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Boris Makarov Anatolii Podkorytov

Real Analysis:Measures,Integrals andApplications

Boris MakarovMathematics and Mechanics FacultySt Petersburg State UniversitySt Petersburg, Russia

Anatolii PodkorytovMathematics and Mechanics FacultySt Petersburg State UniversitySt Petersburg, Russia

Translated from the Russian language edition:

Lekcii po vewestvennomu analizuby B.M. Makarov, A.N. Podkorytov (B.M. Makarov, A.N. Podkorytov)

Copyright BHV-Peterburg (BHV-Petersburg) 2011All rights reserved.

ISSN 0172-5939 ISSN 2191-6675 (electronic)UniversitextISBN 978-1-4471-5121-0 ISBN 978-1-4471-5122-7 (eBook)DOI 10.1007/978-1-4471-5122-7Springer London Heidelberg New York Dordrecht

Library of Congress Control Number: 2013940613

Mathematics Subject Classification: 28A12, 28A20, 28A25, 28A35, 28A75, 28A78, 28B05, 31B05,42A20, 42B05, 42B10

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Diagram of Chapter Dependency

v

Preface to the English Translation

This book reflects our experience in teaching at the Department of Mathematicsand Mechanics of St. Petersburg State University. It is aimed primarily at readersmaking their first acquaintance with the subject.

Lecture courses on measure theory and integration are often confined to abstractmeasure theory, with little attention paid to such topics as integration with respectto Lebesgue measure, its transformation under a diffeomorphism and so onthatis, topics that are more special but no less important for applications. Believingthat such reticence is counterproductive, we choose an approach that avoids it andcombines general notions with classical special cases.

A substantial part of the book is devoted to examples illustrating the obtainedresults both in and beyond the framework of mathematical analysis, in particular, ingeometry. The exercises appearing at the end of almost every section serve the samepurpose.

In the English translation we use three-digit numbers for sections. The first digitrefers to a Chapter, the second to a Section within the Chapter and the third toSubsection. When referencing to a statement we give the number of a Subsectionwhich contains it. E.g., Lemma 7.5.4 would mean a lemma from Sect. 7.5.4.

Comparing with Russian edition, we have extended the book by adding, in par-ticular, the new Sects. 6.1.3 and 6.2.6.

Taking into account the difference between curricula in Russia and the West,as well as the considerable volume of our book, we think it necessary to say sev-eral words about how to use it, and we draw the readers attention to the chapterdependency chart. A reader interested only in an introduction to the foundations ofmeasure theory and integration may prefer to read only those sections of Chaps. 15that are not marked with a . This symbol indicates sections that contain either someillustrative material (e.g., Sects. 2.8, 6.66.7, 7.27.3, 8.7, 10.2, 10.6), or some op-tional information that can be omitted in the first reading (e.g., Sects. 1.6, 4.11,5.55.6, 6.5, 7.4, 8.8, 10.4, 12.112.3), or else material used outside Chaps. 15(Sects. 2.6, 3.4, 4.9). The material of Sects. 1.11.4, 2.12.5, 3.13.2, 4.14.8, 5.15.4 can be taken as a basis for a two semester course on the foundations of measure

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viii Preface to the English Translation

theory and integration. Time permitting, the course can be extended by includingthe material of Sects. 6.16.2, 3.3, 4.9, 6.4.

The book can also be used for courses aimed at students familiar with the notionof integration with respect to a measure. There is a sufficiently wide choice of suchcourses devoted to relatively narrow topics of real analysis. For example:

The maximal function and differentiation of measures (Sects. 2.7, 4.9, 11.2, 11.3). Surface integrals (Sects. 2.6, 8.18.6). Functionals in spaces of measurable and continuous functions (Sects. 11.111.2,

Chap. 12). Approximate identities and their applications (Sects. 7.57.6, Chap. 9). Fourier series and the Fourier transform (Chaps. 9, 10). A course covering only the preliminaries of the theory of Fourier series and

the Fourier transform may be based, for example, on Sects. 9.1.19.1.3, 10.1.110.1.4, 10.3.110.3.6, 10.5.110.5.4.

Acknowledgments We are deeply indebted to Springer for publishing our bookand we are happy to see it reach a much wider audience via its English translation.We are grateful to V.P. Havin who attracted the publishers attention to the Russianedition of our book soon after its publication.

In the process of preparing and publishing the volume, we have been helpedby several people to whom we would like to express our thanks. In particular,the anonymous referees provided us with constructive comments and suggestions.B.M. Bekker, A.A. Lodkin, F.L. Nazarov and N.V. Tsilevich undertook the hard taskof translation. We were also most fortunate to receive feedback from A.I. Nazarov,F.L. Nazarov and O.L. Vinogradov, which helped us to improve the text in manyplaces. Our special thanks go to Joerg Sixt, the Springer Editor, for his invaluablehelp and encouragement in preparing this manuscript.

We also wish to acknowledge the help and support we received from our friends,families and colleagues who read and commented upon various drafts and con-tributed to the translation.

Our special thanks go to O.B. Makarova (who happened to be a granddaugh-ter of the first of co-authors) who very competently and patiently conducted ourcorrespondence related to publication of this book and helped us enormously withproofreading.

The translation was carried out with the support of the St. Petersburg StateUniversity program Function theory, operator theory and their applications6.38.78.2011.

Boris MakarovAnatolii Podkorytov

St Petersburg, Russia

Preface

Measure theory has been an integral part of undergraduate and graduate curriculain mathematics for a long time now. A number of texts in this subject area havebecome well-established and widely used. For example, one might recall books byB.Z. Vulih [Vu], A.N. Kolmogorov and S.V. Fomin [KF], not to mention the classi-cal monograph by P. Halmos [H]. However, books on measure theory typically treatit as an isolated subject, which makes it difficult to include it in a general course inanalysis in a natural and seamless way. For example, the invariance of the Lebesguemeasure is either omitted entirely, or considered as a special case of the invarianceof the Haar measure. Quite often, the question of how Lebesgue measure transformsunder diffeomorphisms is left out. On the other hand, most introductory courses onintegration are still based on the theory of the Riemann integral. As a result, the stu-dents are forced to absorb numerous, however similar, definitions based on Riemannsums corresponding to various situations, such as double integrals, triple integrals,line integrals, surface integrals and so on. They must also overcome the unneces-sary technical complications caused by the lack of a sufficiently general approach.Typical examples of such difficulties include justifying the change of the order ofintegration and taking limits under the integral sign.

For this reason, one often faces a two-tier exposition of the theory of integration,where at the first stage the notion of measure is not discussed at all, and later theelementary topics are never revisited, leaving the task of reconciling the variousapproaches to the student. The authors aim to eliminate this divide and provide anexposition of the theory of the integral that is modern, yet easily integrated into ageneral course in analysis. This encapsulates in a textbook the established practice atthe Department of Mathematics and Mechanics of the University of St. Petersburg.This practice is based on an idea introduced in the early 60s by G.P. Akilov and firstimplemented by V.P. Havin during the academic year 19631964.

The main emphasis of the book is on the exposition of the properties of theLebesgue integral and its various applications. This approach determined the styleof exposition as well as the choice of the material. It is our hope that the reader whomasters the first third of the book will be sufficiently prepared to study any area of

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x Preface

mathematics that relies upon the general theory of measure, such as, among others,probability theory, functional analysis and mathematical physics.

Applications of the theory of integration constitute a substantial part of this book.In addition to some elements of harmonic analysis, they also include geometricapplications, among which the reader will find both classical inequalities, such asthe BrunnMinkowski and isoperimetric inequalities, and more recent results, suchas the proof of Brouwers theorem on vector fields on the sphere based on a changeof variables, the K. Ball inequality and others. In order to illustrate the effectivenessand applicability of the theorems presented, and to give the reader an opportunityto absorb the material in a hands-on fashion, the book includes numerous examplesand exercises of various degrees of difficulty.

Pedagogical considerations caused us to refrain from stating some of the resultsin their full generality. In such cases, references to the appropriate literature areprovided for the interested reader. The notion of surface area is discussed in moredetail than is common in analysis texts. Using a descriptive definition, we prove itsuniqueness on Borel subsets of smooth and Lipschitz manifolds.

It is desirable that the reader be familiar with the notion of an integral of a con-tinuous function of one variable on an interval prior to being exposed to the basicsof measure theory. However, we do not feel that this prerequisite necessarily needsto be fulfilled in the context of the Riemann integral, which we view to be primarilyof historical interest. A possible alternative approach is outlined in Appendix 13.1.

This book is based on a series of lectures delivered by the authors at the De-partment of Mathematics and Mechanics of St. Petersburg State University. Themajority of the material in Chaps. 18 approximately corresponds to the fourth andfifth semester analysis program for mathematics majors in our department. The ma-terial from Chaps. 912 and some other parts of the book was previously includedby the authors in advanced courses and lectures in functional analysis. Some addi-tional information is presented in Appendices 13.213.6. Appendix 13.7, dedicatedto smooth mappings, is included for the sake of completeness.

The reader is expected to have the necessary mathematical background. The stu-dents entering the fourth semester at the Department of Mathematics and Mechanicsof St. Petersburg State University are familiar with multivariable calculus and basiclinear algebra. This prerequisite material is used throughout the book without anyadditional explanations. In Chap. 8, familiarity with the basics of smooth manifoldtheory is assumed. In Appendices 13.2 and 13.3, the rudiments of the theory ofmetric spaces are taken for granted.

The authors have previously encountered texts where a definition or notation,once introduced, is never repeated and is used without any further comments orreferences many pages later. We believe that such manner of presentation, possi-bly appropriate in monographs of an encyclopedic nature, puts too much strain onthe readers memory and attention span. Taking into account the fact that this is atextbook intended for relatively inexperienced readers, many of whom will be en-countering the subject matter for the first time, the authors find it useful to includesome repetitions and reminders. However, they are unable to measure the degree towhich they have succeeded in this direction.

Preface xi

In the process of writing this book, the authors have frequently sought ad-vice from their colleagues. The comments and suggestions of D.A. Vladimirov,A.A. Lodkin, A.I. Nazarov, F.L. Nazarov, A.A. Florinsky and V.P. Havin provedespecially useful. We are grateful to them as well as to A.L. Gromov, who kindlyagreed to produce computer generated graphics and K.P. Kohas, who handled thetype-setting of the book.

The chapters are numbered using Roman numerals. They are divided into sec-tions consisting of subsections which are numbered using two Arabic numerals.The first of these indicates the number of the section, and the second the number ofthe subsection. The subsections in Appendices are numbered by two numerals, oneRoman (Appendix number) and the other Arabic, with the addition of the letter Awhen referencing.

All the assertions contained in a given subsection are numbered in the same wayas the subsection itself. In the case of references within a given chapter, only thenumber of the subsection is indicated. For example, the reference by Theorem 2.1refers to a theorem in subsection 2.1 of a given chapter. When referencing materialfrom another chapter, the number of the chapter is also indicated. For example,the reference Corollary II.3.4 refers to a corollary contained in subsection 3.4 ofChapter II. The enumeration of the formulas is consecutive within each section. Theend of a proof is indicated by black triangle .

Contents

Basic Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvii

1 Measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Systems of Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Volume . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.3 Properties of Measure . . . . . . . . . . . . . . . . . . . . . . . . 171.4 Extension of Measure . . . . . . . . . . . . . . . . . . . . . . . . 231.5 Properties of the Carathodory Extension . . . . . . . . . . . . . . 311.6 Properties of the Borel Hull of a System of Sets . . . . . . . . . . 35

2 The Lebesgue Measure . . . . . . . . . . . . . . . . . . . . . . . . . . 412.1 Definition and Basic Properties of the Lebesgue Measure . . . . . 412.2 Regularity of the Lebesgue Measure . . . . . . . . . . . . . . . . 482.3 Preservation of Measurability Under Smooth Maps . . . . . . . . . 522.4 Invariance of the Lebesgue Measure Under Rigid Motions . . . . . 572.5 Behavior of the Lebesgue Measure Under Linear Maps . . . . . . 622.6 Hausdorff Measures . . . . . . . . . . . . . . . . . . . . . . . . 712.7 The Vitali Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 832.8 The BrunnMinkowski Inequality . . . . . . . . . . . . . . . . . 87

3 Measurable Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 953.1 Definition and Basic Properties of Measurable Functions . . . . . . 963.2 Simple Functions. The Approximation Theorem . . . . . . . . . . 1053.3 Convergence in Measure and Convergence Almost Everywhere . . 1083.4 Approximation of Measurable Functions by Continuous

Functions. Luzins Theorem . . . . . . . . . . . . . . . . . . . . . 117

4 The Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1214.1 Definition of the Integral . . . . . . . . . . . . . . . . . . . . . . . 1224.2 Properties of the Integral of Non-negative Functions . . . . . . . . 1254.3 Properties of the Integral Related to the Almost Everywhere

Notion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

xiii

xiv Contents

4.4 Properties of the Integral of Summable Functions . . . . . . . . . 1354.5 The Integral as a Set Function . . . . . . . . . . . . . . . . . . . . 1434.6 The Lebesgue Integral of a Function of One Variable . . . . . . . . 1484.7 The Multiple Lebesgue Integral . . . . . . . . . . . . . . . . . . . 1654.8 Interchange of Limits and Integration . . . . . . . . . . . . . . . . 1694.9 The Maximal Function and Differentiation of the Integral with

Respect to a Set . . . . . . . . . . . . . . . . . . . . . . . . . . . 1804.10 The LebesgueStieltjes Measure and Integral . . . . . . . . . . . 1884.11 Functions of Bounded Variation . . . . . . . . . . . . . . . . . . 197

5 The Product Measure . . . . . . . . . . . . . . . . . . . . . . . . . . 2055.1 Definition of the Product Measure . . . . . . . . . . . . . . . . . . 2055.2 The Computation of the Measure of a Set via the Measures of Its

Cross Sections. The Integral as the Measure of the Subgraph . . . . 2085.3 Double and Iterated Integrals . . . . . . . . . . . . . . . . . . . . 2155.4 Lebesgue Measure as a Product Measure . . . . . . . . . . . . . . 2255.5 An Alternative Approach to the Definition of the Product

Measure and the Integral . . . . . . . . . . . . . . . . . . . . . . . 2365.6 Infinite Products of Measures . . . . . . . . . . . . . . . . . . . . 239

6 Change of Variables in an Integral . . . . . . . . . . . . . . . . . . . 2436.1 Integration over a Weighted Image of a Measure . . . . . . . . . . 2436.2 Change of Variable in a Multiple Integral . . . . . . . . . . . . . . 2486.3 Integral Representation of Additive Functions . . . . . . . . . . . 2626.4 Distribution Functions. Independent Functions . . . . . . . . . . 2696.5 Computation of Multiple Integrals by Integrating over

the Sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2816.6 Some Geometric Applications . . . . . . . . . . . . . . . . . . . 2896.7 Some Geometric Applications (Continued) . . . . . . . . . . . . 296

7 Integrals Dependent on a Parameter . . . . . . . . . . . . . . . . . . 3077.1 Basic Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . 3077.2 The Gamma Function . . . . . . . . . . . . . . . . . . . . . . . . 3167.3 The Laplace Method . . . . . . . . . . . . . . . . . . . . . . . . 3307.4 Improper Integrals Dependent on a Parameter . . . . . . . . . . . 3597.5 Existence Conditions and Basic Properties of Convolution . . . . . 3767.6 Approximate Identities . . . . . . . . . . . . . . . . . . . . . . . . 384

8 Surface Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3958.1 Auxiliary Notions . . . . . . . . . . . . . . . . . . . . . . . . . . 3958.2 Surface Area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4108.3 Properties of the Surface Area of a Smooth Manifold . . . . . . . . 4178.4 Integration over a Smooth Manifold . . . . . . . . . . . . . . . . . 4348.5 Integration of Vector Fields . . . . . . . . . . . . . . . . . . . . . 4458.6 The GaussOstrogradski Formula . . . . . . . . . . . . . . . . . . 4538.7 Harmonic Functions . . . . . . . . . . . . . . . . . . . . . . . . 4718.8 Area on Lipschitz Manifolds . . . . . . . . . . . . . . . . . . . . 495

Contents xv

9 Approximation and Convolution in the Spaces LLL p . . . . . . . . . . 5079.1 The Spaces L p . . . . . . . . . . . . . . . . . . . . . . . . . . . 5079.2 Approximation in the Spaces L p . . . . . . . . . . . . . . . . . 5159.3 Convolution and Approximate Identities in the Spaces L p . . . . 523

10 Fourier Series and the Fourier Transform . . . . . . . . . . . . . . . 53510.1 Orthogonal Systems in the Space L 2(X,) . . . . . . . . . . . . 53510.2 Examples of Orthogonal Systems . . . . . . . . . . . . . . . . . 54610.3 Trigonometric Fourier Series . . . . . . . . . . . . . . . . . . . . 56110.4 Trigonometric Fourier Series (Continued) . . . . . . . . . . . . . 58210.5 The Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . 59910.6 The Poisson Summation Formula . . . . . . . . . . . . . . . . . 624

11 Charges. The RadonNikodym Theorem . . . . . . . . . . . . . . . . 63911.1 Charges; Integration with Respect to a Charge . . . . . . . . . . . 63911.2 The RadonNikodym Theorem . . . . . . . . . . . . . . . . . . . 64911.3 Differentiation of Measures . . . . . . . . . . . . . . . . . . . . 65511.4 Differentiability of Lipschitz Functions . . . . . . . . . . . . . . 666

12 Integral Representation of Linear Functionals . . . . . . . . . . . . . 67112.1 Order Continuous Functionals in Spaces of Measurable Functions 67112.2 Positive Functionals in Spaces of Continuous Functions . . . . . . 67612.3 Bounded Functionals . . . . . . . . . . . . . . . . . . . . . . . . 687

13 Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69713.1 An Axiomatic Definition of the Integral over an Interval . . . . . . 69713.2 Extension of Continuous Functions . . . . . . . . . . . . . . . . . 70213.3 Regular Measures . . . . . . . . . . . . . . . . . . . . . . . . . . 70813.4 Convexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71313.5 Sards Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 72813.6 Integration of Vector-Valued Functions . . . . . . . . . . . . . . . 73113.7 Smooth Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . 746

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 761

Author Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 763

Subject Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 767

Basic Notation

Logical SymbolsP Q, Q P P implies Q Universal quantifier (for every) Existential quantifier (there exists)Setsx X An element x belongs to a set Xx /X An element x does not belong to a set XA B , B A A is a subset of a set BAB The intersection of sets A and BAB The union of sets A and BAB The union of disjoint sets A and BA \B The difference of sets A and BAB The direct (Cartesian) product of sets A and Bcard(A) The cardinality of a set A{x X |P(x)} The subset of a set X whose elements have the

property P The empty set

Sets of NumbersN The set of positive integersZ The set of integersQ The set of rational numbersR The set of real numbersR= [,+] The extended real lineC The set of complex numbersR

m The arithmetic m-dimensional spaceR+ The set of positive numbersR

m+ The subset of Rm consisting of all points with positivecoordinates

Qm, Zm The subsets of Rm consisting of all points with rational

and integer coordinates, respectively

xvii

xviii Basic Notation

(a, b), [a, b), [a, b] An open, half-open, and closed interval, respectivelya, b An arbitrary interval with endpoints a and binfA (supA) The greatest lower (least upper) bound of a number set A

Sets in Topological and Metric SpacesA The closure of a set AInt(A) The interior of a set AB(a, r) The open ball of radius r > 0 centered at aB(a, r) The closed ball of radius r > 0 centered at aB(r) or Bm(r) The ball B(0, r) in the space Rm

Bm The ball Bm(1)Sm1 The unit sphere (the boundary of Bm) in the space Rmdiam(A) The diameter of a set Adist(x,A) The distance from a point x to a set A

Systems of SetsAm The -algebra of Lebesgue measurable subsets of Rm

Bm The -algebra of Borel subsets of Rm

B(E) The Borel hull of a system EBX The -algebra of Borel subsets of a space XPm The semiring of m-dimensional cellsP Q The product of semirings P and QMaps and Functionsdet(A) The determinant of a square matrix Af+, f The functions max{f,0}, max{f,0}fn f A sequence of functions fn uniformly converges to a

function fJ(x)= det((x)) The Jacobian of a map at a point xsupp(f ) The support of a function fT :X Y A map T acting from X to YT (A) The image of a set A under a map TT 1(B) The inverse image of a set B under a map TT S The composition of maps T and ST |A The restriction of a map T to a set Aesssup

X

f The essential supremum of a function f on a set X

x T (x) A map T sends a point x to T (x)f (E) The graph of a function f :ERPf (E) The region under the graph of a non-negative function f

over a set EE The characteristic function of a set E(x) The Jacobi matrix of a map at a point x The Euclidean norm of a vector, or the norm of a

function in L 2(X,), or the norm in a Banach spacef p The norm of a function f in L p(X,)f = esssup

X

|f |

Basic Notation xix

, The inner product of vectors in a Euclidean space, or offunctions in L 2(X,)

Measures(X,A,) A measure space(X,A) A measurable spacem The volume (Lebesgue measure) of the unit ball in Rm

m m-dimensional Lebesgue measure The product of measures and k k-dimensional area

Sets of FunctionsC(X) The set of continuous functions on a topological space XC0(X) The set of compactly supported continuous functions on

a locally compact topological space XCr(O) (Cr(O;Rn)) The set of r times (r = 0,1, . . . ,+) differentiable

functions (Rn-valued maps) defined on an open subset Oof Rm

C0 (O) The set of infinitely differentiable compactly supportedfunctions defined on an open subset O of Rm

L 0(X,) The set of measurable functions defined on X and finitealmost everywhere with respect to a measure

L p(X,) The set of functions from L 0(X,) satisfying thecondition

X|f |p d

Chapter 1Measure

1.1 Systems of Sets

In classical analysis, one usually works with functions that depend on one or severalnumerical variables, but here we will study functions whose argument is a set. Ourmain focus will be on measures, i.e., set functions that generalize the notions oflength, area and volume. Dealing with such generalizations, it is natural to aim atdefining a measure on a sufficiently good class of sets. We would like this classto have a number of natural properties, namely, to contain, with any two elements,their union, intersection and set-theoretic difference. In order for a measure to beof interest, its domain must also be sufficiently rich in sets. Aiming to satisfy theserequirements, we arrive at the notions of an algebra and a -algebra of sets.

As a synonym for a set of sets, we use the term a system of sets. The setsconstituting a system are called its elements. The phrase a set A is contained ina given system of sets A means that A belongs to A, i.e., A is an element of A.To avoid notational confusion, we usually denote sets by upper case Latin lettersA,B, . . . , and points belonging to these sets by lower case Latin letters a, b, . . . .For systems of sets, we use Gothic and calligraphic letters. The symbol stands forthe empty set.

1.1.1 We assume that the reader is familiar with the basics of naive set theory. Inparticular, we leave the proofs of set-theoretic identities as easy exercises. Some ofthese identities, which will be used especially often, are summarized in the follow-ing lemma for the readers convenience.

Lemma Let A, A ( ) be arbitrary subsets of a set X. Then(1) X \A =

(X \A);

(2) X \A =

(X \A);(3) AA =

(AA).

Equations (1) and (2) are called De Morgans laws. Equation (3) is the distributivelaw of intersection over union. Associating union with addition and intersection with

B. Makarov, A. Podkorytov, Real Analysis: Measures, Integrals and Applications,Universitext, DOI 10.1007/978-1-4471-5122-7_1, Springer-Verlag London 2013

1

http://dx.doi.org/10.1007/978-1-4471-5122-7_1

2 1 Measure

multiplication, the reader can easily see the analogy between this property and theusual distributivity for numbers.

Considering the union and intersection of a family of sets with a countable setof indices , we usually assume that the indices are positive integers. This does notaffect the generality of our results, since for every numbering of (i.e., everybijection n n from the set of positive integers onto ), we have the equalities

A =

nNAn,

A =

nNAn,

which follow directly from the definition of the union and intersection.In what follows, we often write a set as the union of pairwise disjoint subsets.

Thus it is convenient to introduce the following definition.

Definition A family of sets {E} is called a partition of a set E if E arepairwise disjoint and

E =E.

We do not exclude the case where some elements of a partition coincide with theempty set.

A union of disjoint sets will be called a disjoint union and denoted by . ThusA B stands for the union A B in the case where A B =. Correspondingly,

E stands for the union of a family of sets E in the case where all these setsare pairwise disjoint.

We always assume that the system of sets under consideration consists of subsetsof a fixed non-empty set, which will be called the ground set. The complement of aset A in the ground set X, i.e., the set-theoretic difference X \A, is denoted by Ac.Definition A system of sets A is called symmetric if it contains the complement Ac

of every element A A.Consider the following four properties of a system of sets A:

(0) the union of any two elements of A belongs to A;(0) the intersection of any two elements of A belongs to A;( ) the union of any sequence of elements of A belongs to A;() the intersection of any sequence of elements of A belongs to A.

The following result holds.

Proposition If A is a symmetric system of sets, then (0) is equivalent to (0) and( ) is equivalent to ().

Proof The proof follows immediately from De Morgans laws. Let us prove, forexample, that () ( ). Consider an arbitrary sequence {An}n1 of elements of A.Their union can be written in the form

n1An =

(

n1Acn

)c.

1.1 Systems of Sets 3

Since Acn A for all n (by the symmetry of A), it follows from () that the intersec-tion of these complements also belongs to A. It remains to use again the symmetryof A, which implies that A also contains the complement of this intersection, i.e.,the union of the original sets.

The reader can easily establish the remaining implications.

1.1.2 Now we introduce systems of sets that are of great importance for us.

Definition A non-empty symmetric system of sets A is called an algebra if it sat-isfies the (equivalent) conditions (0) and (0). An algebra is called a -algebra(sigma-algebra) if it satisfies the (equivalent) conditions ( ) and ().

Note the following three properties of an algebra A.

(1) ,X A. Indeed, let A A. Then = A Ac A and X = A Ac Adirectly by the definition of an algebra.

(2) For any two sets A,B A, their set-theoretic difference A \ B also belongsto A. This follows from the identity A \ B = A Bc and the definition of analgebra.

(3) If A1, . . . ,An are elements of A, then their union and intersection also belongto A. This property can be proved by induction.

Examples

(1) The system that consists of all bounded subsets of the plane R2 and their com-plements is an algebra (but not a -algebra!).

(2) The system that consists of only two sets, X and , is obviously an algebra anda -algebra. It is often called the trivial algebra on X.

(3) The other extreme case (as compared to the trivial algebra) is the system of allsubsets of X. It is obviously a -algebra.

(4) If A is an algebra ( -algebra) of subsets of a set X and Y X, then the systemof sets {A Y |A A} is an algebra (respectively, -algebra) of subsets of Y .We call it the induced algebra (on Y ) and denote it by A Y .

More generally, if E is an arbitrary system of subsets of a set X and Y X, then{E Y |E E} is called the system induced on Y by E and is denoted by E Y .The part of E Y that consists of the sets belonging to E and lying in Y is denotedby EY . Note that if E is an algebra, then EY is an algebra if and only if Y E .

Proposition Let {A} be an arbitrary family of algebras ( -algebras) con-sisting of subsets of some set. Then the system

A is again an algebra

( -algebra).

Proof The proof is left to the reader.

It is sometimes convenient to consider, along with algebras, related systems ofsets that do not satisfy the symmetry requirement. A system of sets A is called a

4 1 Measure

ring if for any two elements A,B A, the sets AB , AB and A \B also belongto A. A ring that contains the union of any sequence of elements is called a -ring.

Clearly, every algebra ( -algebra) is also a ring ( -ring).

1.1.3 Every system of sets is contained in some -algebra, for example, in the -algebra of all subsets of the ground set X. But this -algebra usually contains toomany sets, and it is often useful to embed the given system of sets into an algebra inthe most economical way, so that the ambient algebra does not contain superfluouselements.

It turns out that every finite collection of subsets {Ak}nk=1 of a set X is a part ofan algebra consisting of finitely many elements. This is obvious if the sets underconsideration form a partition of X. Then all finite unions of these sets, togetherwith the empty set (which, in set theory, is considered the union over an emptyset of indices), constitute an algebra. But if the sets Ak do not form a partition,there is a standard procedure for constructing an auxiliary partition that generatesan algebra containing these sets. This procedure is as follows: to each collection = {1, . . . , n}, where k = 0 or k = 1, we associate the intersection B = A11 Ann , where A0k = Ak and A1k = Ack (= X \ Ak). Note that, by Property (3),the sets B must belong to every algebra containing A1, . . . ,An. The reader caneasily check that the sets B form a partition of X, which we will call the canonicalpartition corresponding to the sets A1, . . . ,An. We encourage the reader to find thesets B in the case where the original collection of sets is already a partition of X.It is clear that B is either contained in Ak (if k = 0), or is disjoint with it. HenceAk =k=0 B . All finite unions of the sets B (together with the empty set) forman algebra containing all Ak . This algebra contains at most 22

nsets (see Exercise 6)

and (like any algebra consisting of finitely many sets) is a -algebra. Clearly, it isthe smallest -algebra containing all Ak .

The description of the sets that constitute the minimal -algebra containing agiven infinite system of sets is very complicated; we will not consider this question,instead restricting ourselves to the proof that such a -algebra exists. This importantresult will often be used in what follows.

Theorem For every system E of subsets of a set X there exists a minimal -algebracontaining E .

This -algebra is called the Borel1 hull of E and is denoted by B(E). It consistsof subsets of the same ground set as E .

Proof Clearly, there exists a -algebra containing E (for example, the -algebra ofall subsets of X). Consider the intersection of all such -algebras. This system ofsets contains E and is a -algebra by Proposition 1.1.2. Its minimality follows fromthe construction.

1mile Borel (18711956)French mathematician.

1.1 Systems of Sets 5

Definition An element of the minimal -algebra containing all open subsets of thespace Rm is called a Borel subset of Rm or merely a Borel set. The -algebra ofBorel subsets of Rm is denoted by Bm.

Remarks

(1) The simplest examples of Borel sets, along with open and closed sets, are count-able intersections of open sets and countable unions of closed sets. They arecalled G and F sets, respectively.

(2) It is not at all obvious that the -algebra Bm does not coincide with the -algebra of all subsets of Rm, but this is indeed the case. Moreover, these -algebras have different cardinalities. One can prove that Bm has the cardi-nality of the continuum, i.e., the same cardinality as Rm while the cardinalityof the -algebra of all subsets of Rm, by Cantors theorem, is strictly greaterthan the cardinality of Rm. We will not dwell on the proofs of these results; thereader can find them, for example, in the books [Bo, Bou].

1.1.4 Before proceeding to the definition of another system of sets, we establish anauxiliary result, which will be repeatedly used in what follows.

Lemma (Disjoint decomposition) Let {An}n1 be an arbitrary sequence of sets.Then

n=1An =

n=1

(

An\n1

k=0Ak

)

(1)

(for uniformity, we assume that A0 =).

Proof Let En = An \n1k=0 Ak . It is clear that these sets are pairwise disjoint: if,say, m< n, then Em Am, while En An \Am.

To verify (1), take an arbitrary point x from

n=1 An. Let m be the smallestof the indices n such that x An, i.e., x Am and x / Ak for k < m. Then x Em n1 En. Thus

n=1 An

n=1(An \

n1k=0 Ak). The reverse inclusion is

trivial.

Note that every finite collection of sets A1, . . . ,AN satisfies a similar identity:

N

n=1An =

N

n=1

(

An\n1

k=0Ak

)

. (1)

The proof is almost a literal repetition of that of the lemma (one can also apply thelemma to the sequence of sets {An}n=1 with An = for n >N ).

Along with algebras and -algebras, it will also be convenient to use systems ofsets that are not so good, but are often more tractable; namely, so-called semirings.

Definition A system of subsets P is called a semiring if the following conditionsare satisfied:

6 1 Measure

(I) P ;(II) if A,B P , then AB P ;

(III) if A,B P , then the set-theoretic difference A \ B can be written as a finiteunion of pairwise disjoint elements of P , i.e.,

A \B =m

j=1Qj, where Qj P.

Example The system P1 of all half-open intervals of the form [a, b), wherea, b R, a b, and the part P1r of P1 that consists of intervals with rationalendpoints, are semirings.

We leave the reader to prove these simple but important facts.

Every algebra is a semiring, but, as one can see from the above example, theconverse is not true. If P is a semiring, then, for arbitrary Y , the systems P Yand PY are, obviously, semirings too. Also, every system of pairwise disjoint setscontaining the empty set is a semiring.

The union and the set-theoretic difference of elements of a semiring P may notbelong to P . However, they have partitions consisting of elements of P . We willprove this result in a slightly stronger form.

Theorem Let P be a semiring and P,P1, . . . . . . ,Pn, . . . P . Then for every Nthe sets P \Nn=1 Pn and

Nn=1 Pn have decompositions of the form

P \N

n=1Pn =

m

j=1Qj, where Qj P; (2)

N

n=1Pn =

N

n=1

mn

j=1Qnj , where Qnj P and Qnj Pn. (3)

Furthermore,

n=1Pn =

n=1

mn

j=1Qnj , where Qnj P and Qnj Pn. (4)

It follows from (3) and (4) that the union of an arbitrary (finite or infinite) se-quence of elements of a semiring can be written as a finite or countable disjointunion of finer sets (i.e., subsets of the original sets) that are pairwise disjoint andstill belong to the semiring.

Proof Formula (2) can be proved by induction. To prove (3) and (4), we use (2) andformulas (1) and (1).

1.1 Systems of Sets 7

Corollary Let P be a semiring of subsets of a set X and R be the system of setsthat can be written as finite unions of elements of P . Then the union, intersection,and set-theoretic difference of two elements of R also belongs to R. If X P (orat least X R), then R is an algebra.

Thus the system R of finite unions of elements of a semiring P is a ring. It isobviously the smallest ring containing P .

Remark Equality (3) can be strengthened as follows: the union of Pn can be writtenin the form

N

n=1Pn =

K

k=1Rk, where R1, . . . ,RK P

and for any k and n the following alternative holds: either Rk is contained in Pn, orthese sets are disjoint.

To prove this for N = 2, use the identityP1 P2 = (P1 \ P2) (P1 P2) (P2 \ P1)

and write each of the differences P1 \ P2 and P2 \ P1 as a disjoint union accordingto the definition of a semiring. The general case can be proved by induction (toprove the inductive step from N to N + 1, replace P1 with Nn=1 Pn in the aboveargument).

1.1.5 Let P and Q be semirings of subsets of sets X and Y , respectively. Considerthe Cartesian product XY and the system PQ of subsets of XY that consistsof the products of elements of P and Q:

P Q= {P Q |P P, Q Q}.We call this system the product of the semirings P and Q.

Theorem The product of semirings is a semiring.

Proof The system P Q obviously satisfies condition I from the definition of asemiring. Let A= P Q and B = P0 Q0, where P,P0 P and Q,Q0 Q. Itfollows from the identity A B = (P P0) (Q Q0) that the system P Qalso satisfies condition II.

To verify condition III, we may assume that B A, i.e., P0 P and Q0 Q(otherwise replace B with B A). Then, by the definition of a semiring, we have

P = P0 P1 Pm and Q=Q0 Q1 Qnfor some P1, . . . ,Pm P and Q1, . . . ,Qn Q. Hence all rectangles Pk Qj ,0 k m, 0 j n, form a partition of the product A= P Q. Removing from

8 1 Measure

them the set B = P0Q0, we obtain a partition of the set-theoretic difference A\Binto elements of the system P Q, as required in condition III.

1.1.6 Now consider two very important examples of semirings of subsets of Rm.We identify the space Rm with the Cartesian product R R (m factors). Thecoordinates of a point x Rm are denoted by the same letter with subscripts. Thusx (x1, x2, . . . , xm). In some cases, we will also canonically identify Rm with theproduct of spaces of smaller dimension: Rm =Rk Rmk for 1 k

1.1 Systems of Sets 9

Unfortunately, neither open nor closed parallelepipeds form a semiring. Hence inwhat follows we are mainly interested in parallelepipeds [a, b) of another form,which we call cells (of dimension m). By definition,

[a, b)=m

j=1[aj , bj )=

{x = (x1, . . . , xm) |aj xj < bj for all j = 1, . . . ,m

}.

If aj = bj for at least one j , then the sets (a, b) and [a, b) are empty. Thus (a, b),[a, b) = if and only if a < b. Note also that the Cartesian product of cells ofdimension m and l is again a cell (of dimension m+ l).Proposition Every non-empty cell is the intersection of a decreasing sequence ofopen parallelepipeds and the union of an increasing sequence of closed paral-lelepipeds.

Proof Let [a, b) be a non-empty cell and h > 0 be a vector such that b h [a, b).Consider the parallelepipeds Ik = (a 1k h, b) and Sk = [a, b 1k h]. Then [a, b)=

k1 Sk =

k1 Ik . The details are left to the reader.

As follows from the proposition, every cell is simultaneously a G and an F set.In particular, every cell is a Borel set.

If all edge lengths of a cell are equal, then it is called a cubic cell. If all verticesof a cell have rational coordinates, we call it a cell with rational vertices. Note thefollowing simple but important fact: every cell with rational vertices is the disjointunion of finitely many cubic cells.

Indeed, since the coordinates of the vertices of such a cell can be written as frac-tions with a common denominator n, it can be split into cubes with edge length 1

n.

The system of all m-dimensional cells will be denoted by Pm, and its part con-sisting of cells with rational vertices, by Pmr .

Theorem The systems Pm and Pmr are semirings.

Proof The proof is by induction on the dimension. In the one-dimensional case,the assertion is obvious (see Example 1.1.4). The inductive step is based on The-orem 1.1.5 and the fact that, by the definition of cells, Pm =Pm1 P1 andPmr =Pm1r P1r . Remark In some cases (see the proof of Theorem 10.5.5), instead of Pmr we needto consider the system PmE consisting of all cells for which the coordinates of allvertices belong to a fixed set E R. As one can easily see, this system is also asemiring.

1.1.7 The next theorem will be repeatedly used in what follows.

Theorem Every non-empty open subset G of the space Rm is the union of a count-able family of pairwise disjoint cells whose closures are contained in G. All thesecells may be assumed to have rational vertices.

10 1 Measure

Proof For each point x G, find a cell Rx Pmr such that x Rx and Rx G.Obviously, G =xG Rx . Since the semiring Pmr is countable, among Rx thereare only countably many distinct cells. Numbering them, we obtain a sequence ofcells Pk (k N) with the following properties:

k=1Pk =G, Pk G for all k N.

To obtain a decomposition of G into disjoint cells with rational vertices, it remainsto use decomposition (4) from Theorem 1.1.4 on the properties of semirings.

Corollary B(Pm)=B(Pmr )=Bm.

Proof The inclusions B(Pmr )B(Pm)Bm are obvious. The reverse inclusionBm B(Pmr ) follows from the definition of Bm, since, by the above theorem, the -algebra B(Pmr ) contains all open sets.

Remark The proof of the theorem remains valid for every semiring PmE providedthat the set E is dense. The corollary also remains valid in this case.

EXERCISES

1. Show that the system of all (one-dimensional) open intervals and the system ofall closed intervals are not semirings.

2. Verify that the circular arcs (including degenerate ones) of angle less than form a semiring; show that without this additional restriction the assertion isfalse.

3. What is the Borel hull of the system of all half-lines of the form (, a), wherea R? Does the answer change if we consider only rational a or if we considerclosed rather than open half-lines?

4. For sets A,B , their symmetric difference is the set AB = (A \B) (B \A).Show that AB = (AB) \ (AB). Give an example of a symmetric systemof sets A that contains the symmetric difference of any two elements A,B A,but is not an algebra. Hint. Assuming that X = {a, b, c, d}, consider the systemof all subsets of X consisting of an even number of points.

5. Let A be the algebra of all subsets of a two-point set. Show that the semiringAA does not contain the complements of one-point sets and hence is not analgebra.

6. Show that the minimal algebra containing n sets has at most 22n

elements. Showthat this bound is sharp.

7. Show that all subsets of Rm that are simultaneously G and F sets form analgebra containing all open sets. Verify that it is not a -algebra (for instance,it does not contain Qm).

8. Refine Theorem 1.1.7 by proving that it suffices to use only cubic cells sat-isfying the additional condition that the diameter of each cell is substantially

1.2 Volume 11

smaller than the distance to the boundary of the set:

diam(P ) C min{x y |x P, y G}

(here C > 0 is a predefined arbitrarily small coefficient).9. Let P1, . . . ,Pn be elements of a semiring P . Show that all elements of the

canonical partition corresponding to these sets, except possibly for the setnk=1 P ck , can be written as disjoint unions of elements of P . Deduce the result

mentioned in Remark 1.1.4.10. A symmetric system of sets E is called a D-system if it contains the unions of

all at most countable families of pairwise disjoint elements A1,A2, . . . E . LetE be a D-system and A,B E . Show that:(a) if A B , then B \A E ;(b) each of the inclusions A B E , A B E and A \ B E implies the

other two.

11. Let a D-system contain all finite intersections of sets A1, . . . ,An. Show that italso contains the minimal algebra generated by these sets.

12. A system F of non-empty subsets of a set X is called a filter (in X) if it con-tains the intersection of any elements A,B F. For example, the system of allneighborhoods of a given point is a filter. A filter U is called an ultrafilter ifevery filter containing U coincides with U. An example of an ultrafilter is thesystem of all sets containing a given point (a trivial ultrafilter).Show that a filter F in X is an ultrafilter if and only if for every set A Xthe following alternative holds: either A or X \ A belongs to F. Using Zornslemma, show that for every filter there exists an ultrafilter that contains it.

1.2 Volume

In this section, we embark on the study of the main topic of this chapter. Namely,we will investigate the properties of so-called additive set functions. The assertionthat some quantity is additive means that the value corresponding to a whole objectis equal to the sum of the values corresponding to the parts of this object for everypartition of the object into disjoint parts. Numerous examples of additive quantitiesappearing in mathematics, as well as their prototypes in mechanics and physics,are well known. They include, in particular, length, area, probability, mass, momentof inertia about a fixed axis, quantity of electricity, etc. In this chapter, we restrictourselves to the study of additive functions with non-negative numerical (possiblyinfinite) values. The properties of additive functions of an arbitrary sign will bestudied in Chap. 11. Let us proceed to more precise statements.

1.2.1 Let X be an arbitrary set and E be a system of subsets of X.

12 1 Measure

Definition A function : E (,+] defined on E is called additive if(AB)= (A)+ (B) provided that A,B E and AB E . (1)

It is called finitely additive if for every set A E and every finite partition of A intoelements A1, . . . ,An of E ,

(A)=n

k=1(Ak). (1)

The sums on the right-hand sides of (1) and (1) always make sense, because thecorresponding terms cannot take infinite values of opposite sign (by definition, >).

Remark If is defined on an algebra (or a ring) A, then the additivity of impliesits finite additivity. This can be proved by induction using (1).

1.2.2 We define the concept to which this paragraph is devoted.

Definition A finitely additive function defined on a semiring of subsets of a setX is called a volume2 (in X) if is non-negative and ()= 0.

According to the definition of an additive function, a volume may take infinitevalues. It is called finite if X belongs to the semiring and (X)

1.2 Volume 13

(3) Let g be a non-decreasing function defined on R. We define a function g on thesemiring P1 as follows: g([a, b))= g(b) g(a). It is a volume, as the readercan easily verify.

(4) Let A be an arbitrary algebra of subsets of a set X, x0 X and a [0,+].Given A A, put

(A)={a if x0 A,0 if x0 /A.

One can easily check that is a volume. We will say that is the volumegenerated by a point mass of size a at x0.

More generally, if the volume of a one-point set {x0} is equal to a > 0, we saythat has a point mass of size a at x0.

To obtain a generalization of the last example, we use the notion of the sum of afamily of numbers. For brevity, a family of non-negative numbers is called positive.Recall that card(E) stands for the cardinality of a set E.

Definition The sum of a positive family {x}xX is the value

sup

{

xEx

E X, card(E)

14 1 Measure

If X is a countable set, a bijection : N X will be called a numbering of Xand denoted by {xn}n1, where xn = (n).

Lemma Let {x}xX be an arbitrary positive family. If the set X is countable, thenfor an arbitrary numbering {xn}n1 of X,

xXx =

n=1xn.

Proof Denote by S1 and S2 the left- and right-hand sides of this equality, respec-tively. On the one hand, for every finite set E X, we have xE x

n=1 xn

(since for every x E, the number x is an element of the series). Hence S1 S2.On the other hand, for every k we have

kn=1 xn S1, by the definition of the

sum of a family, whence S2 S1. Since S1 S2, this completes the proof.

We leave the reader to check that the equality we have proved is valid for the sumof every summable family with a countable set of indices.

Now consider the following example.

(5) Let {x}xX be an arbitrary positive family. Assuming that A is an algebra ofsubsets of X that contains all one-point sets, define a function on A as follows:

(A)=

xAx (A A)

(by definition, we assume that

xx = 0). Note that since (E) = x1 + +xN for every finite set E = {x1, . . . , xN }, we have(A)= sup{(E) |E A, card(E) 0,put

(A)={

0 if A is bounded,

a if A is unbounded.

This volume will be useful for constructing various counterexamples.

1.2.3 We establish the basic properties of volume.

Theorem Let be a volume on a semiring P , and let P,P ,P1, . . . ,Pn P .Then

(1) if P P , then (P ) (P );(2) if

nk=1 Pk P , then

nk=1 (Pk) (P );

(3) if P nk=1 Pk , then (P )n

k=1 (Pk).

1.2 Volume 15

Properties (1) and (2) are called the monotonicity and the strong monotonicityof , respectively; property (3) is called the subadditivity of .

Proof Obviously, the monotonicity of follows from its strong monotonicity, sowe will prove the latter property.

By the theorem on the properties of semirings, the set-theoretic difference P \nk=1 Pk can be written in the form P \

nk=1 Pk =

mj=1 Qj , where Qj P .

Therefore, P = (nk=1 Pk) (m

j=1 Qj), and, by the additivity of ,

(P )=n

k=1(Pk)+

m

j=1(Qj )

n

k=1(Pk).

To prove the subadditivity of , put P k = P Pk . Then P =n

k=1 P k , P k P .By the theorem on the properties of semirings,

P =n

k=1

mk

j=1Qkj ,

where Qkj P and Qkj P k Pk for 1 k n and 1 j mk . It follows fromthe strong monotonicity of that

mkj=1 (Qkj ) (Pk). Therefore,

(P )=n

k=1

mk

j=1(Qkj )

n

k=1(Pk).

Note that if a volume is defined on an algebra (or a ring) A, then (A \ B) =(A) (B) provided that A,B A, B A and (B) < +. Indeed, sinceA \B A, we have (A)= (B)+(A \B).

Remark A volume defined on a semiring P can be uniquely extended to the ringR consisting of all finite unions of elements of P . Indeed, let E =nk=1 Pk , wherePk P . We may assume without loss of generality that the sets Pk are pairwisedisjoint (see Theorem 1.1.4). Put (E)=nk=1 (Pk). We leave the reader to showthat this function is well defined and that is a volume that coincides with on P .

1.2.4 Now let us check that the ordinary volume is indeed a volume in the senseof our definition. Since Pm =P1 Pm1, this is a corollary of the followinggeneral theorem, in which we use the notion of the product of arbitrary semirings(see Sect. 1.1.5).

Theorem Let X,Y be non-empty sets, P,Q be semirings of subsets of these sets,and , be volumes defined on P and Q, respectively. We define a function onthe semiring P Q by the formula

(P Q)= (P ) (Q) for any P P,Q Q

16 1 Measure

(the products 0 (+) and (+) 0 are assumed to vanish).Then is a volume on P Q.

The volume is called the product of the volumes and .

Proof We need to check only the finite additivity of . First consider a partition ofP Q of a special form. Let P and Q be partitioned into disjoint sets:

P = P1 PI , Q=Q1 QJ (Pi P, Qj Q).Then the sets Pi Qj (1 i I , 1 j J ) belong to the semiring P Q andform a partition of P Q, which we will call a grid partition. For such a partition,the desired equality is obvious:

(P Q)= (P )(Q)=I

i=1(Pi)

J

j=1(Qj )=

1iI1jJ

(Pi Qj).

Now consider an arbitrary partition of the set P Q into elements of the semiringP Q:

P Q= (P1 Q1) (PN QN) (Pn P,Qn Q).In general, it is not a grid partition, but, refining it, we can reduce the problem tosuch a partition. Clearly, P = P1 PN and Q =Q1 QN , where thesets P1, . . . ,PN and Q1, . . . ,Qn, respectively, may not be disjoint. However, as weobserved in Sect. 1.1 (see the remark in Sect. 1.1.4), there exist partitions

P =A1 AI (Ai P) and Q= B1 BJ (Bj Q)such that

for all i, n, either Ai Pn or Ai Pn =;for all j,n, either Bj Qn or Bj Qn =.

Since the sets Ai Bj form a grid partition of the product P Q, we have

(P Q)=

1iI1jJ

(Ai Bj ). (2)

On the other hand, it is clear that for every n the families {Ai |Ai Pn} and{Bj |Bj Qn} are partitions of the sets Pn and Qn, respectively. Hence {Ai Bj |Ai Pn, Bj Qn} is a grid partition of the product Pn Qn. Therefore,

(Pn Qn)=

i:AiPnj :BjQn

(Ai Bj ).

1.3 Properties of Measure 17

Rearranging the terms on the right-hand side of (2), we obtain the desired equality:

(P Q)=

1iI1jJ

(AiBj )=

1nN

i:AiPnj :BjQn

(AiBj )=

1nN(PnQn).

Corollary The ordinary volume m is a volume in the sense of Definition 1.2.2.

Proof The proof proceeds by induction on the dimension. The one-dimensional caseis left to the reader. Now the additivity of m follows immediately from the theorem,since Pm =P1 Pm1 and m is the product of the volumes 1 and m1.

EXERCISES In Exercises 13, is a finite volume defined on an algebra A ofsubsets of a set X.

1. Show that for any elements of A,

(AB)= (A)+(B)(AB);(AB C)= (A)+(B)+(C)(AB)(B C)(AC)

+(AB C).Generalize these equalities to the case of four and more sets.

2. Let (X)= 1, and let A1, . . . ,An A. Show that if nk=1 (Ak) > n 1, thennk=1 Ak =.

3. Show that every partition of X into subsets of positive volume is at most count-able.

1.3 Properties of Measure

The key property in the definition of a volume is its finite additivity, i.e., the as-sertion that the volume of a whole object is the sum of the volumes of its partsprovided that the number of these parts is finite. As we will see below, this rulemay be violated if the parts form an infinite sequence. Of course, infinite partitionsarise only as an idealization of real-life situations, so it is hard to provide a naturalscientifically motivated explanation of why we need to consider volumes with sucha strong additivity property, which is called countable additivity.

However, intuitively, a violation of the rule the volume of a whole object isthe sum of the volumes of its parts for a countable set of parts seems to be quiteunnatural if, for example, by the volume we mean the length or the area. It is thecountable additivity that allows one to develop a deep theory that comes close tothe theory of integration. This and the next sections are devoted to the theory ofcountably additive volumes, which is usually called measure theory. It has numerousimportant applications. First of all, it is worth mentioning that measure theory liesat the foundations of modern probability theory.

18 1 Measure

1.3.1 Let us proceed to precise definitions.

Definition A volume defined on a semiring P is called countably additive if forevery set P P and every partition {Pk}k=1 of P into elements of P ,

(P )=

k1(Pk).

A countably additive volume is called a measure.

Using the notion of the sum of a family and Lemma 1.2.2, we can formulate thedefinition of countable additivity in an equivalent, though formally more generalform: a volume defined on a semiring P is countably additive if for every setP P and every countable partition {P} of P into elements of P ,

(P )=

(P).

Countable additivity does not follow from finite additivity, so that not every vol-ume is a measure. In particular, the volume from Example (6) of Sect. 1.2.2 is not ameasure, as the reader can easily check.

Examples

(1) The ordinary volume is a measure (see Theorem 2.1.1).(2) Consider the volume g([a, b)) = g(b) g(a) defined in Example (3) of

Sect. 1.2.2. Its countable additivity means, in particular, that if [b0, b) =n=0[bn, bn+1), where bn b, bn < bn+1, then g([b0, b)) =n=0 g([bn, bn+1)). Since ([bn, bn+1)) = g(bn+1) g(bn), this is equiva-

lent to the condition g(bn) n g(b).

Thus for g to be countably additive, it is necessary that the function g becontinuous from the left.

Given an arbitrary increasing function g, one can obtain a measure by settingg([a, b))= g(b0)g(a0), where g(a0) and g(b0) are the left limitsof g at the points a and b, respectively. We will prove the countable additivityof g in Theorem 4.10.2. It implies, in particular, that the continuity of thefunction g from the left is not only a necessary, but also a sufficient conditionfor the volume g to be a measure.

(3) The volume generated by a positive point mass (see Example (4) in Sect. 1.2.2)is a measure.

(4) Let X be an arbitrary set and A be a -algebra of subsets of X containing allone-point sets. We define a function on A as follows:

(A)={

the number of points in A if A is finite;+ if A is infinite.

1.3 Properties of Measure 19

We leave the reader to verify that the function thus defined is indeed a mea-sure. It is called the counting measure.

(5) Let us verify that the volume constructed in Example (5) of Sect. 1.2.2 iscountably additive, i.e., that is a measure.

Indeed, let A=k=1 Ak , where A,Ak A. It is clear that for every n N,

(A) (

n

k=1Ak

)

=n

k=1(Ak),

whence (A)

k=1 (Ak). On the other hand, if E is an arbitrary finite subsetof A, then for some n we have E nk=1 Ak . Therefore,

(E) (

n

k=1Ak

)

=n

k=1(Ak)

k=1(Ak).

It follows that

(A)= sup{(E) |E A, card(E)

20 1 Measure

By Theorem 1.1.4, P can be written in the form

P =

k1

nk

j=1Qkj (Qkj P).

Furthermore,nk

j=1 Qkj P k . Hence, by the strong monotonicity of a volume,nkj=1 (Qkj ) (P k) (Pk). Using the countable additivity, we obtain

(P )=

k1

nk

j=1(Qkj )

k1(Pk),

as required.Now let us prove that countable subadditivity implies countable additivity. Let

{Pk}k=1 P be a partition of a set P P . By the countable subadditivity of ,

(P )

k1(Pk). (2)

On the other hand, the strong monotonicity of a volume implies that (P ) nk=1 (Pk) for every n N. Passing to the limit as n, we see that (P ) k1 (Pk). Together with (2), this proves the countable additivity of .

The last theorem implies a result that we will often use in what follows.

Corollary Let be a measure defined on a -algebra A. Then a countable unionof sets of zero measure is again a set of zero measure.

Indeed, if en are sets from A that have zero measure, then their union also belongsto A and (

n1 en)

n1 (en)= 0.

1.3.3 We will check that for a volume defined on the algebra, countable additivityis equivalent to a property analogous to continuity.

Theorem A volume defined on an algebra A is a measure if and only if it iscontinuous from below, i.e.,

the conditions A,Ak A, Ak Ak+1, A=

k1Ak

imply that (Ak) k (A). (3)

Remark If the algebra A from the statement of the theorem is a -algebra, then thecondition A A in the definition of continuity from below can be omitted, becauseit follows from the equality A=k1 Ak .

1.3 Properties of Measure 21

Proof Let be a countably additive volume and A, Ak be sets satisfying con-ditions (3). Putting B1 = A1, Bk = Ak \ Ak1 for k > 1, we see that Bk A,Bk Bj = for k = j (j, k N), and

Ak =k

j=1Bj , A=

j1Bj .

Therefore, (Ak)=kj=1 (Bj ) and

(A)=

j1(Bj )= lim

k

k

j=1(Bj )= lim

k(Ak).

Now let us prove that continuity from below implies countable additivity. Let{Ej }j=1 A be a partition of a set A A. Put Ak =

kj=1 Ej . Then

Ak A, Ak Ak+1, A=

k1Ak,

and (Ak)=kj=1 (Ej ). Since is continuous from below, we obtain

(A)= limk(Ak)= limk

k

j=1(Ej )=

j1(Ej ).

1.3.4 Recall that a volume defined on a semiring P of subsets of a set X is calledfinite if X P and (X)

22 1 Measure

Proof (1) (2). Let Ak be sets satisfying conditions (4). Put B = A1 \ A, Bk =A1 \Ak . Then Bk Bk+1 and B =k1 Bk . By the continuity of a measure frombelow,

(A1)(Ak)= (Bk) k (B)= (A1)(A),

i.e., (Ak) k (A).

The implication (2) (3) is trivial. Let us prove that (3) (1). Let {Ej }j=1 A be a partition of a set A A. Put Ak =j=k+1 Ej . Then Ak A, since Ak =A \k

j=1 Ej , and the sets Ak obviously satisfy all conditions (4). Hence (Ak) k 0.

Furthermore, A = Ak kj=1 Ej . Thus (A) = (Ak) +k

j=1 (Ej ),(Ak)

k 0, and, consequently, (A)=

j1 (Ej ), as required.

Corollary Every measure is conditionally continuous from above. The latter meansthat the conditions A, Ak A, Ak Ak+1, k1 Ak = A and (Am) < + forsome m imply that (Ak)

k (A).

To prove this, it suffices to consider the restriction of the measure to the in-duced algebra AAm (see Example (4) in Sect. 1.1.2) and use the continuity fromabove of the obtained finite measure.

Remarks

(1) If a volume is infinite, then continuity from above does not imply countableadditivity (see Exercise 1).

(2) If a volume is defined on a semiring, then in Theorems 1.3.3 and 1.3.4 only theonly if parts are true (see Exercise 2).

In what follows, we usually consider measures defined on -algebras. The collec-tion consisting of three objectsa set X, a -algebra A of subsets of X, and a mea-sure defined on Ais usually denoted by (X,A,) and is called a measure space.The sets for which the measure is defined, i.e., the elements of the -algebra A, arecalled measurable, or, more precisely, measurable with respect to A.

EXERCISES

1. Show that the infinite volume from Example (6) in Sect. 1.2.2 (a = +) isconditionally continuous from above, but is not a measure.

2. Let X = [0,1) Q, and let P be the system of all sets P of the form P [a, b)Q, where 0 a b 1. Put (P )= b a. Show that P is a semiringand is a volume that is continuous from above and from below, but is not ameasure.

3. Let (X,A,) be a measure space, and let Ek be measurable sets such thatk=1 (Ek)

1.4 Extension of Measure 23

An = {x X |x Ek for exactly n values of k},Bn = {x X |x Ek for at least n values of k}.

Show that the sets An,Bn are measurable and

n=1n(An)=

n=1(Bn)=

n=1(En).

4. Using the counting measure on N, show that continuity from above at the emptyset does not follow from countable additivity.

5. Show that a finite volume defined on an algebra A is countably additive pro-vided that it is continuous from below at X, i.e., the conditions Ak Ak+1,

k1 Ak =X, Ak A imply (Ak) k (X).

6. Show that for a -finite measure, every partition into sets of positive measure isat most countable.

7. Assume that a measure is such that there exist arbitrarily (finitely) many pairwisedisjoint subsets of positive measure. Show that there exists an infinite family ofsuch subsets.

1.4 Extension of Measure

1.4.1 Although we have considered characteristic properties of measures, with theexception of the counting measure, we still have not produced a non-trivial exampleof a measure defined on a -algebra.

The reason is that we are presently able to define measures only on poor sys-tems of sets, such as most semirings. Due to the tractability of these systems, it iscomparatively easy to define volumes on them (see Examples (1)(3) in Sect. 1.2.2).But we cannot yet define measures on wider systems of sets, e.g., on -algebras, ex-cept for several quite trivial cases. This situation is, of course, highly unsatisfactory.

Indeed, even if we know that the ordinary volume in m-dimensional space de-fined on the semiring of cells is countably additive (this will be proved in Theo-rem 2.1.1), we certainly cannot consider the problem of constructing a measure onR

m completely solved, since it is highly dubious whether a measure on Euclideanspace that cannot be used to measure pyramids, balls, and other important bod-ies has any value; and this is exactly the situation we find ourselves in. The verytractability of semirings, their being poor in sets, which allowed us, in the casesconsidered above, to easily define volumes on them, now demonstrates its down-sides. Thus we must learn to construct measures on richer systems of sets. Thisproblem is difficult even if we restrict ourselves to the -algebra of Borel sets of thereal line and try to assign a length to every Borel set (speaking more formally, try toextend the one-dimensional ordinary volume to the Borel -algebra). It was the so-lution of this problem suggested by Lebesgue4 in 1902 that marked the beginning of

4Henri Lon Lebesgue (18751941)French mathematician.

24 1 Measure

measure theory. This result, inspired by the needs of several areas of mathematics,was a major breakthrough in the theory of integration.

Lebesgues construction of an extension of the length (the one-dimensional or-dinary volume) to a measure defined on a -algebra of subsets of the real line wasbased on clear geometric considerations. It splits into several steps. First, Lebesgueassigns a measure m(G) to all open sets G R, where m(G) is the sum of thelengths of the intervals constituting G. Then he introduces a quantity called theouter measure; for an arbitrary set E R, it is defined by the formula

me(E)= inf{m(G) |GE, G is an open set}.

The inner measure mi(E) of a bounded set E is equal to mi(E) = m() me( \E), where is an arbitrary interval containing E. A bounded set is calledmeasurable if its inner and outer measures coincide. The common value of the in-ner and outer measures of a measurable set E is declared to be the measure of E.Then one checks that the system of measurable sets contained in a fixed interval is a -algebra and that the constructed measure is countably additive. Thus Lebesguesmethod of extending a measure is not altogether direct. It contains an importantintermediate step, the construction of the outer measure. So to speak, we cross achasm in two jumps. A detailed realization of this program (which is described ina slightly modified form, e.g., in [N]) is not at all easy.

Along with some advantages (first of all, the geometric clarity of the construc-tion), this approach also has its disadvantages. Of course, since every open subset ofa Euclidean space is the union of a sequence of cells, the analogy we should followin order to extend a measure from the semiring of cells is clear. However, it is stillnot clear how one should act to extend a measure defined on a semiring of subsetsof a ground set that has no topology and, consequently, no open sets. This questionis all the more relevant, because in the axiomatization of probability theory in theframework of measure theory, the ground set is the space of elementary events,which is not necessarily a topological space.

Later, due mainly to Carathodorys5 results, it became clear that the crucialelements of Lebesgues construction are the following two facts. First, that the outermeasure is countably subadditive, and, second, that it can be constructed withoutinvolving open sets, i.e., without using the topology. For this (bearing in mind that anopen set is the union of a sequence of cells), one should only interpret the inclusionE G used in the one-dimensional case as the fact that E can be covered by asequence of elements of the semiring. This observation allows one to construct theouter measure for an arbitrary measure, regardless of whether or not the ground setis a topological space.

The method suggested by Carathodory shows that it is useful, especially froma technical point of view, not to restrict ourselves to additive functions, but insteadto consider countably subadditive functions defined on all subsets of the ground set.These functions are now called outer measures. Here we must warn the reader that

5Constantin Carathodory (18731950)a German mathematician of Greek origin.

1.4 Extension of Measure 25

the terminology is slightly confusing: in general, an outer measure is not a measurein the sense of Definition 1.3.1.

The key point of Carathodorys construction is the fact that every outer measuregives rise in a natural way to a -algebra (which in non-degenerate cases is quitewide) on which this outer measure is additive and hence countably additive. Thusevery outer measure generates a measure. Since outer measures are much easierto construct, this approach turns out to be useful not only for extending measures,but also in other cases when we need to find a measure with given properties. Wewill encounter such examples when proving the existence of the surface area (whichreduces to constructing the Hausdorff measure of appropriate dimension) and whendescribing positive functionals on the space of continuous functions (Sect. 12.2).

We preface a detailed description of Carathodorys method with the definitionof outer measures and the study of their basic properties.

1.4.2 Here we will consider subsets of a fixed non-empty set X, which we call theground set. Recall that by Ac we denote the complement of a set A X, i.e., theset-theoretic difference X \A.

Definition 1 Let A(X) be the -algebra of all subsets of the ground set X. An outermeasure on X is a function : A(X)[0,+] such that:I. ()= 0 and

II. (A)

n=1 (An) if A

n=1 An.Property II is called countable subadditivity.

We mention two simple properties of outer measures.

(1) An outer measure is finitely subadditive, i.e., the inclusion A A1 ANimplies that (A) (A1)+ + (AN).

This property follows immediately from the countable subadditivity of if weassume that the sets An are empty for all n >N .

(2) An outer measure is monotone, i.e., the inclusion A B implies that (A) (B).

This is a special case of property 1 (corresponding to N = 1).As we will see below, outer measures naturally appear in various situations (see

Sects. 2.1, 2.6, 12.2). Here we only mention that an example of an outer measure isany measure defined on all subsets of the ground set, in particular, a discrete measure(see Example (5) in Sect. 1.3.1).

The next definition is motivated by our desire to single out an algebra of sets onwhich an outer measure is additive. If A and E are such sets, then

(E)= (E A)+ (E \A). (1)To construct a desired system of sets, we let it contain those subsets A of the groundset that split every set E additively. Thus we arrive at the following definition.

26 1 Measure

Definition 2 Let be an outer measure on X. A set A is called measurable, or,more exactly, -measurable if (1) holds for every set E X.

The system of all -measurable sets will be denoted by A .

Let us illustrate this definition by the following informal example. Consider acommuter rail system divided into fare zones. Let X be the collection of intervalsbetween neighboring stations. An arbitrary collection of intervals (a subset of X)will be called a path. If the price of a trip along a connected path is proportional tothe number of zones through which it travels, and for an unconnected path it is thesum of the prices of the connected components, then the price of a trip is an outermeasure on the set of intervals. A path is measurable if and only if it consists ofentire zones.

Note that since E = (EA)(E\A) and an outer measure is countably subaddi-tive, the inequality (E) (E A)+ (E \A) always holds. Hence, to verify (1),it only suffices to establish the inequality

(E) (E A)+ (E \A), (1)and usually we will do exactly this.

Remark If (A)= 0, then (EA)= 0, and hence (1) holds for every set E. Thusall sets of zero outer measure are measurable.

1.4.3 The main result of this subsection is the following theorem.

Theorem Let be an outer measure on X. Then A is a -algebra and the restric-tion of to this -algebra is a measure.

Proof First of all, observe that the system of -measurable sets is symmetric, i.e.,together with every set A it also contains its complement Ac. This follows from thefact that, in view of the identity E \A= E Ac , condition (1) can be written in asymmetric form: (E)= (E A)+ (E Ac).

Now let us prove that A is an algebra of sets. According to Definition 1.1.2, itsuffices to check that A contains the union of any two elements of A .

Let A,B A , and let E be an arbitrary set. Using successively the measurabilityof A and B , we obtain

(E)= (E A)+ (E \A)= (E A)+ ((E \A)B)+ ((E \A) \B).The third term on the right-hand side of this inequality is obviously equal to (E \(AB)), and the sum of the first two terms can be estimated using the subadditivityof :

(E A)+ ((E \A)B) ((E A) ((E \A)B))= (E (AB)).Thus

(E) (E (AB))+ (E \ (AB)),

1.4 Extension of Measure 27

i.e., the union AB satisfies (1) for every set E. Hence AB A for any A andB in A . So, A is an algebra.

If A and B are disjoint measurable sets, then (E(AB))A=EA and (E(AB)) \A=E B for an arbitrary set E. Hence (E (AB))= (E A)+(EB). Then, by induction, for every n N, for pairwise disjoint sets A1, . . . ,Anand an arbitrary set E,

(

E n

j=1Aj

)

=n

j=1(E Aj). (2)

Taking E =X, we see that the outer measure is additive on A :

(n

j=1Aj

)

=n

j=1(Aj ). (2)

Now let us check that A is a -algebra. For this we must show that A con-tains the union A=j=1 Aj of an arbitrary sequence of measurable sets Aj . Firstassume that the sets Aj are pairwise disjoint. Then for every set E and every n itfollows from (2) that

(E)= (

E n

j=1Aj

)

+ (

E \n

j=1Aj

)

=n

j=1(E Aj)+

(

E \n

j=1Aj

)

n

j=1(E Aj)+ (E \A).

Passing to the limit as n and using the countable subadditivity of , we obtain

(E)

j=1(E Aj)+ (E \A)

(

j=1(E Aj)

)

+ (E \A)

= (E A)+ (E \A).Thus we have confirmed that A satisfies (1), so that A A .

The general case can be reduced to that considered above by using a disjointdecomposition (see Lemma 1.1.4): A =j=1 Bj , where B1 = A1 and Bj = Aj \(A1 Aj1) for j 2 (the sets Bj are measurable, since A is an algebra).

It remains to prove the second claim of the theorem. Let be the restriction of to A . It follows from (2) that is a volume. It is countably subadditive, since is. By Theorem 1.3.2, is a measure.

The remark at the end of Sect. 1.4.2 suggests to single out the measures satisfyingan important additional property. In view of monotonicity, it is natural to expect thatevery subset of a set of zero measure also has zero measure. However, this is not

28 1 Measure

always the case, because this subset may not be measurable (for instance, if themeasure is defined only on Borel sets). Measures for which subsets of sets of zeromeasure also have zero measure are of special interest.

Definition A measure defined on a semiring P is called complete if the condi-tions E P and (E)= 0 imply that every subset E of E also belongs to P (and,consequently, (E)= 0).

Using this definition and the remark from Sect. 1.4.2, we can refine the theoremby saying that an outer measure generates a complete measure. In other words, wehave the following corollary.

Corollary The restriction of an outer measure to the -algebra A is a completemeasure.

1.4.4 We now proceed to the description of Carathodorys method of extending ameasure. Like Lebesgues original construction, it consists of two steps. At the firststep, given a measure 0, we construct an auxiliary function that extends 0from the original semiring to the system of all subsets. It is no longer countably ad-ditive, but we can prove that it has a weaker property, countable subadditivity, so that is an outer measure. At the second step, we restrict the constructed outer mea-sure to the system of -measurable sets; as a result, according to Theorem 1.4.3,we obtain a new measure defined on a -algebra. To verify that this measure is anextension of 0, it remains to show that the original semiring is contained in the -algebra of -measurable sets. Let us proceed to the realization of this program.

Let 0 be a measure defined on a semiring P of subsets of a set X. For everyset E X, put

(E)= inf{

j=10(Pj )

E

j=1Pj , Pj P for all j N

}

(3)

(if E cannot be covered by a sequence of elements of P , we put (E)=+).Note that instead of {Pj }j1 in (3) we may consider an arbitrary countable fam-

ily {P}, since the sum 0(P) coincides with

j=1 0(Pj ) for everynumbering of .

Theorem The function defined by formula (3) is an outer measure that coincideswith 0 on P .

We will say that is the outer measure generated by 0.

Proof Let E P . Then the sequence E,,, . . . is a cover of E by elementsof P . It follows that (E) 0(E). On the other hand, if E j=1 Pj , wherePj P for all j N, then 0(E) j=1 0(Pj ) by the countable subadditivity

1.4 Extension of Measure 29

of a measure (Theorem 1.3.2). Since {Pj }j1 is an arbitrary sequence, it followsthat 0(E) (E). Thus (E)= 0(E); in particular, ()= 0.

It remains to show that is countably subadditive, i.e., that

(E)

n=1(En)

if E n=1 En. We may assume that the right-hand side is finite, since otherwisethe inequality is trivial. Fix an arbitrary > 0, and for every n find sets P (n)j P (j N) such that

En

j=1P

(n)j and

j=10

(P

(n)j

)

30 1 Measure

and the subadditivity of ,

(E)= 0(E)= 0(E P)+N

j=10(Qj )= (E P)+

N

j=1(Qj )

(E P)+(

N

j=1Qj

)

= (E P)+(E \ P).

Thus in the case under consideration (4) is proved.To prove (4) in the general case, we may assume that (E) 0 and choose sets Pj P such that E j=1 Pj and

j=1 0(Pj )

j=10(Pj )

j=1

((Pj P)+(Pj \ P)

).

Using the countable subadditivity and monotonicity of , we obtain

(E)+ > ((

j=1Pj

)

P)

+((

j=1Pj

)

\P)

(EP)+(E \P).

Since is arbitrary, this implies (4). Thus we have proved the -measurability ofevery set P P , and hence the inclusion P A .

The measure constructed in the theorem is called the Carathodory extensionof 0. Since such an extension always exists, we may always assume without lossof generality that a measure under consideration is defined on a -algebra.

We draw the readers attention to the fact that the theorem not only guarantees theexistence of an extension, but provides formula (3), i.e., a method for computing theextended measure from the original measure 0. Of course, since these measurescoincide on P , we can also rewrite formula (3) for measurable sets, replacing 0by , in the form

(A)= inf{

j=1(Pj )

A

j=1Pj , Pj P for all j N

}

. (3)

We will often use this equality in what follows.In conclusion, observe that the repeated application of the Carathodory exten-

sion procedure yields the same result as the first one. To check this, let us show thatthe measures 0 and generate the same outer measure. Indeed, the right-hand side

1.5 Properties of the Carathodory Extension 31

of (3) does not increase if we replace the semiring P by the -algebra A and themeasure 0 by the measure . This means that the outer measure generated by isnot greater than . To obtain the reverse inequality, it suffices to observe that

(E)

j=1(Aj )=

j=1(Aj )

for every cover of E by sets Aj from the -algebra A .

EXERCISES

1. We define a function on subsets of the set X = {1,2,3} as follows:()= 0, (X)= 2, (E)= 1 otherwise.

Show that is an outer measure. Which sets are -measurable?2. Let E be an arbitrary system of sets containing , and let : E[0,+] be a

non-negative function with ()= 0. Put

(E)= inf{

j=1(Ej )

E

j=1Ej , Ej E for all j N

}

(in the case where E cannot be covered by a sequence of elements of E , weassume that (E) = +). Show that is an outer measure, and that it is anextension of if and only if the function is countably subadditive.

3. Let be an outer measure. Show that a set A is -measurable if and only if(B C)= (B)+ (C) for any sets B and C satisfying the conditions B Aand C A=.

1.5 Properties of the Carathodory Extension

We keep the notation of the previous section and assume that is the Carathodoryextension of a measure 0 defined on a semiring P and is the outer measuregenerated by 0. We will call -measurable sets just measurable and denote the -algebra of measurable sets by A.

1.5.1 We begin with the main question of this section: do there exist extensionsof 0 other than the Carathodory extension? This breaks down into two questions.First, does the measure 0 have an extension to a -algebra wider than A? Secondly,do there exist other extensions of 0 to the algebra A or to some part of this algebra,for example, the Borel hull of the semiring P?

We leave the first question aside. One can prove (see [Bo, Vol. 1]), that it isusually possible to further extend the measure , but such an extension is neither

32 1 Measure

motivated by any application nor even by the needs of pure mathematics. The -algebra A is usually so wide that one has no need to extend it.

The second question is of quite a different nature, and the importance of theanswer to it cannot be overestimated. It is of crucial importance to know whether anextension of the original measure at least to the minimal -algebra generated by thesemiring P is unique. As we will show, in a wide class of cases (in particular, forall finite measures), the answer to this question is positive. The existence of non-standard extensions should be considered as a pathology, which usually appears insome artificial situations; we will encounter them only in several counterexamples.

The exte