Measure and Integral

584 Pages·1978·25.333 MB·English

by  Konrad Jacobs

MEASURE AND INTEGRAL
KONRAD JACOBS
Mathematisches Institut
Universität Erlangen-Nürnberg
Erlangen, Federal Republic of Germany
With an Appendix by
Jaroslav Kurzweil
Prague, Akademie Ved CSSR
®
ACADEMIC PRESS New York San Francisco London 1978
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United Kingdom Edition published by
ACADEMIC PRESS, INC. (LONDON) LTD.
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Library of Congress Cataloging in Publication Data
Jacobs, Konrad, Date
Measure and integral.
(Probability and mathematical statistics series ; )
1. Measure theory, 2. Integrals, Generalized.
I. Title.
QA312.J23 515\42 78-210
ISBN 0-12-378550-2
AMS (MOS) 1970 Subject Classifications: 28-02,
28 A 30, 52 A 05, 60 G 05
PRINTED IN THE UNITED STATES OF AMERICA
PREFACE
Although there is an abundance of excellent textbooks on measure
theory and an output of more than four books on that subject from
Erlangen alone (HAUPT-AUMANN-PAUC [1], BAUER [4, 5], ANGER-
BAUER [1], KÖLZOW [4]), there seems to be no graduate text and refer-
ence book that covers all the material needed by a senior student of the
field or a research worker in probability, ergodic theory, or functional
analysis. The purpose of the present book is to fill this gap at least to a
certain extent. I dare say this although a number of topics have deliber-
ately not been included in the present volume.
Anyone who considers the immense wealth of present-day measure
theory and its rapid adaptation to more and more purposes will realize
that it is nearly impossible, if not undesirable, to present the theory in a
form from which one can draw without thinking. Only a thorough training
in the basic ideas and their applications can provide a reader with the
capability he needs. I have therefore resolved to display every basic idea
once or twice in the full proof of some fundamental theorem and to put
many important applications, special cases, and extensions in the form of
additional remarks and, in particular, of exercises. The reader is therefore
invited to pay special attention to the exercises. Experience will show
whether this resolution was wise in itself and appropriately realized in
detail.
In spite of this basic decision I have tried to prepare the text in such a
fashion that a solid nucleus of the story can be simply read from the
definitions, lemmas, propositions, and theorems.
The great variety of definitions in measure theory has forced me to
make some decisions concerning notation and assumptions. I have always
hated to be forced to scan hundreds of pages preceding a theorem in order
IX
x PREFACE
to find the definitions of all the relevant notation and assumptions. The
reader will find that I have been very patient in stating assumptions and
notations over and over again. Moreover, I have made up my mind to use
a somewhat lengthy notation so as to remind the reader of the key words.
When I had to decide between a very succinct notation like Mh and a
compound like loc mbleb(n, ÖS00, R), after some deliberation I chose the
latter since it quickly tells the reader more about "where he is." Also, it
proved to be necessary to choose among types of set systems and set
functions. I found it feasible to avoid the values ± ^c of set functions when
possible. Thus the notion of a local σ-ring (also called δ-ring) came into
focus. This brought me into a desirable neighborhood of one of the most
well-developed branches of the theory: measure theory in locally compact
spaces. Another decision was to abstain entirely from the term measure
as meaning a σ-additive set function. I have rigorously used the word
content (resp. σ-content) for such functions since extensions to various
types of set systems, however fundamental they may be, are routine
nowadays. The term measure, in conformity with Bourbaki, is strictly
reserved for linear forms throughout this book. Throughout, I have tried
to emphasize that σ-measures and σ-contents are merely two aspects of
the same matter.
It may be useful to acquaint the reader with some details of the notation
employed in this book. The "local" viewpoint is signaled mostly by an
upper index (M). Thus 38°° is the symbol for a local σ-ring (=8-ring), and
^00(Ω, R) denotes the continuous real functions vanishing outside compact
sets for a topological space Ω. Wherever an upper index 00 seems unfitting,
a prefix loc is used. Shortened words frequently figure as symbols; thus
cont denotes the space of contents (and cont°" the space of σ-contents),
meas the space of measures, and mble the space of measurable functions. It
was, of course, undesirable to replace all well-established symbols by such
agglomerations. Still, % refers to continuous mappings and i? integrable
functions. The indicator (^characteristic) function of a set £ is denoted by
\ . I indulge frequently in a certain abuse of notation, which can be
E
exemplified by saying that "g Ξΐ/Ί^Λ^ · · · \ Ο" means as much as "f f ,
u 2
...€?,/, ^/ ^ · · ·, lim^oo/n = 0 pointwise." The arrow => means
2
"implies," and the double arrow O means "iff" (= "if and only if"). I
distinguish strictly between sets and families; a set of sets is often called a
set system.
σ stands for countable operations and countable continuity properties,
and r stands for the corresponding uncountable items. By tradition, σ also
stands for countable unions, and δ for countable intersections; F sets are
a
countable unions of closed sets, and G sets are countable intersections of
s
open sets (in a topological space, of course). R denotes the set of all real
numbers, Q the set of all rational numbers, Z the set of all integers, and N
the set {1,2,...} of all natural numbers.
PREFACE xi
The book falls into two parts. Chapters I—III are meant to have the
character of a graduate textbook, and students are invited to work
through these carefully. Afterward they should know by heart, e.g., how
the machinery of σ-extension works and how a σ-content is derived from
a σ-measure. From Chapter IV onward a more summary and advanced
style is employed, and various branches of measure and content theory
are presented in a way that tries to lead quickly to basic information but
trusts more than in Chapters I—III in the ability of the reader to fill in
obvious details. I would be very much satisfied if there were some
mathematicians who found some of Chapters IV-XVI to be a concise
introduction to the respective topic.
The contents of the book may be sketched as follows. After the prelimi-
nary Chapter 0 on general mathematical notions and notations, the text
proper begins with Chapter I, where are introduced the notions, on the
one hand, of rings, fields, σ-rings, σ-fields, and, above all, local σ-rings,
and, on the other hand, of positive contents and σ-contents defined on
these objects. The extension problem is touched upon in a section on
Eudoxos extension, and then fully formulated; the treatment includes the
proof of uniqueness of an extension (after Dynkin). Next are considered
positive measures and σ-measures on elementary domains of real functions
(= vector lattices satisfying Stone's axiom). The extension problem is
tackled once again, in the form of Riemann integration, and then more
clearly formulated. The classical Hahn-Banach theorem is proved and
Banach limits are introduced in the form of a major exercise. Chapter II
takes up Carathéodory's extension theory for cr-contents, along with some
routine techniques for further extensions, such as passing from a local
cr-ring to the generated σ-ring or σ-field, and completion.
In Chapter III is presented the Danieh extension theory for positive
σ-measures (= positive σ-continuous linear forms on elementary do-
mains). The basic theorems of integration theory are proved. It is shown
how to pass forth and back between σ-contents and σ-measures. The
extension theory for r-measure is sketched. The Hubert space L2 is
introduced. Measurable functions and their integration are studied. Inte-
gration of complex-valued functions is treated.
Chapter IV deals with the transform of σ-contents and σ-measures by
measurable mappings and kernels. The rudiments of ergodic theory—in
particular, ergodic theorems—are given.
In Chapter V are studied contents and measures in topological spaces
from the viewpoint of regularity. An abstract regularity theory is given
and applied to the standard cases of compact, locally compact, and Polish
spaces.
Chapter VI deals with contents and measures in product spaces. It
turns out that employment of local σ-rings makes all assumptions about
σ-finiteness obsolete so long as finite product spaces are considered. With
xii PREFACE
the theory of infinite product spaces probability theory enters. The exis-
tence of infinite product probabilities is proved, as well as Kolmogorov's
theorem about the extension of compatible marginal contents. The prob-
abilistic theory of independence is given. The last section deals with
Markovian semigroups and path spaces.
Chapter VII deals with rather general set functions. The values ±°° are
occasionally allowed for these (as in part of Chapter II). It is shown how
to split a superadditive function into an additive and a purely superaddi-
tive part, and an additive one into a σ-additive and a purely additive part.
The Vitali-Hahn-Saks theorem is proved, and total variation is exam-
ined.
Chapter VIII is devoted to a thorough study of the vector lattice of
signed contents. How to carry out the vector lattice operations is shown,
as is how to obtain Hahn and Lebesgue decompositions. The Radon-
Nikodym theorem is proved for a σ-content of finite total variation that is
absolutely continuous with respect to an arbitrary positive σ-content, and
an integrable density is obtained. With this as a basis, probability theory is
considered next; conditional expectations are introduced and the rudi-
ments of martingale theory, including the submartingale convergence
theorem, are presented. In the last problem is discussed the question of
proving a Radon-Nikodym theorem without the assumption of finite total
variation; disjoint bases (which always exist in the locally compact case)
and locally integrable functions are introduced, and the Radon-Nikodym
theorem is proved under the assumption that a disjoint base exists.
In Chapter IX Riesz band decompositions in general vector lattices are
studied and applied to vector lattices of signed measures. Absolute con-
tinuity for signed measures is introduced in a purely vector lattice theoret-
ical fashion: Lebesgue decompositions are obtained. In the last section
these results are linked with the corresponding results about signed con-
tents (Chapter VIII).
Chapter X presents, with the Radon-Nikodym theorem as a basic
tool, the theory of the spaces Lp and their duality.
Chapter XI is concerned with the weak topology for measures and
contents in topological spaces. A general theory is followed by applica-
tions to Polish spaces, which are significant in probability theory.
In Chapter XII Haar measure in locally compact groups is studied. The
compact case is treated separately, as a prelude, so to speak; it is shown
that continuous functions on a compact group are almost periodic, and a
combinatorial theorem is used in order to establish a mean value for
almost periodic functions, and thus a two-sided invariant Haar measure,
on compact groups. The idea of using a generalized marriage theorem also
in the locally compact case goes back to Kakutani and von Neumann and
is carried out in Halmos" handwritten notes of von Neumann's lectures on
measure theory delivered in 1940-1941. I have found this approach too
PREFACE xiii
unwieldy and instead followed HEWITT-ROSS [1], for the locally compact
case, who in turn follow CARTAN [1]. In this way, Zorn's lemma is
completely avoided. Examples of Haar measures are given. We do not
penetrate more deeply into Haar measure theory, but the interested
reader is referred to HALMOS [4], BERBERIAN [1], LOOMIS [2], HEWITT-
ROSS [1], and NACHBIN [1].
Souslin sets, analytic sets, and the extension of capacities are the theme
of Chapter XIII. Kuratowskfs theorem on the measurability of "for-
ward" images of measurable sets, the continuum hypothesis for Borel
sets in Polish spaces, Choquet's capacitability theorem, and the measur-
able choice theorem are the main results. Following MEYER [2], we
present a capacity approach to σ-content extension.
Chapter XIV is devoted to the Hanen-Neveu theory of atoms and
conditional atoms and an entropy theory that is free of regularity assump-
tions. Most of the applications lie in ergodic theory but are not treated
here.
Chapter XV starts with an introduction to the theory of topological
vector spaces and proves the Krein-Milman theorem and Choquet's
barycentric representation theorem, including simplexes and uniqueness.
I have tried to present the quickest approaches to these matters, thus
discarding a good portion of the beautiful machinery to be found in
CHOQUET [5]. As an application of the Krein-Milman theorem,
Lyapunov's convexity theorem is given. Choquefs theorem is applied to
ergodic decompositions. Bochner's theorem on positive definite functions
would provide a beautiful application of Choquet's theory, but is left out
here for technical reasons (see CHOQUET [5]). The rest of the chapter is
devoted to Choquet's theory of barycentric representation in convex com-
pact sets.
The Krein-Milman theorem is one of the tools of lifting theory as
treated here. Its rudiments, along with some of their applications, are
presented in Chapter XVI. The existence of liftings in the presence of a
disjoint base is proved, and the existence of strong lifting under separabil-
ity assumptions follows. Kernel representations of linear mappings, con-
ditional probabilities, disintegration of measures, and Strassen's theorem
are among the applications set forth. The reader is referred to IONESCU-
TULCEA [1] for further studies.
The book has two appendixes. Appendix A, an outline of the theory of
the Perron-Ward integral and related topics, was kindly contributed by J.
Kurzweil (Prague), the outstanding specialist in the field. Appendix B, by
myself, treats the topic of contents and σ-contents with given marginals
along the lines of a recent paper by HANSEL-TROALLIC [1], who introduce
the max-flow-min-cut theorem of Ford and Fulkerson in order to obtain
most of the basic theorems in an easy way.
General topology and functional analysis have been confined to a sub-
xiv PREFACE
sidiary role throughout this book. However, while the reader is assumed
to have a solid background (the specifics are given in Section 0.2),
functional analysis is treated in some detail when it is needed. Thus the
Hahn-Banach theorem in its classical form is proved as Theorem 1.6.1,
and Section XV. 1 serves as a rapid course in topological vector spaces,
including the Krein-Milman theorem. I have tried, however, to restrict
the use of general topology in Chapters I— III to a few points, and pre-
ferred substituting Rn for a general locally compact space in many of the
examples.
It can be seen from this outline that some rudiments of topological
vector space theory are contained in this book. They are proved where
needed, but not too many of their consequences are followed up since
they play only a subsidiary role here.
There is a considerable list of topics in measure theory that have
deliberately not been included in this volume: measure and category (see
OXTOBY [2, 3]), measure algebras (see CARATHÉODORY [2], KAPPOS [1, 2]),
partitions in Lebesgue spaces (ROKHLIN [1]), vector-valued measures (see
DINCULEANU [1]) and integration of vector-valued functions (see
DINCULEANU [1] and IONESCU-TULCEA [1]), contents and measures in
topological vector spaces, and other themes. They were left out for lack of
time and space. Moreover, I think that this book enables a reader to
handle any one of the monographs and papers mentioned above. I beg the
reader to test his knowledge by reading classics like HAUSDORFF [1],
CARATHÉODORY [1], and SAKS [3, 4], reputed textbooks like HALMOS [4],
BERBERIAN [1], HAUPT-AUMANN [1], BAUER [4, 5], and ANGER-BAUER
[1], historically important papers listed in the bibliography, and mono-
graphs or treatises like IONESCU-TULCEA [1], KÖLZOW [4], DUNFORD-
SCHWARTZ [1], CHOQUET [5], PARRY [1], FREMLIN [1], and BICHTELER
[1].
Note added in proof: (1) The most exciting result of 1978 is: V. Losert
(Wien), An example of a compact Hausdorff space with no strong lifting,
Math. Annalen (to appear). (2) A fine little textbook has appeared: H.
Michel, "Mass- und Integrationstheorie I." VEB Deutscher Verlag der
Wissenschaften, Berlin, 1978. (3) A reader interested in vector-valued
measures should consult the book: J. Diestel and J. J. Uhl, Jr., "Vector
Measures." American Math. Soc, Providence, Rhode Island, 1977.
Any comments and corrections will be highly appreciated.
Reference numbers and headings follow the following scheme: The
book is divided into a chapter entitled "Basic Notions and Notations,"
which bears the number 0, and sixteen subsequent chapters, labeled with
the Roman numbers I-XVI. Each chapter falls into at least two
sections—1, 2, etc.; in some cases these split into further subsections.
The definitions, remarks, exercises, lemmas, propositions, theorems, etc.
that make up a section are numbered in the order of their appearance;
PREFACE xv
each of their numbers is preceded by the number of the section, and
followed by the word definition, etc. Thus Section 2 consists, e.g., of
2.1. Definition, 2.2. Remark, 2.3. Lemma, 2.4. Theorem,.... A definition,
theorem, etc., may itself have several sections. These are numbered by the
double number of the definition followed by simple or double "local" num-
bers in ordinary print. Often the numbers denoting the sections of a proof of a
theorem parallel these ordinary numbers in the theorem. Thus a 2.4.
Theorem may have sections 2.4.1.1, 2.4.1.2, 2.4.2.1,. . . , and the proof is
then subdivided into 1.1, 1.2, 2.1, ... .
When citing, we let the word definition, etc., precede all numbers. The
numbers refer to the present chapter if no roman number appears.
Otherwise the roman numeral is included. Thus it should be clear where
to find Theorem IX.2.4.1.1. It is the same as Theorem 2.4.1.1 if we are in
Chapter IX.
I thank J. Kurzweil (Prague) for his contribution of an appendix to this
book. I thank D. Kölzow and J. Lembcke (both Erlangen) for the informa-
tion that V. Losert (Vienna) has constructed recently an example of a
compact measure space that has no strong lifting. My thanks go to E.
Lukacs and Academic Press for accepting this book in spite of its size.
Academic and the printer are to be congratulated for their careful editorial
and typsetting work. My special thanks go to Frau H. Zech who carefully
and swiftly typed the manuscript from my handwritten notes.
CHAPTER 0
BASIC NOTIONS AND NOTATION
In this preliminary chapter we present the basic facts about sets, families,
orderings, etc. which will be used throughout the book without particular
citation. No proofs are given and no completeness is aimed at. The purpose
of this chapter is only to fix certain standards of wording in such a way
that it will be easy for us and the reader to adapt the wording to all
situations which might occur. The reader is advised to read this chapter
and to check his own understanding of it carefully before he sets out to go
ahead with any of the subsequent chapters.
1. SETS, RELATIONS, AND MAPPINGS
Throughout this book we rely on what is nowadays called naïve set
theory. For an account of it, see KAMKE [1], HALMOS [2, 3]. (For nonnaïve
(namely, axiomatic) set theory, see, e.g., TAKEUTI and ZARING [1, 2];
JECH [1].) The purpose of the present section is mainly to recall and
standardize notation. We also rely on standard logic and use, e.g., the
implication arrow => in the usual way: If P, Q are statements, then P=>Q
means "if P, then g" or "Q whenever P." For this we write sometimes P(Q).
A set is, naively, an entity that is a certain totality of other entities called
its elements (sometimes, points), and the set is said to consist of its elements.
To specify a set means to exactly specify which entities constitute its
elements. This can be done by listing (possibly multiply) the elements or
writing a precise description of them between braces. Thus {1} is the set
whose only element is the natural number 1, and so is, by the way,
{1, 1} = {1, 1, 1}; {1, 2, 3, 4} is the set whose elements are exactly the
natural numbers 1, 2, 3 and 4; {1, 2, ..., 100} = {1, ..., 100} is the set of
1