(atlantis studies in mathematics 2) alexander b. kharazishvili-topics in measure theory and real...

7/28/2019 (Atlantis Studies in Mathematics 2) Alexander B. Kharazishvili-Topics in Measure Theory and Real Analysis-Atlantis

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ATLANTIS STUDIES IN MATHEMATICSVOLUME 2

SERIES EDITOR: J. VAN MILL


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Atlantis Studies in Mathematics

Series Editor:

J. van Mill, VU University Amsterdam, Amsterdam, the Netherlands

(ISSN: 1875-7634)

Aims and scope of the series

The series Atlantis Studies in Mathematics (ASM) publishes monographs of high quality

in all areas of mathematics. Both research monographs and books of an expository nature

are welcome.

All books in this series are co-published with World Scientific.

For more information on this series and our other book series, please visit our website at:

www.atlantis-press.com/publications/books

AMSTERDAM PARIS

c ATLANTIS PRESS / WORLD SCIENTIFIC


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Topics in Measure Theory

and Real Analysis

Alexander B. Kharazishvili

A. Razmadze Mathematical Institute

Tbilisi

Republic of Georgia

AMSTERDAM PARIS


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Atlantis Press

29, avenue Laumiere

75019 Paris, France

For information on all Atlantis Press publications, visit our website at:www.atlantis-press.com

Copyright

This book, or any parts thereof, may not be reproduced for commercial purposes in any

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information storage and retrieval system known or to be invented, without prior permission

from the Publisher.

Atlantis Studies in Mathematics

Volume 1: Topological Groups and Related Structures - A. Arhangelskii, M. Tkachenko

ISBN: 978-90-78677-20-8

ISSN: 1875-7634

c 2009 ATLANTIS PRESS / WORLD SCIENTIFIC

e-ISBN: 978-94-91216-36-7


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Preface

This book is concerned with questions of classical measure theory and related topics of

real analysis. At the beginning, it should be said that the choice of material included in the

present book was completely dictated by research interests and preferences of the author.

Nevertheless, we hope that this material will be of interest to a wide audience of mathe-

maticians and, primarily, to those who are working in various branches of modern math-

ematical analysis, probability theory, the theory of stochastic processes, general topology,

and functional analysis. In addition, we touch upon deep set-theoretical aspects of the top-

ics discussed in the book; consequently, set-theorists may detect nontrivial items of interestto them and find out new applications of set-theoretical methods to various problems of

measure theory and real analysis. It should also be noted that questions treated in this book

are related to material found in the following three monographs previously published by

the author.

1) Transformation Groups and Invariant Measures, World Scientific Publ. Co., London-

Singapore, 1998.

2) Nonmeasurable Sets and Functions, North-Holland Mathematics Studies, Elsevier, Am-sterdam, 2004.

3) Strange Functions in Real Analysis, 2nd edition, Chapman and Hall/CRC, Boca Raton,

2006.

For the convenience of our readers, we will first, briefly and schematically, describe the

scope of this book.

In Chapter 1, we consider the general problem of extending partial real-valued functions

which, undoubtedly, is one of the most important problems in all of contemporary math-

ematics and which deserves to be discussed thoroughly. Since the satisfactory solution to

this task requires a separate monograph, we certainly do not intend on entering deeply into

v


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vi Topics in Measure Theory and Real Analysis

various aspects of the problem of extending partial functions, but rather we restrict our-

selves to several examples that are important for real analysis and classical measure theory

and vividly show the fundamental character of this problem. The corresponding examplesare given in Chapter 1 and illustrate different approaches and appropriate research methods.

Notice that some of the examples presented in this chapter are considered in more details

in subsequent sections of the book.

Chapter 2 is devoted to a special, but very important, case of the extension problem for

real-valued partial functions. Namely, we discuss therein several variants of the so-called

measure extension problem and we pay our attention to purely set-theoretical, algebraic and

topological aspects of this problem. In the same chapter, the classical method of extendingmeasures, developed by Marczewski (see [234] and [235]), is presented. Also, a useful

theorem is proved which enables us to extend any -finite measure on a base set E to

a measure on the same E, such that all members of a given family of pairwise disjoint

subsets ofE become -measurable (see [1] and [13]). This theorem is then repeatedly

applied in further sections of the book.

In Chapters 3 and 4 we primarily deal with those measures on E which are invariant or

quasi-invariant with respect to a certain group of transformations ofE. It is widely known

that invariant and quasi-invariant measures play a central role in the theory of topological

groups, functional analysis, and the theory of dynamical systems. We discuss some gen-

eral properties of invariant and quasi-invariant measures that are helpful in various fields

of mathematics. First of all, we mean the existence and uniqueness properties of such

measures. The problem of the existence and uniqueness of an invariant measure naturally

arises for a locally compact topological group endowed with the group of all its left (right)

translations. In this way, we come to the well-known Haar measure. The theory of Haar

measure is thoroughly covered in many text-books and monographs (see, for instance, [80],

[83], [182], [202]), so we leave aside the main aspects of this theory. But we present the

classical Bogoliubov-Krylov theorem on the existence of a dynamical system for a one-

parameter group of homeomorphisms of a compact metric space E. More precisely, we

formulate and prove a significant generalization of the Bogoliubov-Krylov statement: the

so-called fixed-point theorem of Markov and Kakutani ([93], [168]) for a solvable group of

affine continuous transformations of a nonempty compact convex set in a Hausdorff topo-

logical vector space. In the same chapters, we distinguish the following two situations: the

case when a given topological spaceE is locally compact and the case whenE is not locally

compact. The latter case involves the class of all infinite-dimensional Hausdorff topolog-


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Preface vii

ical vector spaces for which the problem of the existence of a nonzero -finite invariant

(respectively, quasi-invariant) Borel measure needs a specific formulation. Some results in

this direction are presented with necessary comments.Chapter 5 is concerned with measurability properties of real-valued functions defined on

an abstract space E, when a certain class Mof measures on E is determined. We introduce

three notions for a given function facting from E into the real line R. Namely, fmay be

(a) absolutely nonmeasurable with respect to M,

(b) relatively measurable with respect to M,

(c) absolutely (or universally) measurable with respect to M.

We examine these notions and show their close connections with some classical construc-

tions in measure theory. It should be pointed out that the standard concept of measurability

of fwith respect to a fixed measure on E is a particular case of the notions (b) and (c).

In this case, the role ofM is played by the one-element class {}.

In Chapter 6 we discuss, again from the measure-theoretical point of view, some properties

of the so-called step-functions. Since step-functions are rather simple representatives of

the class of all functions (namely, the range of a step-function is at most countable), it is

reasonable to consider them in connection with the measure extension problem. It turns out

that the behavior of such functions is essentially different in the case of ordinary measures

and in the case of invariant (quasi-invariant) measures.

In Chapter 7, we introduce and investigate the class of almost measurable real-valued func-

tions on R. This class properly contains the class of all Lebesgue measurable functions

on R and has certain interesting features. A characterization of almost measurable func-

tions is given and it is shown that any almost measurable function becomes measurable

with respect to a suitable extension of the standard Lebesgue measure on R.

Chapter 8 focuses on several important facts from general topology. In particular, Kura-

towskis theorem (see, for instance, [58], [101], [149]) on closed projections is presented

with some of its applications among which we especially examine the existence of a co-

meager set of continuous nowhere differentiable functions in the classical Banach space

C[0,1]. Also, we prove a deep theorem on the existence of Borel selectors for certain

partitions of a Polish topological space, which is essentially used in the sequel.

In Chapter 9 the concept of the weak transitivity of an invariant measure is considered

and its influence on the existence of nonmeasurable sets is underlined. Here it is vividly

shown that some old ideas of Minkowski [173] which were successfully applied by him in


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viii Topics in Measure Theory and Real Analysis

convex geometry and geometric number theory, are also helpful in constructions of various

paradoxical (e.g., nonmeasurable) sets. Actually, Minkowski had at hand all the needed

tools to prove the existence of those subsets of the Euclidean space Rn

(n 1), which arenonmeasurable with respect to the classical Lebesgue measure n on R

n.

Chapter 10 covers bad subgroups of an uncountable solvable group (G, ). The term bad,

of course, means the nonmeasurability of a subgroup with respect to a given nonzero -

finite invariant (quasi-invariant) measure on G. We establish the existence of such sub-

groups ofG and, moreover, show that some of them can be applied to obtain invariant

(quasi-invariant) extensions of. So, despite their bad structural properties, certain non-

measurable subgroups ofG have a positive side from the view-point of the general measureextension problem.

The next two chapters (i.e., Chapters 11-12) are devoted to the structure of algebraic sums

of small (in a certain sense) subsets of a given uncountable commutative group (G,+).

Recall that the first deep result in this direction was obtained by Sierpinski in his classical

work [219] where he stated that there are two Lebesgue measure zero subsets of the real line

R, whose algebraic (i.e., Minkowskis) sum is not Lebesgue measurable. Let us stress that

in [219] the technique of Hamel bases was heavily exploited and in the sequel such an ap-proach became a powerful research tool for further investigations. We develop Sierpinskis

above-mentioned result and generalize it in two directions. Namely, we consider the purely

algebraic aspect of the problem and its topological aspect as well. The difference between

these two aspects is primarily caused by two distinct concepts of smallness of subsets

ofR.

In Chapters 13 and 14 we turn our attention to Sierpinski-Zygmund functions [225] and

study them from the point of view of the measure extension problem. It is well known

that the restriction of a Sierpinski-Zygmund function to any subset of R of cardinality

continuum is discontinuous. This circumstance directly implies that a Sierpinski-Zygmund

function is nonmeasurable in the Lebesgue sense and, moreover, is nonmeasurable with

respect to the completion of an arbitrary nonzero -finite diffused Borel measure on R. In

addition, no Sierpinski-Zygmund function has the Baire property.

We give two new constructions of Sierpinski-Zygmund functions.

(1) The construction of a Sierpinski-Zygmund function which is absolutely nonmeasurable

with respect to the class of all nonzero -finite diffused measures on R. (Notice that

this result needs some extra set-theoretical axioms.)


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Preface ix

(2) The construction of a Sierpinski-Zygmund function which is relatively measurable with

respect to the class of all translation-invariant extensions of the Lebesgue measure

on R. (This result does not need any additional set-theoretical hypotheses.)

Chapters 15-17 are similar to each other in the sense that the main topics discussed therein

are connected with different constructions of nonseparable extensions of-finite measures.

Among the results presented in these chapters, let us especially mention:

(i) the construction (assuming the Continuum Hypothesis) of a nonseparable extension

of the Lebesgue measure without producing new null-sets;

(ii) the construction (also under some additional set-theoretical axioms) of nonseparable

invariant extensions of-finite invariant measures by using their nontrivial ergodic

components;

(iii) the construction (assuming again the Continuum Hypothesis) of a nonseparable non-

atomic left invariant -finite measure on any uncountable solvable group.

In Chapter 18, we consider universally measurable additive functionals. The universal

measurability is treated here in a generalized sense, namely, a real-valued functional fon

a Hilbert space E is universally measurable if and only if for any -finite Borel measure

given on E, there exists an extension of such that f becomes measurable with

respect to . It is established that there are universally measurable additive functionals

which are everywhere discontinuous. This result may be regarded as a counter-version to

the well-known statement (see, e.g., [97], [153], [154]), according to which any universally

measurable (in the usual sense) additive functional on E is necessarily continuous.

Chapter 19 is devoted to certain strange subsets of the Euclidean plane R2. We discuss

various properties of these paradoxical sets from the measure-theoretical view-point. In

particular, a subset ZofR2 is constructed which is almost invariant under the group of all

translations ofR2, is 2-thick (where 2 stands for the two-dimensional Lebesgue measure

on R2) and, in addition, has the property that for each straight line l in R2, the intersection

lZ is of cardinality strictly less than the cardinality of the continuum. By using this set

Z, we define translation-invariant extensions of2 for which no analogue of the classical

Fubini theorem can be valid.

The final chapter is connected with certain restrictions of functions acting from R into R.

The first examples of those restrictions of measurable functions, which are defined on suf-

ficiently large subsets ofR and have various nice properties, were given in widely known


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x Topics in Measure Theory and Real Analysis

statements of real analysis. For instance, in accordance with the classical Luzin theorem

(see, e.g., [16], [26], [65], [80], [161], [183], and [192]), every real-valued Lebesgue mea-

surable function restricted to a certain set of strictly positive -measure becomes continu-ous (and an analogous purely topological result holds true in terms of the Baire property

and category). We touch upon some other results in this direction. In particular, it is proved

that every Lebesgue measurable function g : R R is monotone on a nonempty perfect

subset of R. At the same time, such a perfect set does not need to be of strictly posi-

tive -measure. The last circumstance is shown by considering Jarniks [88] continuous

nowhere approximately differentiable function whose existence is a rather deep theorem of

real analysis (cf. [33], [34], and [167]).

In general, we tried to present the material in a self-contained form completely accessible

to graduate and post-graduate students. Moreover, for the readers convenience, six Ap-

pendices are attached to the main text of this book, which can be read independently of the

material covered in the chapters.

In Appendix 1 some auxiliary set-theoretical facts and constructions are considered, which

are essential in various sections of the book. Namely, elements of infinite combinatorics

(e.g., infinite trees and Konigs lemma), several delicate set-theoretical statements, and the

existence of an uncountable universal measure zero subset ofRare discussed.

Appendix 2 is devoted to various theorems on the existence of measurable selectors. Re-

sults of this type are important and attractive and have found applications in numerous

branches of modern mathematics. We begin with the Choquet theorem on capacities (see

[24], [52], [187]) and show its close connection with statements about measurable selectors.

Also, we present the following two fundamental results in this topic: the theorem of Kura-

towski and Ryll-Nardzewski [151] and, as one of its consequences, the Luzin-Jankov-von

Neumann theorem (see, e.g., [99]).

In Appendix 3 deep properties of-finite Borel measures on metrizable topological spaces

are examined. It is proved that if the topological weight of a metric space E is not measur-

able in the Ulam sense, then any -finite Borel measure on E admits a separable support

(cf. [192]). This important fact is essentially used in Chapter 12.

Appendix 4 contains a detailed proof of the existence of a continuous function f: R R

which is nowhere approximately differentiable. As mentioned above, the first example of

such a function was constructed by Jarnik in his remarkable work [88]. A function of this

type is needed for considerations in Chapter 20.


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Preface xi

In Appendix 5 general properties of commutative groups (primarily, infinite commutative

groups) are examined and one useful theorem, due to Kulikov, on the algebraic structure

of such groups is proved. This theorem is utilized in several parts of the main text of thebook; see especially Chapters 10 and 11.

Appendix 6 presents elements of classical descriptive set theory. Namely, we touch upon

certain properties of Borel and analytic (Suslin) subsets of uncountable Polish spaces and

apply those properties to the question of measurability of sets or functions. (In this con-

text, Appendix 2 is also helpful.) The so-called separation principle, first introduced and

extensively studied by Luzin and Sierpinski, receives special attention. Of course, our pre-

sentation of this material is concise and superficial. The standard monographs or text-books

devoted to classical descriptive set theory are [99], [148], [150], [160], and [162]; see also

Martins article in [10].

Finally, we would like to note that all sections of the book, including the Appendices, are

provided with exercises which contain additional information concerning the questions un-

der discussion. Some of the exercises are quite easy but some of them involve difficult

mathematical facts and need intensive efforts for their solution. These more difficult exer-

cises are marked by the symbol and we recommend that the reader solve them in order to

better understand the subject. Finally, we also state in the text several unsolved problems

which are motivated by (or closely connected with) topics presented in this book.

The Bibliography consists of 251 titles and contains only the most relevant ones. Of course,

it is far from being complete but rather provides a basic orientation to the subject in order

to stimulate further interest of our readers in various questions of measure theory and real

analysis.

A.B. Kharazishvili


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Contents

Preface v

1. The problem of extending partial functions . . . . . . . . . . . . . . . . . . 1

2. Some aspects of the measure extension problem . . . . . . . . . . . . . . . 19

3. Invariant measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

4. Quasi-invariant measures . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

5. Measurability properties of real-valued functions . . . . . . . . . . . . . . . 79

6. Some properties of step-functions connected with extensions of measures . . 97

7. Almost measurable real-valued functions . . . . . . . . . . . . . . . . . . . 111

8. Several facts from general topology . . . . . . . . . . . . . . . . . . . . . . 125

9. Weakly metrically transitive measures and nonmeasurable sets . . . . . . . 145

10. Nonmeasurable subgroups of uncountable solvable groups . . . . . . . . . 159

11. Algebraic sums of measure zero sets . . . . . . . . . . . . . . . . . . . . . 177

12. The absolute nonmeasurability of Minkowskis sum of certain universal

measure zero sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199

13. Absolutely nonmeasurable additive Sierpinski-Zygmund functions . . . . . 215

14. Relatively measurable Sierpinski-Zygmund functions . . . . . . . . . . . . 227

15. A nonseparable extension of the Lebesgue measure without new null-sets . 241

16. Metrical transitivity and nonseparable extensions of invariant measures . . 257

17. Nonseparable left invariant measures on uncountable solvable groups . . . 269

18. Universally measurable additive functionals . . . . . . . . . . . . . . . . . 281

19. Some subsets of the Euclidean plane . . . . . . . . . . . . . . . . . . . . . 297

20. Restrictions of real-valued functions . . . . . . . . . . . . . . . . . . . . . 313

Appendix 1. Some set-theoretical facts and constructions . . . . . . . . . . . . 339

Appendix 2. The Choquet theorem and measurable selectors . . . . . . . . . . 359

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xiv Topics in Measure Theory and Real Analysis

Appendix 3. Borel measures on metric spaces . . . . . . . . . . . . . . . . . . 383

Appendix 4. Continuous nowhere approximately differentiable functions . . . 397

Appendix 5. Some facts from the theory of commutative groups . . . . . . . . 411Appendix 6. Elements of descriptive set theory . . . . . . . . . . . . . . . . . 423

Bibliography 447

Subject Index 457


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Chapter 1

The problem of extending partial functions

There are several general concepts and ideas in contemporary mathematics which play a

fundamental role in almost all of its branches. Among ideas of this kind, the concept of

extending a given partial function is of undoubted interest and of paramount importance for

various domains of mathematics. For instance, every working mathematician knows that

the problem of extending partial functions is considered and intensively studied in universal

algebra, general and algebraic topology, mathematical and functional analysis, as well as

otherfields. Of course, this problem has specific features in any of the above-mentioned

disciplines and it frequently needs special approaches or appropriate research tools whichare suitable only for a given situation and are applicable to concrete mathematical objects,

for example, groups, topological spaces, ordered sets, differentiable manifolds, and other

structures.

However, this problem can also be examined from the abstract view-point and method-

ological conclusions of a general character can be made. Below, we touch upon different

aspects of the problem and illustrate them by relevant examples. Some of those examples

will be envisaged more thoroughly in subsequent sections of this book. The main goal

of our preliminary consideration is to demonstrate how the problem of extending partial

functions accumulates ideas from different areas of modern mathematics.

The best known example of this type is the famous Tietze-Urysohn theorem which states

that every real-valued continuous function defined on a closed subset of a normal topo-

logical space (E,T) can be extended to a real-valued continuous function defined on the

whole space E (see, for instance, [58], [101], and [148]).

Another example of this kind is the classical Hahn-Banach theorem which states that a

continuous linear functional defined on a vector subspace of a given normed vector space

(E, || ||) can be extended to a continuous linear functional having the same norm and

defined on the whole E (see any text-book of functional analysis, for instance, [56], [57],

A.B. Kharazishvili, Topics in Measure Theory and Real Analysis, Atlantis Studies in Mathematics 2, 1

DOI 10.1007/978-94-91216-36-7_1, 2009 Atlantis Press/World Scientific


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2 Topics in Measure Theory and Real Analysis

or [209]).

The third example of this sort, although far from the main topics of topology and analysis,

is the problem of extending a given partially recursive function to a recursive function.As known, the latter should be defined on the set N (= ) of all natural numbers. The

existence of a partially recursive function that does not admit an extension to a recursive

function is crucial for basic statements of mathematical logic and the theory of algorithms.

It suffices to mention Godels incompleteness theorem of the formal arithmetic (see, for

instance, [10] and [215]).

Obviously, many other interesting and important examples can be pointed out in this con-

text.

The present book contains selected topics of measure theory which is a necessary part of

modern mathematical and functional analysis. As is well known, ordinary measures are

real-valued functions defined on certain classes of subsets of a given base set Eand having

the countable additivity property. Of course, in contemporary mathematics the so-called

vector-valued measures and operator-valued measures are also extremely important and are

used in many questions of analysis and the theory of stochastic processes, but we do not

touch them in our further considerations. Here we would like to stress especially that topics

presented in this book are primarily concentrated around the measure extension problem

which plays a significant role in numerous questions of real analysis, probability theory,

and set-theoretical topology. Actually, the measure extension problem will be central for

us in most sections of the book.

Consequently, it is reasonable to begin our preliminary discussion by considering several

facts from mathematical analysis, which are closely connected to extensions of partial

real-valued functions. Some of the facts listed below are fairly standard and accessible

to average-level students. But among the presented facts the reader will also encounterthose which are more important and deeper and which find applications in various domains

of modern mathematics.

Let Rdenote the real line and let X be an arbitrary subset ofR.

A function f: X Ris called a partial function acting from Rinto R.

For this f, we may write f: R R saying that f is a partial function whose domain is

contained in R. As usual, we denote

dom(f) = X.

IfX= R, then we obviously have an ordinary function f: R R.


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The problem of extending partial functions 3

The symbol ran(f) denotes the range of a partial function f, i.e.,

ran(f) = {f(x) :x dom(f)}.

IfY is any subset ofR, then the symbol f|Y stands for the restriction of a partial function

fto Y.

As a rule, people working in classical mathematical analysis are often interested in the

following general question.

Does there exist an extension f : R R of a partial function f: R R such that f is

defined on the whole Rand has certain nice properties?

In particular, we may require that f should be differentiable or continuous or semicontin-

uous or monotone or convex or Borel measurable or Lebesgue measurable or should have

the Baire property. (Notice that the Baire property can be regarded as a topological version

of measurability; extensive material about this property is contained in remarkable books

[148], [176], and [192].)

An analogous question arises in a more general situation, e.g., for partial functions facting

from subsets of an abstract set E into R, where E is assumed to be endowed with some

additional structure. In such a case an extension

f : E R

with dom(f) = E must preserve a given structure on E or should be compatible, in an

appropriate sense, with this structure. It is clear that questions of the above-mentioned kind

quite frequently arise in mathematical analysis, general topology, and abstract algebra.

Therefore, this topic is of interest for large groups of the working mathematicians.

Below, we have made a small list of results in this direction, have commented on each of

them or have given a necessary explanation, and have referred the reader to other related

works in which extensions of partial functions are considered more thoroughly (see, [58],

[70], [83], [101], and [148]).

For the convenience of potential readers, the material below is presented in the form of

examples of statements about extensions of real-valued partial functions. By the way, we

think that in various lecture courses for students it is useful to provide them with additional

information concerning extensions of partial functions. Such an approach essentially helps

them to see more vividly deep connections and interactions between distinct fields of con-

temporary mathematics. In addition, the students should know that the general problem of

extending partial functions is important for all mathematics because this problem almost

permanently occurs in different mathematical branches and finds numerous applications.


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We start with a fundamental result of purely set-theoretical flavor.

Example 1. Let {Xi : i I} be an arbitrary family of nonempty sets. A partial selector of

(for) {Xi : i I} is any family {xi : i J} where J is a subset ofIand xi Xi for all indices

i J. IfJ= I, then {xi : i J} is said to be a selector of (for) the given family {Xi : i I}.

The natural question arises whether every partial selector of{Xi : i I} can be extended to

a selector of the same family. As is well known, this question is solved positively if and

only if the Axiom of Choice (denoted by AC) is assumed.

At the same time, even in a very special case

card(I) = 2c, (i I)(card(Xi) = 2),

where c denotes the cardinality of the continuum, the question of the existence of a selector

of{Xi : i I} is highly nontrivial. For instance, as shown by Sierpinski, in this case the

positive answer to the question necessarily implies the existence of a subset ofRwhich is

not measurable in the Lebesgue sense (see [223] and references therein; cf. also [90]).

The formal Zermelo-Fraenkel set theory (denoted usually by ZF) is the standard system

of set-theoretical statements (axioms) without the Axiom of Choice (see, e.g., [10], [91],

[145], [150], and [215]). The symbol ZFC stands for the theory ZF & AC. There are

many set-theoretical assertions equivalent (within ZF) to AC, for instance, the Zorn lemma,

Zermelos theorem on the existence of a well-ordering of any set, the equality a2 = a for

all infinite cardinal numbers a, and so on. (In this connection, see especially [90], [150],

and [223].)

It is remarkable that some nontrivial and interesting equivalents ofAC can be formulated

in concrete mathematical disciplines. For example, in general topology we have the fun-

damental Tychonoff theorem stating that the product of any family of quasicompact spaces

is a quasicompact space, too. As demonstrated by Kelley [100], this theorem is equivalent

to AC within ZF theory. In linear algebra we have a very important theorem stating that

every vector space (over an arbitrary field) possesses at least one basis. As shown by Blass

[14], this theorem is also equivalent to AC within ZF theory.

Additionally, the importance ofAC in classical mathematical analysis is well known (see,

e.g., the old extensive work by Sierpinski [217]). It suffices to remind that even a proof of

the equivalence of the two standard definitions, due to Cauchy and Heine respectively, of

the continuity at a point x dom(f) of a partial function f: RRneeds some weak formofAC.

The following simple result can be included in a beginner lecture course of real analysis.


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Example 2. Let a partial function f: R Rbe given. Then it admits a continuous exten-

sion f : RRdefined on Rif and only if for each open interval ]a,b[ R, the restriction

of fto the set dom(f) ]a,b[ is uniformly continuous. More generally, let {]ai,bi[ : iI}be a family of open intervals in Rsuch that

R= {]ai,bi[ : i I}.

We can assert that a partial function f: R R admits a continuous extension f with

dom(f) = Rif and only if the restrictions of this fto all sets

dom(f) ]ai,bi[ (i I)

are uniformly continuous.

The proof of this fact is quite easy. It suffices to take into account that any continuous

real-valued function defined on a closed bounded subinterval ofRis uniformly continuous.

Also, it is not hard to give an example of a partial function f: RRfor which there exists

a countable family {[ai,bi] : i I} of segments such that

R= {[ai,bi] : i I}

and all restrictions f|(dom(f) [ai,bi]) are uniformly continuous, but fdoes not admit a

continuous extension f with dom(f) = R.

The next example is far from trivial, however.

Example 3. Let X and Y be two metric spaces and let Y be complete. Suppose also that

f:X Y is a continuous partial mapping. Then there exists a continuous partial mapping

f : X Y extending fand defined on some G-subset ofX; consequently, f can be

extended to a Borel mapping acting from the wholeX into Y.

This result is due to Lavrentiev and has numerous applications in descriptive set theory and

general topology (see [58], [148], [157], and Exercise 13 for Chapter 8 of this book).

Example 4. Recall that a partial function f: R R is upper (respectively, lower) semi-

continuous if for each t R, the set

{x dom(f) : f(x) < t}

(respectively, the set {x dom(f) : f(x) > t}) is open in dom(f). For fwith dom(f) = R,

this definition is equivalent to the following: fis upper (respectively, lower) semicontinu-ous if and only if

limsupyxf(y) = f(x)


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(respectively, liminfyxf(y) = f(x)) for all x R(see, e.g., [58], [101], [148], and [183]).

It is interesting to notice that every bounded upper (lower) semicontinuous partial function

admits a bounded upper (lower) semicontinuous extension defined on R. Let us formulatea more precise result in this direction.

First, recall that a partial function g: R R is locally bounded from above (from below)

if for each pointx R, there exists a neighborhoodU(x) such that g|U(x) is bounded from

above (from below).

A partial function g: R R is locally bounded if it is locally bounded from above and

from below simultaneously.

We also recall that an upper semicontinuous function can take its values from the set R

{} and a lower semicontinuous function can take its values from the set R {+}.

(These assumptions are convenient in numerous topics of mathematical analysis.)

Now, let f: R R{,+} be any partial function. The following two assertions are

equivalent:

(a) fadmits an upper (lower) semicontinuous extension f whose domain coincides with

the closure of dom(f);

(b) fis upper (lower) semicontinuous and locally bounded from above (from below).

The equivalence of (a) and (b) implies the validity of the next two statements.

(i) Let f: R [a,b] be a partial upper semicontinuous function. Then there exists an upper

semicontinuous function f : R [a,b] extending f.

(ii) Let f: R [a,b] be a partial lower semicontinuous function. Then there exists a lower

semicontinuous function f : R [a,b] extending f.

It should be mentioned that the same results hold true in a more general situation, namely,

for partial semicontinuous bounded functions acting from a normal topological space E

into the real line R. Of course, in this generalized case, the Tietze-Urysohn theorem shouldbe applied to E in order to obtain the required result.

The next example deals with monotone extensions of partial functions acting from R into

R.

Example 5. Let f: R R be a partial function increasing on its domain. It is easy to

show that fcan always be extended to an increasing function f defined on some maximal

(with respect to inclusion) subinterval of R. Let us denote the above-mentioned maximal

subinterval by T and let

a= inf(T), b = sup(T).

Ifa = and b = +, then f is the required increasing extension of fdefined on the


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whole R. Ifa = , then in view of the maximality ofT, we must have

inf{f(t) : t T} =

and, therefore, fcannot be extended to an increasing function acting from Rinto R. Anal-

ogously, ifb = +, then in view of the maximality ofT, we must have

sup{f(t) : t T} = +

and therefore fcannot be extended to an increasing function acting from Rinto R. We see

that in both of these cases our partial function fis not locally bounded.

A similar result holds true for any decreasing partial function f: R R. We thus obtain

a necessary and sufficient condition for extending a given partial function to a monotonefunction acting from Rinto R. Namely, the following two assertions are equivalent:

(a) fis extendable to a monotone function f with dom(f) = R;

(b) fis monotone and locally bounded.

Of course, there is no problem connected with extending monotone partial functions if we

admit infinite values of functions under consideration (cf. Example 4). Indeed, in such a

case any monotone partial function

f: R R{,+}

can be extended to a monotone function f with dom(f) = R.

Example 6. Let f: RRbe a partial function. Suppose that fis Borel on its domain, i.e.,

for every Borel setBR, the pre-image f1(B) is a Borel subset of dom(f) where dom(f)

is assumed to be endowed with the induced topology. It can be proved that falways admits

a Borel extension f : R R with dom(f) = R. However, the proof of this fact is far

from being easy. It needs a certain classification of all Borel partial functions acting from

R into R. This classification is due to Baire (see [3], [4], and [5]). According to it, any

Borel partial function fhas its own Baire order= (f), where is some countable

ordinal number. For instance, the equality (f) = 0 simply means that fis continuous on

its domain. Taking into account the above-mentioned classification and Lavrentievs result

mentioned earlier (see Example 3), the existence of f can be established by using the

method of transfinite induction on (for more details, see [148] or Exercise 16 for Chapter

8).

Example 7. Let f: R Rbe a partial function. The following two assertions are equiva-

lent:

(a) fadmits an extension f defined on Rand measurable in the Lebesgue sense;


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(b) there exists a Lebesgue measure zero set A Rsuch that the restriction of fto the set

dom(f)\A is a Borel function on dom(f)\A.

This fact can be proved by using the well-known Luzin criterion for the Lebesgue measur-ability of real-valued functions (see, e.g., [161], [183], and [192]).

A parallel fact holds true for partial functions having the Baire property. Similarly to the

measurability in the Lebesgue sense, for a partial function f: R R, the following two

assertions are equivalent:

(c) fadmits an extension f defined on Rand having the Baire property;

(d) there exists a first category set BRsuch that the restriction of fto the set dom(f)\B

is a Borel function on dom(f)\B.

In connection with the latter fact, let us remark that ifX is a second category subset of

R, then there always exists a function f: X R which cannot be extended to a function

f : R R having the Baire property. This deep result is due to Novikov (see [188]). It

is essentially based on the Axiom of Choice and some special facts from descriptive set

theory (e.g., the separation principle for analytic sets). A detailed discussion of this result

is also given in Chapter 14 of [122].

Example 8. Ifg: R R is a Lebesgue measurable function (respectively, function hav-ing the Baire property), then there exists a nonempty perfect set P R such that g|P is

monotone on P(see Chapter 20 of this book).

Sierpinski and Zygmund proved in their joint work [225] that there exists a function

h : R R

satisfying the following condition: for any set X Rof cardinality continuum, the restric-

tion h|X is not continuous.

In particular, this condition readily implies that for any setXRof cardinality continuum,

the restriction h|X is not monotone on X. Indeed, it suffices to use the fact that the set of

all discontinuity points of any monotone partial function acting from R into R is at most

countable.

Assuming the Continuum Hypothesis or, more generally, Martins Axiom (see [10], [40],

[67], [91], and [145]), we have the following two statements.

(a) IfX Ris of second category, then h|X cannot be extended to a function on Rhaving

the Baire property (cf. Novikovs result mentioned above).

(b) IfX Ris of strictly positive outer Lebesgue measure, then h|X cannot be extended to

a function on Rmeasurable in the Lebesgue sense.


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Notice that the validity of (a) and (b) does not need the full power of Martins Axiom.

Actually, it suffices to assume that any subset of R of cardinality strictly less than c is of

first category (respectively, of Lebesgue measure zero) in R.

Statement (a) directly implies that no Sierpinski-Zygmund function has the Baire property.

Statement (b) directly implies that no Sierpinski-Zygmund function is measurable in the

Lebesgue sense. Moreover, we can assert that a Sierpinski-Zygmund function is nonmea-

surable with respect to the completion of any nonzero -finite diffused (i.e., vanishing at

all singletons) Borel measure on R(see Exercise 2 for Chapter 13).

In addition to the above, no Sierpinski-Zygmund function is countably continuous. (A

partial function f: RRis called countably continuous if dom(f) admits a representation

in the form dom(f) = {Xn : n< }, where all restrictions f|Xn (n< ) are continuous.)

Sierpinski-Zygmund functions have other interesting and important properties. Many

works were devoted to functions of Sierpinski-Zygmund type (see [7], [42], [123], [124],

[148], [185], and [199]). In the sequel, we will be dealing with Sierpinski-Zygmund

functions possessing some additional properties which are of interest from the measure-

theoretical point of view and are closely connected with the measure extension problem

(see Chapters 13 and 14).

Example 9. Consider the set R as a vector space over the field Q of all rational numbers.

Let f: Q Q denote the identity mapping. Since Q is a vector subspace of R (actually,

Q can be treated as a line in the infinite-dimensional vector space R) and fis a partial

linear functional, it admits a linear extension

f : R Q

with dom(f) = R. Of course, the construction of f is not effective because it is based on

the Axiom of Choice or on the Zorn lemma (cf. Exercise 5 for this chapter). In fact, the

obtained extension f is a solution of the Cauchy functional equation, that is we have

f(x+y) = f(x) + f(y)

for all x Rand y R. At the same time, taking into account the relation

ran(f) = Q,

we see that f is discontinuous at all points ofRor, equivalently, f is a nontrivial solution

of the Cauchy functional equation (cf. [18], [82], [108], [143], and [176]). It is well

known that all nontrivial solutions of the Cauchy functional equation are nonmeasurable


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with respect to the Lebesgue measure on Rand do not possess the Baire property (see, for

instance, [143]). In our case, it can be observed that the set

X= {x R: f(x) = 0}

is not Lebesgue measurable and does not have the Baire property. Notice also that among

nontrivial solutions of the Cauchy functional equation, we can encounter some Sierpinski-

Zygmund functions (see Chapters 13 and 14). In addition, it should be pointed out that

certain nontrivial solutions of the Cauchy functional equation are successfully applied in

some deep geometrical questions concerning equidecomposability of polyhedra lying in

a finite-dimensional Euclidean space (see [18] where Hilberts third problem and related

topics are discussed in detail).

Example 10. In fact, the preceding example is purely algebraic. Another example of a

similar type is the following. Let (G,+) be a commutative group and let (H,+) be a

divisible commutative group, i.e., any equation of the form

nx = h (n N\{0}, h H)

has a solution in H. Suppose that a partial homomorphism : G H is given, which

means that is a homomorphism from some subgroup ofG into H. Then we can assert

that there always exists a homomorphism : GH extending (see [70], [83], [152],

or Exercise 2 in Appendix 5).

The above-mentioned result is extremely useful in all theory of commutative groups. For

instance, by applying it, we may readily prove that every commutative group can be isomor-

phically embedded in some divisible commutative group. Since all divisible commutative

groups admit a visual algebraic characterization (cf. [70] or [152]), the importance of this

result becomes quite evident.

An analogous statement concerning extensions of partial continuous homomorphisms can

be formulated in the case of a compact commutative group (G,+) and the one-dimensional

unit torus (S1, ) which is a canonical representative of the class of all divisible commutative

compact groups (see [83], [177], and [202]).

Example 11. We may expect that if a partial function f: R R is defined on a small

(in an appropriate sense) subset of R, then fadmits extensions with nice properties as

well as extensions with bad properties. Indeed, if dom(f) is of Lebesgue measure zero

(respectively, is offirst category), then ftrivially can be extended to a Lebesgue measurable

function (respectively, to a function possessing the Baire property). Thus, in both of these

cases, we come to extensions with nice properties.


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On the other hand, it is possible to present an example of an extension of a partial func-

tion which is defined on a small subset of R, with an extremely bad property from the

measure-theoretical view-point. For this purpose, let usfi

rst introduce two auxiliary no-tions which play a significant role in the sequel.

We recall that a set X R is universal measure zero if there exists no nonzero -finite

diffused Borel measure on X.

Evidently, ifX R is countable, then X is universal measure zero. The question of the

existence of uncountable universal measure zero subsets of R was posed many years ago

and turned out to be closely connected with some rather delicate set-theoretical construc-

tions due to Hausdorff, Mahlo, Luzin and other authors. For instance, every Luzin subset

of R is uncountable and universal measure zero (various interesting properties of Luzin

sets are considered in [143], [148], [159], [176], and [192]; see also Chapters 5, 12 and

Appendix 1).

We shall say that a functiong: RRis if there exists no nonzero-finite diffused measure

on Rsuch that gis measurable with respect to . (It is assumed in this definition that the

domain ofmay be an arbitrary -algebra of subsets ofR, containing all singletons.)

We thus see that absolutely nonmeasurable functions (if they exist) are of more pathological

nature than well-known examples of Lebesgue nonmeasurable real-valued functions.

It can be proved that the following two assertions are equivalent:

(a) a functiong: R Ris absolutely nonmeasurable;

(b) the set ran(g) is universal measure zero and for each t R, the set g1(t) is at most

countable.

For the proof of the equivalence (a) (b), see Chapter 5.

Starting with this result, it is not hard to show that an absolutely nonmeasurable function

acting from Rinto Rexists if and only if there exists a universal measure zero subset ofRof cardinality continuum. Also, we can infer the validity of the next statement.

Under the Continuum Hypothesis (or, more generally, under Martins Axiom), for a partial

function f: R R, the following two assertions are equivalent:

(c) fadmits an absolutely nonmeasurable extension f with dom(f) = R;

(d) the set ran(f) is universal measure zero and the sets f1(t) are countable for all t R.

From the equivalence of (c) and (d) we also obtain, under the same additional set-theoretical

assumptions, that any partial function f: RRdefined on a countable subset ofRadmits

an extension to an absolutely nonmeasurable function defined on the whole R.

Let us remark that according to Blumbergs fundamental theorem [15], any function acting


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from Rinto Rcan be regarded as an extension of some continuous partial function whose

domain is a countable everywhere dense subset of R (for the proof, see [15] or Exercises

21 and 22 of Chapter 8).The existence of a Sierpinski-Zygmund function h : R R shows that under the Contin-

uum Hypothesis, there is no uncountable set X R such that h|X is continuous and, con-

sequently, h cannot be considered as an extension of a continuous partial function defined

on an uncountable subset of R.

On the other hand, according to a recent result of Roslanowski and Shelah [208], it is

consistent with ZFC that any function f: R R may be regarded as an extension of a

continuous function defined on a Lebesgue nonmeasurable subset of R which, obviously,

is necessarily uncountable.

Let us also notice that under Martins Axiom, there exist additive absolutely nonmeasurable

functions which simultaneously are Sierpinski-Zygmund functions. One construction of

such functions will be given later in this book (see Chapter 13). It is based on the fact that

there exists a generalized Luzin subset of R which simultaneously is a vector space over

the field Q of all rational numbers.

The next example is concerned with extensions of real-valued partial functions of two vari-

ables.

Example 12. Let (= 1) denote the Lebesgue measure on the real line R(= R1). Con-

sider a function of two real variables

: R [0,1] R.

Recall that satisfies the Caratheodory conditions if the following two relations hold:

(i) for each x R, the function (x, ) : [0,1] Ris continuous;

(ii) for each y [0,1], the function(,y) : R Ris -measurable.

Functions of this type play a prominent role in mathematical analysis, the theory of ordinary

differential equations, optimization theory, and probability theory.

It is well known that if satisfies the Caratheodory conditions, then is measurable

with respect to the product -algebra dom()B([0,1]), where B([0,1]) denotes the

-algebra of all Borel subsets of[0,1].

Now, take a partial function of the form

F : R [0,1] R,

i.e., suppose that dom(F) R [0,1]. In addition, suppose that F is measurable with

respect to the product -algebra dom()B([0,1]). Then the following two assertions

are equivalent:


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(a) F admits an extension F : R [0,1] R satisfying the Caratheodory conditions and

such that dom(F) = R [0,1];

(b) for each x R, the partial functionF(x, ) is uniformly continuous on its domain.This result is rather deep. (The reader may compare it with the simplest Example 2.)

Indeed, to establish the equivalence of (a) and (b), we have to apply the Choquet theorem on

capacities and the theorem on measurable selectors due to Kuratowski and Ryll-Nardzewski

(see Appendix 2).

It should be mentioned that the same result remains true if we replace the Lebesgue measure

space (R,dom(),) by an arbitrary -finite complete measure space (,dom(),),

simultaneously replace the segment [0,1] by a compact metric space Y and consider partial

functions of the form

F :Y R.

For more details, see Exercise 13 of Appendix 2. Notice also that the standard Wiener

process

W : R[0,1] [0,1] R,

which is regarded as a reasonable mathematical model of the Brownian motion, yields a

good example of a function of two variables, satisfying the Caratheodory conditions. Here

R[0,1] stands, as usual, for the space of all real-valued functions defined on the segment

[0,1], and this space is assumed to be equipped with the completion of the Wiener proba-

bility measure w. More precisely, we are dealing with a certain set R[0,1] such that:

(c) w() = 1;

(d) for any t [0,1], the partial functionW(, t) is measurable with respect to w;

(e) for any , the trajectoryW(, ) is continuous on [0,1].

Extensive information about stochastic processes and corresponding probability measures

may be found in [16], [26], [52], [187], [205], and [226].

Now, we would like to consider a more difficult special case of the problem of extending

partial functions. In fact, our last example will be concerned with extensions of real-valued

partial set-functions.

We recall that a set-function is any function whose domain is some family of sets. Equiva-

lently, we may say that a set-function fis a function whose domain is a subset of the power

set P(E) of some base set E. Thus, fcan be treated as a partial function acting from theset E =P(E). Of course, partial functions of the form

f:P(E) R


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are of prime interest in this book because they include measures and measure type func-

tionals on E, for instance, capacities in the Choquet sense (see Appendix 2).

Example 13. The Lebesgue measure is a real-valued function defined on some class of

subsets ofR. It is well known that the class dom() is properly contained in P(R), i.e.,

there are -nonmeasurable sets in R. However, the proof of this fact needs uncountable

forms of the Axiom of Choice. Thus, may be regarded as a partial function acting from

the power set ofRinto R{+}, i.e., is a partial function of the form

:P(R) R{+}.

Several constructions of

-nonmeasurable subsets ofRare presented in the literature which

give us Vitali sets, Bernstein sets, nontrivial ultrafilters in the set N of all natural num-

bers, and other pathological sets of real numbers. Compare also Example 9, where the

-nonmeasurable set X associated with a certain nontrivial solution of the Cauchy func-

tional equation was pointed out.

It is natural to ask whether there exists a measure on R extending and defined on the

family of all subsets of R. This problem was originally posed by Banach and then was

studied by many famous mathematicians. Under some additional set-theoretical assump-

tions (e.g., the Continuum Hypothesis or Martins Axiom), the answer is negative (see, for

instance, [67], [69], [78], [79], [91], [192], [222], [224], and [238]). But it seems that this

question is undecidable within the standard system of axioms of contemporary set theory,

e.g., within the Zermelo-Fraenkel formal system ZFC.

It can be shown (see [1], [13], and Chapter 2) that for any countable disjoint family {Xi :

i I} of subsets ofR, there exists a measure on Rextending and such that

{Xi : i I} dom().

We thus see that the countable additivity property can be preserved by under the assump-

tion that the points Xi (i I) onto which we extend are pairwise disjoint. Actually, it

was established in [1] and [13] that the same result remains valid for any -finite measure

given on an abstract set E and for any disjoint family {Yj : j J} of subsets ofE. In

particular, we easily obtain from this result that for every finite family {Z1,Z2,...,Zn} of

subsets ofE, there exists a measure extending and satisfying the relation

{Z1,Z2,...,Zn} dom(

).

However, if we have an arbitrary infinite sequence {Z1,Z2,...,Zn,...} of subsets ofE, then

we cannot assert, in general, that is extendable to a measure for which all these subsets


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become -measurable. In other words, sometimes there are countably many points in

the power set ofE, which all together do not admit further extensions of a given nonzero

-finite measure . For instance, we always have such bad points Z1,Z2,...,Zn,... in thepower set ofEwhere E is an arbitrary uncountable universal measure zero subset ofR.

It should be noticed that the existence of an uncountable universal measure zero set E R

is well known and can be proved within ZFC theory. In Appendix 1 of the present book

we give a construction ofE starting with the classical Sierpinski partition of the product

set 11, where 1 denotes, as usual, the least uncountable cardinal number.

In this context, the works [197] and [250] should also be mentioned, in which analogous

and stronger results are obtained stating the existence of uncountable universally small

subsets ofR.

Finishing this chapter, we hope that the examples just considered and concerning extensions

of partial real-valued functions are sufficiently illustrative to show the reader the importance

of the problem of extending partial functions. We will continue our discussion of this

problem in the following sections. As already said, the measure extension problem touched

upon in Example 13 will be of special interest in our further considerations.

EXERCISES

1. Give a proof of the statement presented in Example 1. Namely, verify that the following

two assertions are equivalent within ZF theory:

(a) the Axiom of Choice;

(b) if{Xi : i I} is an arbitrary family of nonempty sets, then any partial selector of{Xi :

i I} can be extended to a selector of this family.

Moreover, by using AC demonstrate that every infinite set contains a countably infinite

subset. (Note that this fact is deducible with the aid of some weak forms of AC but is not

provable within ZF theory.)

2. Let ]a,b[ be an open subinterval of R and let f: ]a,b[ R be a convex function on

]a,b[. Show that the following two statements are equivalent:

(1) there exists a convex extension f : RRof fdefined on the whole R;

(2) fadmits a continuous extension defined on [a,b] and this extension has the right-hand

side derivative at a and the left-hand side derivative at b.

Notice that the above-mentioned derivatives are assumed to be finite.

3. Give a proof of the equivalence (a) (b), where (a) and (b) are assertions from Example

4.


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4. Give a proof of the equivalence (a) (b), where (a) and (b) are assertions from Example

5.

5. Show that the set X= {x R : f(x) = 0} described in Example 9 is not Lebesgue

measurable and does not possess the Baire property.

6. Let (E,d) be a metric space and let f: E R be a partial function satisfying the

Lipschitz condition with a Lipschitz constant L, that is

|f(x) f(y)|Ld(x,y) (x dom(f),y dom(f)).

Prove that there exists an extension f :E Rof falso satisfying the Lipschitz condition

with the same constant L and defi

ned on the whole space E.

7. Let f: R R be a function and let x R. Recall that t R is a Dini right derived

number of fat x if there exists a sequence {hn : n N} of strictly positive real numbers

such that

limn+hn = 0, t= limn+(f(x+hn) f(x))/hn.

In this case, the notation tD+f(x) is frequently used.

Let f: R R be a continuous function. Suppose that for any x R, this f admits a

nonnegative Dini right derived number at x. Show that fis increasing on R.

Starting with this result, prove that the following two assertions are equivalent for a contin-

uous function facting from Rinto R:

(a) fsatisfies the Lipschitz condition;

(b) there is a constant L 0 such that for any point x R, the function fhas at least one

Dini right derived number whose absolute value does not exceed this L.

8. Let Bn denote the unit ball in the Euclidean space Rn and let Sn1 denote the boundary

ofBn. Show that the following two assertions are equivalent:

(a) the identity embedding ofSn1 into Bn does not admit a continuous extension defined

on the whole Bn;

(b) Bn has the fixed-point property which means that for every continuous mapping f:

BnBn, there exists a point x Bn such that f(x) =x.

As is well known, assertion (b) is valid and was first proved by Brower (for the proof,

see [56], [58], and [148]). Consequently, (a) is valid, too. Browers theorem (b) is very

deep, has numerous applications, and was generalized in many directions (see, e.g., [56]

and [57]). One of important generalizations of this theorem is due to Kakutani and states

the existence offixed-points for certain set-valued mappings.


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9. Show that every partial isometry f: Rn Rn admits an extension to an isometry f :

Rn Rn. Find necessary and sufficient conditions under which f is uniquely determined

by f.On the other hand, for any infinite-dimensional real Hilbert space E, give an example of a

partial isometry g: EEwhich does not admit an extension to an isometry of the whole

E.

10. A partially ordered set (E,) is called complete if for any set X E, there exists

sup(X) (equivalently, there exists inf(X)).

Let (E,) be a complete partially ordered set and let : EE be a monotone mapping

(i.e., is either increasing or decreasing). Show that there exists a fixed-point of; in otherwords, show that

(y E)((y) = y).

For this purpose, assume without loss of generality that is increasing and put

Y= {x E : x (x)}, y = sup(Y).

Check that (y) = y, i.e., y is the desired fixed-point of.

This result is due to Tarski and is known as Tarskisfi

xed-point theorem.

11. Let a function : R [0,1] R satisfy the Caratheodory conditions (see Example

12 of this chapter). Prove that is measurable with respect to the product -algebra

dom()B([0,1]).

12. Let (E,S,) be a -finite measure space and let

S0 = {XS : (X) < +}.

Consider inS

0 (which is a -ring of subsets ofE) the equivalence relationR(X,Y) defi

nedby the formula

R(X,Y) (XY) = 0,

where the symbol stands for the operation of symmetric difference of two sets, that is

XY= (X\Y) (Y\X).

Let S0/R denote the corresponding quotient set and let, for any XS0, the symbol [X]

denote the R-equivalence class containingX. Verify that the function

d([X], [Y]) = (XY) ([X] S0/R, [Y] S0/R)

is a metric on S0/R such that the metric space (S0/R,d) is complete.


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This (S0/R,d) is usually called the metric space canonically associated with the given

measure . The topological weight of this space (see Appendix 3) is a certain inner char-

acteristic of. Denote it by w().Check that ifw() , then w() is equal to the topological weight of the Hilbert space

L2() consisting of all real-valued square integrable (with respect to ) functions on E.

A measure is called separable if w() . Otherwise, is called a nonseparable

measure.

Demonstrate that ifS is a countably generated -algebra of subsets ofE, then is a

separable measure. Also, give an example which shows that the converse assertion is not

valid.

Observe that the completion of a separable measure is separable, too. Infer from this fact

that the Lebesgue measure n on the Euclidean space Rn is separable.

As mentioned in Example 13 of this chapter, the one-dimensional Lebesgue measure

= 1 cannot be extended (within ZFC theory) to a universal measure defined on P(R).

However, there exist nonseparable extensions of which are invariant under the group of

all translations ofR(and even under the group of all isometries ofR).

Nonseparable -finite measures with some specific properties will be considered in Chap-

ters 15, 16, and 17.


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Chapter 2

Some aspects of the measure extension problem

In this chapter, we concisely present several versions of the general measure extension

problem which is a special case of the problem of extending partial real-valued functions,

envisaged in the preceding chapter. Undoubtedly, the measure extension problem is impor-

tant in many questions of analysis and probability theory. Among various aspects of this

problem the following three should be especially distinguished: purely set-theoretical, al-

gebraic, and topological. Below, we will touch upon each of the above-mentioned aspects

and will show their specific features. Since the measure extension problem has its origins

in real analysis, namely, in classical theory of the Lebesgue measure (= 1) on the realline R (= R1), we also schematically consider several constructions of proper extensions

of and compare such extensions to each other. However, a more detailed consideration

of those constructions will be given in subsequent sections.

Let E be a set, S be an algebra of subsets ofE, and let be a nonzero -finite measure

on S. The general measure extension problem is to extend onto a maximally large class

of subsets ofE. This problem was originally formulated at the end of the 19th century,

within the theory of real-valued functions. As is well known, it was partially solved by

Lebesgue [158] at the beginning of the 20th century. Namely, starting with the classical

Jordan measure, Lebesgue gave a construction of his measure for the real line R and,

in a similar way, he introduced his measure n for the n-dimensional Euclidean space Rn

(see [80], [158], [183], [192], and [210]). Actually, Lebesgues above-mentioned construc-

tion extends the one-dimensional (respectively, multi-dimensional) Riemann integral to the

one-dimensional (respectively, multi-dimensional) Lebesgue integral. The latter integral

allows operation with a sufficiently large class of real-valued functions and turns out to be

a necessary research tool in various problems of modern analysis and probability theory.

Then Caratheodorys fundamental theorem followed, which deals with -finite measures

given on abstract measurable spaces of type (E,S). Let us recall the formulation of this

A.B. Kharazishvili, Topics in Measure Theory and Real Analysis, Atlantis Studies in Mathematics 2,

DOI 10.1007/978-94-91216-36-7_ , 2009 Atlantis Press/World Scientific

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classical result.

Theorem 1. Any -finite measure defined on an algebra S of subsets of E admits a

unique measure extendingand defined on the generated-algebra (S).

Here we omit the standard proof of Caratheodorys theorem (see [16], [56], [65], [80], [83],

[194], and [210]). This theorem shows that, without loss of generality, we can consider only

those -finite measures which are defined on -algebras of sets.

Afterwards, the measure extension problem found important applications in many other do-

mains of mathematics, such as axiomatic set theory, general topology, functional analysis,

probability theory, and the theory of stochastic processes.

A sufficiently general method of extending measures was suggested by Marczewski (Szpil-

rajn) in his classical works [234] and [235]. We would like to describe this method because

it will be used many times in our further considerations.

Namely, let E be a set, be a nonzero -finite complete measure on some -algebra of

subsets ofE, and letI be a -ideal of subsets ofE such that

(YI)((Y) = 0),

where usually stands for the inner measure associated with . Denote

S= dom()

and consider the -algebraS of subsets ofE, generated bySI, i.e.,

S = (SI).

Obviously, any set ZS can be represented in the form

Z= (XY1)\Y2,

where XS and both Y1 and Y2 are some members ofI. Let us put

(Z) = ((XY1)\Y2) = (X).

It can easily be checked that the functional is well defined on S in the sense that the

value (Z) does not depend on the above-mentioned representation ofZ, and turns out

to be a measure onS extending . Also, it directly follows from the definition of that

(YI)(

(Y) = 0).

Clearly, the measure is a proper extension of if and only if there exists a -

nonmeasurable set belonging to the -idealI.


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Some aspects of the measure extension problem 21

Observe that this construction generalizes the standard procedure of obtaining the comple-

tion of.

If we have a single set YE with (Y) = 0, then we can extend a given measure to ameasure in such a way that (Y) = 0. Indeed, it suffices to consider the -ideal I of

subsets ofE, generated by the one-element family {Y}, that is

I= {Z: ZY},

and to apply to thisI the Marczewski method described above.

The construction of is, in fact, purely set-theoretical because no specific properties of

(E,S,) are utilized here. Notice that a purely set-theoretical, or abstract, aspect of themeasure extension problem was deeply investigated by famous representatives of the Polish

mathematical school (Banach, Kuratowski, Sierpinski, Ulam, Tarski, and Marczewski).

They obtained a number of fundamental results in this direction which have stimulated

further remarkable investigations in the subject by other mathematicians (see, e.g., [69],

[91], [144], [150], and [231]). The structure of so-called large cardinals was deeply studied

in those investigations.

In particular, according to Ulams classical theorem [238], it is consistent with ZFC theory

that the domain of any extension of a nonzero -finite diffused (i.e., vanishing at all

singletons) measure given on an uncountable set E cannot coincide with the power set

P(E) ofE. Consequently, there always exists a set XE such that X dom(). Then

can be extended to a measure so that X becomes -measurable and, in general, there

are various possibilities to construct such an extension (see Exercise 1 for this chapter).

Thus, by starting with the assumption that there are no large cardinals, we can infer that

there are no maximal extensions of the original nonzero -finite diffused measure . Ex-

tensive information about large cardinals and their combinatorial properties can be found

in [10], [69], [91], [145], and [150] (see also Appendix 1).

For our purposes, we need to look at one important result on extensions of -finite mea-

sures, obtained in the works [1] and [13].

Theorem 2. Let E be a set, be a -finite measure on E and let {Xi : i I} be an

arbitrary disjoint family of subsets of E. Then there exists a measure on E extending

and satisfying the relation

{Xi : i I} dom().


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Proof. Obviously, we may suppose (without loss of generality) that the given family {Xi :

i I} is a partition ofE.

Let us first consider the case where the partition {Xi : i I} is countable, i.e., we have

card(I) card(N) = .

Let {ti : i I} be an injective family of real numbers and let f: E R be a step-function

such that ran(f|Xi) = {ti} for any i I. Clearly, it suffices to show that there exists an

extension of for which fbecomes measurable.

Without loss of generality, we may assume that eitherI= {1,2,...,m} orI= N = . Under

this assumption, let us put:

Xn = a -measurable hull ofXn, where n I;

Yn = Xn \ ({X

k : k< n}), where n I.

Obviously, the family of sets {Yn : n I} is a disjoint covering ofE. Define a new function

f0 : E R

by putting f0(x) = tn if and only if x Yn. Since I is countable and all sets Yn (n I) are

-measurable, it immediately follows from the definition of f0 that f0 is a -measurable

function. Let us show that

({x E : f(x) = f0(x)}) = 0.

Suppose otherwise, that is

({x E : f(x) = f0(x)}) > 0.

Then there exists an index n I such that

(Yn{x E : f(x) = f0(x)}) > 0.

On the other hand, it is easy to verify the following inclusion:

Yn{x E : f(x) = f0(x)} Xn \Xn,

which gives a contradiction with the definition of the measurable hull Xn of Xn. The con-

tradiction obtained establishes the required equality

({x E : f(x) = f0(x)}) = 0.

But, by virtue of the Marczewski method of extending -finite measures, the set

X = {x E : f(x) = f0(x)}


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can be made measurable with respect to some extension of and, moreover, this proce-

dure can be made so that (X) = 0. Consequently, fbecomes measurable with respect to

the same .Let now {Xi : iI}be an arbitrary partition ofE. Using the -finiteness ofwhich implies

the validity of the so-called countable chain condition, it is not difficult to show that there

exists a set JI satisfying the following two relations:

(a) card(I\J) ;

(b) for any countable set J0 J, we have ({Xj : j J0}) = 0.

Starting with (b) and applying again the Marczewski method, we first make all the sets

Xj (j J) to be measurable with respect to some extension of. Notice, by the way,

that according to this method, (Xj) = 0 for all j J. Finally, we apply our previous

argument to the countable disjoint family {Xi : i I\J} and extend to a measure

such that

{Xi : i I\J} dom().

It is clear now that = turns out to be the required extension ofwhich ends the proof

of the theorem.

Remark 1. Theorem 2 shows, in particular, that having any finite family of subsets ofE,

we can always extend to a measure , which makes all these subsets to be -measurable

(cf. also Exercise 1). On the other hand, it is well known that an analogous assertion fails to

be true for some countable families of subsets ofE, where card(E) = 1 (see Appendix 1).

Moreover, the existence of a Luzin set L R with card(L) = c readily implies that there

is a countably generated -algebra of subsets of R containing all Borel sets in R and not

admitting a nonzero -finite continuous (i.e., diffused) measure.

Recall that the existence of a Luzin set necessarily needs additional set-theoretical axioms.

In this context, it should be underlined that the existence of other small subsets ofRhaving

cardinality 1, can be established within the theory ZFC (see [69], [79], [113], [148],

[172], [197], and [250]). Some questions of this type will be discussed in more details in

our further considerations (see especially Appendix 1).

Another aspect of the measure extension problem has an algebraic (more precisely, group-

theoretical) flavor. Namely, suppose that E is an arbitrary set and denote by Sym(E) the

family of all bijections ofE onto itself. Obviously, Sym(E) becomes a group with respect

to the standard composition operation . This group is usually called the symmetric group

ofE. Any subgroup G ofSym(E) is called a group of transformations ofE.


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We shall say that the pair (E,G) is a space equipped with a transformation group if G is a

subgroup ofSym(E).

Spaces equipped with transformation groups can be frequently met in algebra, geometry,topology, mathematical analysis, the theory of dynamical systems, and in otherfields of

mathematics.

Let G be a group of transformations ofE and let L be a family of subsets ofE.

We shall say that L is invariant underG or, in short, G-invariant if

(g G)(XL)(g(X) L).

We shall say that a measure on E is invariant underG or, in short, G-invariant if dom()

is a G-invariant class of subsets ofE and

(g G)(X dom())((g(X)) = (X)).

More generally, we shall say that a measure on E is quasi-invariant underG or, in short,

G-quasi-invariant if dom() is a G-invariant class of subsets ofE and

(g G)(X dom())((g(X)) = 0 (X) = 0).

In many situations a base setE is a group with respect to some algebraic operation , i.e., the

pair(E, ) is an abstract group. Then the group G of all left (respectively, right) translationsof E is canonically isomorphic to (E, ) and can be identified with E. So we may write

(G,) instead of (E, ). In such a case, we deal with left invariant (left quasi-invariant)

measures on E and, analogously, with right invariant (right quasi-invariant) measures on

the same E. Actually, in many cases it suffices to consider only left invariant (left quasi-

invariant) measures on groups.

Obviously, if (E, ) is a commutative group, then the concepts of left invariant (left quasi-

invariant) and right invariant (right quasi-invariant) measures do not differ from each other.

Suppose that an uncountable group (G, ) is given and suppose that G is equipped with

a nonzero -finite left G-invariant (or, more generally, nonzero -finite left G-quasi-

invariant) measure . As stated by Kharazishvili [104] and Erdos and Mauldin [60], the

domain of such a cannot be identical with the family P(G) of all subsets of G. Notice

especially that this statement does not need any additional set-theoretical assumptions (see

proof sketched in Exercises 9, 10 and 11 of Appendix 1). In view of the above-mentioned

statement, it is natural to ask whether there exists a left G-invariant (left G-quasi-invariant)

measure on G properly extending . It is consistent with ZFC theory that the an-

swer to this question is positive (see Exercise 20 for Chapter 3) but it is still unknown

whether the same result can be obtained within ZFC. However, without using additional


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set-theoretical hypotheses, for some sufficiently wide classes of uncountable groups this

question has a positive answer even in terms of certain subgroups of the original group. In

particular, if (G, ) is an arbitrary uncountable solvable group, then there always exists a-nonmeasurable subgroup H of G such that can be extended to a left G-invariant (left

G-quasi-invariant) measure for which we have H dom() and (H) = 0. A more

detailed information about this relatively recent result can be found in Chapter 10.

Marczewskis method of extending -finite measures successfully works in the cases of

invariant and quasi-invariant measures but needs a slight modification (cf. [41], [42], [45],

[108], [115], [195], [242], [247], and [249]).

Namely, let E be a set, G be a group of transformations of E, and let be a -finite com-

plete G-invariant (G-quasi-invariant) measure on some G-invariant-algebra of subsets of

E. Suppose that a G-invariant -idealI of subsets ofE is given such that

(YI)((Y) = 0).

Again, let us denote

S= dom()

and consider the -algebraS of subsets ofE, generated bySI. It can easily be seen

that S is also a G-invariant -algebra of sets. We already know that any set ZS can

be represented in the form

Z= (XY1)\Y2,

where XS and both Y1 and Y2 are some members ofI. Let us put

(Z) = ((XY1)\Y2) = (X).

Again, it is not difficult to check that the functional is well defined in the sense that

the value (Z) does not depend on the above-mentioned representation ofZ, and turns

out to be a G-invariant (G-quasi-invariant) measure onS extending . In addition to this

circumstance, the equality (Y) = 0 is valid for all sets Y belonging toI.

Consider now a particular case, when we have a single set Y E with (Y) = 0 which

satisfies the relation

(g G)((g(Y)Y) = 0),

where stands for the outer measure associated with (any se

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