Spaces of Measures [Reprint 2011 ed.] 9783110853995, 9783110087840

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de Gruyter Studies in Mathematics 4 Editors: Heinz Bauer · Peter Gabriel

Corneliu Constantinescu

Spaces of Measures

w DE

G

Walter de Gruyter · Berlin · New York 1984

Author

Dr. Corneliu Constantinescu Professor at the Eidgenössische Technische Hochschule Zürich Department of Mathematics

CIP-Kurztitelaufnahme

der Deutschen

Bibliothek

Constantinescu, Corneliu: Spaces of measures / Corneliu Constantinescu. Berlin ; New York : de Gruyter, 1984. (De Gruyter studies in mathematics ; 4) ISBN 3-11-008784-7

Library of Congress Cataloging in Publication Data Constantinescu, Corneliu. Spaces of measures. (De Gruyter studies in mathematics ; 4) Bibliography: p. Includes index. 1. Spaces of measures. I. Title. II. Series. QA312.C578 1984 515.4'2 84-5815 ISBN 3-11-008784-7

© Copyright 1984 by Walter de Gruyter & Co., Berlin. All rights reserved, including those of translation into foreign languages. No part of this book may be reproduced in any form - by photoprint, microfilm, or any other means - nor transmitted nor translated into a machine language without written permission from the publisher. Printed in Germany. Cover design: Rudolf Hübler, Berlin. Typesetting and Printing: Tutte Druckerei GmbH, Salzweg-Passau. Binding: Lüderitz & Bauer, Berlin.

Preface

The theory of spaces of measures may be considered as a chapter of measure and integration theory, if we accept a rather comprehensive definition for the latter theory. In contrast to the other chapters, in which attention is concentrated generally on isolated measures, the properties of sets of measures are of interest for the theory of spaces of measures. Nikodym's convergence theorem may illustrate this situation quite well. This theorem states that if (μ„)π6Ν is a sequence of real (σ-additive) measures defined on a σ-ring such that (μ„(Λ))πεΝ converges for every A eW, then the limit function A i—• lim μη{Α) is a measure as well; moreover, the set FI-» 00 {μη\ΐϊΕ Ν} is equi-ff-additive. Here we have a typical example of a topological property of the set of all real measures of iH. The greatest part of the properties of the spaces of measures is of topological nature and so the topological methods are the main tools in this theory. Besides the σ-additivity there is another weaker property, called exhaustivity, which compels attention in this field. A function μ defined on a σ-ring with values in a topological commutative group is called exhaustive if (μ(Λ„))ηεΙΗ converges to 0 for any disjoint sequence (A„)neN in 9?. The greatest part of the theorems of the theory of spaces of measures consists of two parallel formulations, one for the measures, the other for the exhaustive finitely additive functions, the latter one being in general weaker. An important case occurs when the measures in question are defined on a ring of subsets of a Hausdorff space and when they satisfy some compatibility conditions with the topology; in this case the theorems may be strongly reinforced. As an application of this theory we mention integration theory itself (e.g. the theorem stating that L1 is weakly sequentially complete is a consequence of Nikodym's convergence theorem quoted above) and the theory of vector lattices, especially of AZ-spaces. An M-space is a vector lattice Ε endowed with a topology generated by a set of semi-norms ρ satisfying the conditions x,yeE,

|*| < \y\ =>

p{x) 0, y > 0 => p(x ν y) = sup

(p(x),p(y)).

The origins of the theory of spaces of measures can be traced back to the period between the two world wars. The theorems, classical ones nowadays, are associated with the names Orlicz-Pettis, Vitali-Hahn-Saks, Nikodym (the convergence theorem and the boundedness theorem), and Phillips. After the war, especially during the seventies, an almost explosive development took place in this field, the above mentioned theorems being generalized in all kinds of directions. U p to the present

6

Preface

time no unified bibliography on this subject has existed. The literature is spread out and, in part, difficult to trace. One aim of this book is to collect a part of this literature and to present it in a unified manner. The book also contains the applications of this theory to the study of vector lattices and of the M-spaces. The author would like to thank the editors of the "de Gruyter Studies in Mathematics" for accepting this book as part of their series. My special thanks go to Wolfgang Filter and to Peter von Siebenthal for reading the manuscript and for contributing improvements. Corneliu Constantinescu Zurich, March 29, 1983

Contents

Introduction

9

Logical connections between sections

12

Notations and Terminology

13

1. Set theory 2. Order relations 3. Topological spaces 4. Uniform spaces

13 13 14 14

Chapter 1: Topological preliminaries §1.1 § 1.2 §1.3 §1.4 § 1.5 § 1.6 §1.7 § 1.8

Sets of filters Sets of filters on topological and uniform spaces Φ-continuous and uniformly Φ-continuous maps Filters defined by sets of sequences Sets of sequences on topological and uniform spaces (5-stable filters Mioritic spaces tf>rcontinuous and uniformly Φί-continuous maps

17 19 22 25 31 42 43 46

Chapter 2: Spaces of functions § 2.1 Uniformities on spaces of functions § 2.2 Uniformities on spaces of functions defined by sets of sequences §2.3 Smulian spaces § 2.4 Constructions with spaces of functions §2.5 Spaces of parametrized functions

53 67 75 85 87

Chapter 3: Spaces of supersummable families §3.1 The set %(I,G) §3.2 Structures on §3.3 Spaces of supersummable families § 3.4 Spaces of supersummable families of functions §3.5 Supersummable families in special spaces

91 95 102 123 130

8

Contents

Chapter 4: Spaces of measures §4.1 §4.2 § 4.3 § 4.4 §4.5 §4.6 §4.7 § 4.8 § 4.9 § 4.10 §4.11

Measures and exhaustive maps Spaces of measures and of exhaustive additive maps Vitali-Hahn-Saks theorem and Phillips lemma Weak topologies on spaces of measures Spaces of measures on topological spaces Measures with parameter Bounded sets Bounded sets and measures on topological spaces Spaces of integrals Supersummable families of functions and their integrals Measurability considerations

155 186 199 210 223 240 244 255 263 273 280

Chapter 5: Locally convex lattices § 5.1 § 5.2 §5.3 §5.4 § 5.5 §5.6 §5.7 §5.8 § 5.9

Order summable families Order continuous maps Spaces of order continuous group homomorphisms Vector lattices Duals of vector lattices Spaces of vector valued measures Quasi M-spaces M-spaces Strict M-spaces

289 304 315 321 325 341 361 386 405

References

429

Index

439

Notations

443

Introduction

The theory of spaces of measures developed from the following classical theorems. 1. Nikodym's convergence theorem (O. Nikodym (1931) [1] II, (1933) [3] page 427). Let 9i be a σ-ring, and let (μ„)π 6 N be a sequence of real-valued (α-additive) measures on 91 such that (μ η (/ί)) πε ^ converges for every A e 9ί. Then the measures μη (η e Ν) are equi-tr-additive and the map 9Ϊ —»· iR,

A

lim μ (Α) η -» oc π

is also a measure. 2. Orlicz-Pettis theorem (W. Orlicz (1929) [1] Satz 2; B.J. Pettis (1938) [1] Theorem 2.32). Let 9t be a σ-ring, Ε a Banach space, and μ a map of 91 into E. If μ is a measure with respect to the weak topology of Ε then μ is also a measure with respect to the norm topology of E. 3. Boundedness theorem of Nikodym (O. Nikodym (1931) [1] I, (1933) [2] page 418). Let 9Ϊ be a σ-ring, and let Jl be a set of real-valued measures on 91. If sup \μ{Α) \ < 00 piejt for every A e 9t then sup \μ(Α)\ < 00 . fiejf AeSl 4. Vitali-Hahn-Saks theorem (G. Vitali (1907) [1] Teorema, page 147; H. Hahn (1922) [1] XXI; S. Saks (1933) [1] Theorem 5). Let 9? be a σ-ring, and let μ be a positive real-valued measure on 91. Let (μπ)πεκι be a sequence of μ-absolutely continuous, real-valued measures on 9ί such that (μ η (Λ)) πεΝ converges for every A e 9Ϊ. Then (μ„)πεΝ is equi^-absolutely continuous, and the limit function is also μ-absolutely continuous. 5. Phillips' lemma (R.S. Phillips (1940) [1] Lemma 3.3). Let (μ„)„6ΐΜ be a sequence of finitely additive bounded real-valued functions on the power set of Ν such that (^„(Λ/))ηεΝ converges to 0 for every Μ a [M. Then Hm Z l / U { m } ) l = 0. " CC m = ΓΚΙ

10

Introduction

The first four theorems give information about spaces of measures, the last one about a space of exhaustive additive maps. For instance let us reformulate Nikodym's convergence theorem in the language of spaces of measures. Let be the space of all real-valued functions on a σ-ring 9? endowed with the product topology, i. e. with the topology of pointwise convergence. Any real-valued measure on 9Ϊ may be considered as a point in the space IR*. In general the set of all real-valued measures on SR is not a closed set in IR* but Nikodym's convergence theorem says that it is sequentially closed (in fact a stronger result holds). The other theorems can also be formulated in a similar way. The results concerning spaces of measures rely primarily on two theorems. In order to present these two theorems we denote by G a Haudorff topological additive (i.e. commutative) group, and we call a sequence in G supersummable if every one of its subsequences is summable. The announced theorems read as follows. a) If (xJneiw i s

a

supersummable sequence from G, then the map

^(IN) —> G,

Α ι • Σ xn tie A

is continuous. Here ^β(ΙΝ) denotes the power set of IN endowed with the compact topology obtained by identifying ^β(ΙΝ) with {0,1 via the map {0,1}" -

$(IN),

/ h - { « e IN | / ( « ) = 1}.

b) If ((* mn )„ eN ) me iN is a sequence of supersummable sequences from G such that (Σ

xmJmeiNj

converges for every Λ(>>))),

(these spaces are discussed in the second part of chapter 5), etc. In all of these spaces the methods for establishing the properties in question are similar. Certain constructions on these spaces of functions appear again and again. Repeating these constructions would not only be irritating but it would also be unproductive since the number of repetitions is considerable. Therefore it seemed reasonable to extract the common features of all these constructions and formulate them as theorems that can be quoted later. The resulting abstract construction is carried out in chapter 2. The author is aware of the difficulties connected with any step in the direction of abstractness, especially when the reason for the abstraction or the way in which this abstract theory applies is not clear. We advise the readers who find it hard to study chapter 2 because of such motives (and the same advice holds for chapter 1) to start by searching the historical remarks for the more or less classical results (the list of references can be used for a purposeful tracking of these historical remarks, since the reader will find for each paper the places where it is cited). The historical remarks will lead back to the corresponding theorems, the formulations and the proofs of which will lead rapidly to the first two chapters of the book. Via this backtracking the reader will find out where and in what connection use is made of the various notions and deliberations that are presented abstractly in chapters 1 and 2, and this may facilitate the study of these chapters. In the present context no other choice seemed practicable. Some results of chapter 5 deserve special mention since they offer a nice example of applications of the theory of spaces of measures. The main result of Grothendieck's thesis (1953, [2], Theoreme 1 and Theoreme 6) consists of the proof that the Banach space of continuous real functions on a compact space possesses the DPproperty and the D-property. In fact these properties (and even stronger ones) are possessed by any M-space. The proof given in this book uses the fact that large parts of the dual of an M-space may be identified (via Choquet's theorem on simplexes) with spaces of measures. The above mentioned properties of the ΛΖ-spaces are nothing other than the reflection of the corresponding properties of the spaces of measures. It is assumed that, besides possessing the usual knowledge of a graduate student, the reader is also familiar with some special topological objects, namely filters, nets, special kinds of topological spaces (Hausdorff, regular, completely regular and so on: reference book N. Bourbaki [1]), knows elementary facts abort topological groups (reference book N. Bourbaki [1]) and has some knowledge about locally

12

Introduction

convex spaces (reference book H.H. Schaefer [1]). Section 4.9 and all the other sections which use it require (real) integration theory. Starting with section 5.5 we use the theory of vector lattices and of locally convex vector lattices (reference book Ch.D. Aliprantis, O. Burkinshaw [1]), and from section 5.6 on the theory of duality in measure theory (reference book C. Constantinescu [5]). There are certain sections in which some notation occurs very frequently. In order to avoid too many repetitions we have preferred to adopt the following convention: throughout the book it is assumed that, besides the hypotheses explicitely formulated, the hypotheses stated at the beginning of the section also hold. Logical connections between sections

Notations and Terminology

In general we shall use the notation and terminology of N. Bourbaki. 1. Set theory If X is a set, then ^P(X) will denote the power set of X, i.e. the set of subsets of X. We will denote by Yx the set of maps of X into Y for any sets X, Y. Let Χ, Y be sets, S be a subset of X, 5 be a filter on X, and φ e Yx. We shall denote by · φ(χ),

and by 3iel,

ι' 3l 6 I,

l which is a contradiction. b => c. Let Μ be a map of Φ 0 into (ΛΓ) with M(©) e © for any © e Φ 0 . Assume that for any finite subset ψ of Φ 0 we have (J A/(©) φ Then ©elf ft' := {A\ U M(©) M e g , Ψ finite b). By the hypothesis x ( © ) n j > ( © ) e Φ. We get >(©)) is a Cauchy filter and this is a contradiction. • Proposition 1.3.10. Let Χ, Y be uniform spaces, Φ be a set offilters on Χ, φ be a uniformly Φ-continuous map of X into Y and let 0r e Φ. Then for any entourage V of Y

24

Chapter 1: Topological preliminaries

there exist Ae% and an entourage U of X such that (φ (χ), φ (.y)) e V for any (x, y)e(AxA)n U. Let U be the uniformity of X and / be the set U χ 5 endowed with the upper directed order relation < defined by {U,A) where denotes the section filter of J. Hence, © € Φ{Θ). The last assertions follow from the first one and from Corollary 1.1.5 and Proposition 1.1.7. • Proposition 1.4.3. Let X be a set, Abe a subset of X and Θ be a set of sequences in X. Then A is a !„_!}.

Hence, there exists ine I with ιη >ιη- α and f{in) = xn. It follows that (x„)neiM belongs to Θ. Assume now that any sequence in A belongs to Θ. Let < be the coarsest preorder relation on A (i. e. χ < y for any x, ye A) and / be the inclusion map A —> X. Then (.A,f) is a Θ-net in X and f®)={B\AczBkn_l}e%.

Hence there exists (/„, kn) e I χ ΓΝ with Then

) n e N is a subsequence of (x„) neN belonging to Θ.

•

§ 1.4 Filters defined by sequences

27

Proposition 1.4.5. Let X be a set, A be a subset of X and Θ be a set of sequences in X such that any sequence in X belongs to 0 if it possesses a subsequence belonging to 0. Then the following assertions are equivalent: a) A is a b is trivial; b => c. Let (.vn)neIN be a sequence in A. Then the elementary filter of (.xn)n€thJ belongs to Φ(Θ) and by Proposition 1.4.4 it possesses a subsequence belonging to 0 . By the hypothesis (-Χη)Π6 N e Θ. c => a. follows from Proposition 1.4.3. • Proposition 1.4.6. Let X be a set, Θ be a set of sequences in X, A be the set of χ e X such that the constant sequence (x)netK| e Θ, (/,/) be a Θ-net in X and g e Φ(Θ). Then / ( / ) c A and A eft. Let xef(I). Then there exists lei so that f(i) = x. The constant sequence (i)nelsJ in / being increasing it follows that the constant sequence (jf)„eN belongs to Θ. Hence, xeA and /(/)c=>d 2) itself 3) itself

if to if to

and

c => e => f ;

every sequence in X which possesses a subsequence belonging to Θ belongs Θ then d => a&f; every sequence in X' which possesses a subsequence belonging to Θ' belongs Θ' then f => d;

4) if the hypotheses of 2) and 3) are fulfilled, then the assertions a), b), c), d), e), f ) coincide. 1) a

b => c is trivial, c => d follows from Proposition 1.4.2. c => e follows

from Proposition 1.1.9. e => f follows from Corollary 1.1.8. 2) d => a. By Proposition 1.4.7 {x„ \n e fNJ} is a (0)-set and by d) {>), (p(x',y))eU

for

any (x, y), (x',y)sA,

ultrafilters of Φ such that ^ η © e Φ for any

and let Ψ be a set of

© e Ψ. Then Ψ c Φ.

Let FT Ε Ψ and assume $ Φ Φ- Then there exists an entourage U of Ζ such that for any A e ^ there exist x{A),

χ '{A) e X, y(A) e Y with

z(A):=(x(A),y(A))eA,

z'(A)

y(A)) e Α, (φ(ζ(Α)),

•.= (x'(A),

φ(ζ'(Α)))φ

U.

Let us order 5 by the converse inclusion relation and let (5 be an ultrafilter on $ finer than the section filter of 5· Then ζ ((5), ,z'((5) are ultrafilters on Χ χ Y finer than 5- By Proposition 1.1.3 a => b they belong to Ψ. Thus, Hence, there exists

ßez(®)nz'((5)

Let C, C ' e ©

{x,y),{x',y)eB.

and therefore there exists

such that

{φ{χ, y), φ{χ', y))eU

for any

z ' ( C ' ) c i ) . Then

CnC'e©

such that z(C)'„)„ e N in Χ χ Y such that for any entourage U of Ζ there exist m, η e Ν with m κ, h(i') = h(i") and (φ(/(ι% φ(/(ι")))φ U. Since i 2n + t or M/(0),

4-set (Proposition 1.5.5). A. Weil proved (1938) ([1] Theoreme VIII) that any /} is empty. Hence, ( / ( 0 ) „ £ W e Θ5(Χ) and t h e r e f o r e / ( g ) e Φ5(Χ). • The following Corollary will not be used in the sequel. Corollary 1.5.14. The following uniform space X: α)

δεΦ5(Χ)/

b)

δ6Φ5(Ζ);

c) %eιη, (f(l'),f(l"))