical Library, vol. 38. North-Holland Publishing Co., Amsterdam, 1987 & Dover Books 6. K.P.S. Bhaskara Rao, M. Bhaskara Rao, Theory of Charges - A Study of Finitely Additive Measures, vol. 109, Pure and Applied Mathematics (Academic Press, London, 1983) 7. N.H. Bingham, Finite additivity vs countable additivity: De Finetti and Savage. J. Électron. Hist. Probab. Stat. 6(1), 35 (2010) 8. V.I. Bogachev, Measure Theory, vol. 1 (Springer, Berlin, 2007) 9. A.G. Chentsov, Finitely Additive Measures and Relaxations of Extremal Problems. Monographs in Contemporary Mathematics (Plenum, New York, 1996) 10. E.F. Collingwood, A.J. Lohwater, The Theory of Cluster Sets (Cambridge University Press, Cambridge, 1966) 11. J.B. Conway, A Course in Functional Analysis, 2nd edn. (Springer, New York, 2007) 12. N. Dunford, J.T. Schwartz, Linear Operators, vol. I (Wiley Interscience, New York, 1958) 13. G. Fichtenholz, Sur les fonctions d’ensemble additives et continues. Fund. Math. 7(1), 296– 301 (1925) 14. G. Fichtenholz, L. Kantorovitch, Sur les opérations lináires dans lespace des fonctions bornés. Stud. Math. 5, 69–98 (1934) 15. G.B. Folland, Real Analysis, 2nd edn. (Wiley, New York, 1999) 16. I. Fonseca, G. Leoni, Modern Methods in the Calculus of Variations: L p Spaces, Springer Monographs in Mathematics (Springer, New York, 2007) 17. J.B. Garnett, Bounded Analytic Functions (Revised First Edition, Springer, New York, 2007) 18. J. Heinonen, Lectures on Analysis on Metric Spaces (Springer, New York, 2001) © The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 J. Toland, The Dual of L ∞ (X, L, λ), Finitely Additive Measures and Weak Convergence, SpringerBriefs in Mathematics, https://doi.org/10.1007/978-3-030-34732-1

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Index

Symbols (X, ) a topological space, 9 (X ∞ , ∞ ) one-point compactification, 18 ((G, τ ), 57 Aα (u), 70 B ∗ , 22 C(G, τ ), 3 C ∗ (G, τ ), C(G, τ )∗ , 87 C0 (X, ), 18 C D ( f, z 0 ), 84 EF, 14 Gδ closed, 24 set, 24 space, 24 G δ -space, 25 H ∞ (D), 84 L p -spaces, 17 L p (X, L, λ), 17 X ∞ , one-point compactification, 2 L ∞ (X, L, λ), 18 L ∞ (X, L, λ)∗ , 27 L ∗∞ (X, L, λ), 3, 27, 39, 44 (L) purely finitely additive measures, 36 (L), 13, 31 ba(L) finitely additive measures, 31 ba(L) = (L) ⊕ (L), 38 χ A characteristic function, 28 ∞ , 45 ∞ (N), 21, 45, 46, 71, 83 (†), v, 3, 16, 31, 42 Lˆ , 14 λ-finite intersection property, 44 G 0-1 finitely additive measures, 3, 41 Hausdorff topology, 57 weak* closed in L ∗∞ (X, L, λ), 64

G(x0 ): G localised at x0 ∈ X ∞ , 78 U ultrafilters, 3 D, 84 T, 84 D Dirac measures, 5, 66 U, 42 ν1  ν2 , partial ordering, 33 ν + , ν − , |ν|, 34 ν1  ν2 , absolute continuity, 35 ν1 ⊥ ν2 , 35 ν1 ∨ ν2 , ν1 ∧ ν2 , sup and inf, 33 M, 9 ω-almost everywhere, 42 ωU , Uω , 43 ∂(E, F), 14 ≺, 7 , 7 B, 9 L, 8 M, 8 N null sets, 10 T , 57 W , 63 σ -additive measures, 31 σ -additivity, 10 σ -algebra, 8 σ -compact, 18 ∼, 14  collection of open sets, 9 , weak convergence, 1 ℘ (S) the power set of S, 7 ∗

, 22 x∞ , point at infinity in (X ∞ , ∞ ), 18 (V), 1 (W), 1

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 J. Toland, The Dual of L ∞ (X, L, λ), Finitely Additive Measures and Weak Convergence, SpringerBriefs in Mathematics, https://doi.org/10.1007/978-3-030-34732-1

97

98 A Absolute continuity, 35 Algebra, 8 Algebras and σ -algebras of sets, 8 Atom, 11 Axiom of choice, 8

B Banach limit, 21, 53, 84 Banach space, 19 Base, 23 Blaschke product, 86 Borel σ -algebra, 9, 77 measure, 77 sets, 9

C Canonical decomposition, 38 Characteristic function, 10, 14 Closed G δ -set, 62, 79 Cluster set, 84 Cluster value, 77 Completely separable, 24, 60, 81 Complex function theory, 84 Convergence in measure, 10 λ-almost everywhere, 10 Countable, 7 Counting measure on N, 11

D Darboux property, 11 Denumerable, 7 Dirac measure, 11 Directional limit of u at x0 , 82 Discrete topology, 19 Dual space, 1, 19

E Equality almost everywhere, 42 Equivalence class, 8 relation, 8 Essential range, 52, 77 localised, 81 Extended real numbers, 7 Extreme point, 22, 64

Index F Filter, 42 Fine structure at x0 of u ∈ L ∞ (X, B, λ), 81 Finitely additive measures, 31 First axiom of countability, 23 Functionals, bounded, 19 Function spaces, 17

H Hahn decomposition, 13 Hardy spaces, complex, 84 Hausdorff topology, 24

I Inner function, 84 Integration, 47 with respect to ν ∈ G, 51 Isolated, 60 Isometric isomorphism, 58

J Jordan decomposition, 13

L Lebesgue σ -algebra, 12 decomposition, 17 integrable, 16 integral, 15 measure, 12 point, 69 Lemma Urysohn, 24, 90 Zorn, 3, 7, 8, 42–44 Local base, 23 Localising G, 77 complex function theory, 84 essential range, 81 weak convergence, 80 Locally compact, 18, 77

M Maximal, 8, 42 Maximal element, 8 Maximum norm, 18 Measurable functions, 9 sets, 9

Index space, 9 Measure, 10 σ -finite, 10 Borel outer, inner regular, 11 finite, 10 Lebesgue, 12 space, 10 Measure space complete, 10

N Normal topology, 24 Null sets, 10

O One-point compactification, 18 Oscillatory functions, 74

P Partial ordering, 33 Poisson kernel, 85 Positive and negative parts, 13 Purely finitely additive measures, 36

R Real measure, 12 Reconciling representations, 87 Regular Borel measure, 11 inner, 11 outer, 11 real Borel measure, 13 Regular topology, 24

S Second axiom of countability, 24 Separable, 22, 23 Sequential weak continuity, 67 Shargorodsky, 5, 84 Shift operator, 53 Simple function, 10, 73 Singleton, 7 Singular, 35 Splitting measurable sets, 14 Step functions, 73 Sub-base, 22, 23, 57, 63 Symmetric difference, 14

99 T Theorem Alaoglu, 22, 64, 68 Alexandroff, 91 Baire’s category, 3, 14, 15 Banach–Steinhaus, 21 Dini, 71 Dominated convergence, 1 Hahn–Banach, 5, 20, 28, 44, 69 Krein–Milman, 23 Lebesgue–Radon–Nikodym, 16, 90 Lusin, 19 Mazur, 21, 69 Monotone convergence, 49 Nikodym’s convergence, 32 Radon–Nikodym, 17, 50 Rainwater, 23 Riesz representation, 1, 20, 27, 87 Stone–Weierstrass, 21, 59 Valadier–Hensgen, 3, 5, 54, 56 Weierstrass approximation, 21 Yosida–Hewitt representation, v, 3, 27, 50, 87 Totally disconnected, 60 Total ordering, 7 Total variation, 13 Translations, 74

U Ultrafilter, 42, 43 Upper bound, 8

W Weak* closed convex hull, 23 convergence, 22 sequential compactness, 22 topology, 22 topology on L ∗∞ (X, L, λ), 63 Weak convergence at a point, 77, 80 criterion, 70 of sequences, 21, 67

Y Yosida–Hewitt decomposition, 89