Table Of ContentREAL ANALYSIS AND PROBABILITY
Thismuchadmiredtextbook,nowreissuedinpaperback,offersaclearexpo-
sitionofmodernprobabilitytheoryandoftheinterplaybetweentheproperties
ofmetricspacesandprobabilitymeasures.
The first half of the book gives an exposition of real analysis: basic set
theory,generaltopology,measuretheory,integration,anintroductiontofunc-
tional analysis in Banach and Hilbert spaces, convex sets and functions,
and measure on topological spaces. The second half introduces probability
basedonmeasuretheory,includinglawsoflargenumbers,ergodictheorems,
the central limit theorem, conditional expectations, and martingale conver-
gence.AchapteronstochasticprocessesintroducesBrownianmotionandthe
Brownianbridge.
The new edition has been made even more self-contained than before;
it now includes early in the book a foundation of the real number system
and the Stone-Weierstrass theorem on uniform approximation in algebras
of functions. Several other sections have been revised and improved, and
theextensivehistoricalnoteshavebeenfurtheramplified.Anumberofnew
exercises,andhintsforsolutionofoldandnewones,havebeenadded.
R.M.DudleyisProfessorofMathematicsattheMassachusettsInstituteof
TechnologyinCambridge,Massachusetts.
CAMBRIDGESTUDIESINADVANCEDMATHEMATICS
EditorialBoard:
B.Bollobas,W.Fulton,A.Katok,F.Kirwan,P.Sarnak
Alreadypublished
17 W.Dicks&M.Dunwoody Groupsactingongraphs
18 L.J.Corwin&F.P.Greenleaf RepresentationsofnilpotentLiegroupsandtheir
applications
19 R.Fritsch&R.Piccinini Cellularstructuresintopology
20 H.Klingen IntroductorylecturesonSiegelmodularforms
21 P.Koosis ThelogarithmicintegralII
22 M.J.Collins Representationsandcharactersoffinitegroups
24 H.Kunita Stochasticflowsandstochasticdifferentialequations
25 P.Wojtaszczyk Banachspacesforanalysts
26 J.E.Gilbert&M.A.M.Murray CliffordalgebrasandDiracoperatorsin
harmonicanalysis
27 A.Frohlich&M.J.Taylor Algebraicnumbertheory
28 K.Goebel&W.A.Kirk Topicsinmetricfixedpointtheory
29 J.F.Humphreys ReflectiongroupsandCoxetergroups
30 D.J.Benson RepresentationsandcohomologyI
31 D.J.Benson RepresentationsandcohomologyII
32 C.Allday&V.Puppe Cohomologicalmethodsintransformationgroups
33 C.Souleetal. LecturesonArakelovgeometry
34 A.Ambrosetti&G.Prodi Aprimerofnonlinearanalysis
35 J.Palis&F.Takens Hyperbolicity,stabilityandchaosathomoclinicbifurcations
37 Y.Meyer Waveletsandoperators1
38 C.Weibel Anintroductiontohomologicalalgebra
39 W.Bruns&J.Herzog Cohen-Macaulayrings
40 V.Snaith ExplicitBrauerinduction
41 G.Laumon CohomologyofDrinfeldmodularvarietiesI
42 E.B.Davies Spectraltheoryanddifferentialoperators
43 J.Diestel,H.Jarchow,&A.Tonge Absolutelysummingoperators
44 P.Mattila GeometryofsetsandmeasuresinEuclideanspaces
45 R.Pinsky Positiveharmonicfunctionsanddiffusion
46 G.Tenenbaum Introductiontoanalyticandprobabilisticnumbertheory
47 C.Peskine Analgebraicintroductiontocomplexprojectivegeometry
48 Y.Meyer&R.Coifman Wavelets
49 R.Stanley EnumerativecombinatoricsI
50 I.Porteous Cliffordalgebrasandtheclassicalgroups
51 M.Audin Spinningtops
52 V.Jurdjevic Geometriccontroltheory
53 H.Volklein GroupsasGaloisgroups
54 J.LePotier Lecturesonvectorbundles
55 D.Bump Automorphicformsandrepresentations
56 G.Laumon CohomologyofDrinfeldmodularvarietiesII
57 D.M.Clark&B.A.Davey Naturaldualitiesfortheworkingalgebraist
58 J.McCleary Auser’sguidetospectralsequencesII
59 P.Taylor Practicalfoundationsofmathematics
60 M.P.Brodmann&R.Y.Sharp Localcohomology
61 J.D.Dixonetal. Analyticpro-Pgroups
62 R.Stanley EnumerativecombinatoricsII
63 R.M.Dudley Uniformcentrallimittheorems
64 J.Jost&X.Li-Jost Calculusofvariations
65 A.J.Berrick&M.E.Keating Anintroductiontoringsandmodules
66 S.Morosawa Holomorphicdynamics
67 A.J.Berrick&M.E.Keating CategoriesandmoduleswithK-theoryinview
68 K.Sato Levyprocessesandinfinitelydivisibledistributions
69 H.Hida ModularformsandGaloiscohomology
70 R.Iorio&V.Iorio Fourieranalysisandpartialdifferentialequations
71 R.Blei Analysisinintegerandfractionaldimensions
72 F.Borceaux&G.Janelidze Galoistheories
73 B.Bollobas Randomgraphs
REAL ANALYSIS AND
PROBABILITY
R. M. DUDLEY
MassachusettsInstituteofTechnology
The Pitt Building, Trumpington Street, Cambridge, United Kingdom
The Edinburgh Building, Cambridge CB2 2RU, UK
40 West 20th Street, New York, NY 10011-4211, USA
477 Williamstown Road, Port Melbourne, VIC 3207, Australia
Ruiz de Alarcón 13, 28014 Madrid, Spain
Dock House, The Waterfront, Cape Town 8001, South Africa
http://www.cambridge.org
©R. M. Dudley 2004
First published in printed format 2002
ISBN 0-511-04208-6 eBook (netLibrary)
ISBN 0-521-80972-X hardback
ISBN 0-521-00754-2 paperback
Contents
Preface to the Cambridge Edition page ix
1 Foundations; Set Theory 1
1.1 Definitions for Set Theory and the Real Number System 1
1.2 Relations and Orderings 9
*1.3 Transfinite Induction and Recursion 12
1.4 Cardinality 16
1.5 The Axiom of Choice and Its Equivalents 18
2 General Topology 24
2.1 Topologies, Metrics, and Continuity 24
2.2 Compactness and Product Topologies 34
2.3 Complete and Compact Metric Spaces 44
2.4 Some Metrics for Function Spaces 48
2.5 Completion and Completeness of Metric Spaces 58
*2.6 Extension of Continuous Functions 63
*2.7 Uniformities and Uniform Spaces 67
*2.8 Compactification 71
3 Measures 85
3.1 Introduction to Measures 85
3.2 Semirings and Rings 94
3.3 Completion of Measures 101
3.4 Lebesgue Measure and Nonmeasurable Sets 105
*3.5 Atomic and Nonatomic Measures 109
4 Integration 114
4.1 Simple Functions 114
*4.2 Measurability 123
4.3 Convergence Theorems for Integrals 130
v
vi Contents
4.4 Product Measures 134
*4.5 Daniell-Stone Integrals 142
5 L p Spaces; Introduction to Functional Analysis 152
5.1 Inequalities for Integrals 152
5.2 Norms and Completeness of Lp 158
5.3 Hilbert Spaces 160
5.4 Orthonormal Sets and Bases 165
5.5 LinearFormsonHilbertSpaces,InclusionsofLpSpaces,
and Relations Between Two Measures 173
5.6 Signed Measures 178
6 Convex Sets and Duality of Normed Spaces 188
6.1 Lipschitz, Continuous, and Bounded Functionals 188
6.2 Convex Sets and Their Separation 195
6.3 Convex Functions 203
*6.4 Duality of Lp Spaces 208
6.5 Uniform Boundedness and Closed Graphs 211
*6.6 The Brunn-Minkowski Inequality 215
7 Measure, Topology, and Differentiation 222
7.1 Baire and Borelσ-Algebras and Regularity of Measures 222
*7.2 Lebesgue’s Differentiation Theorems 228
*7.3 The Regularity Extension 235
*7.4 The Dual of C(K) and Fourier Series 239
*7.5 Almost Uniform Convergence and Lusin’s Theorem 243
8 Introduction to Probability Theory 250
8.1 Basic Definitions 251
8.2 Infinite Products of Probability Spaces 255
8.3 Laws of Large Numbers 260
*8.4 Ergodic Theorems 267
9 Convergence of Laws and Central Limit Theorems 282
9.1 Distribution Functions and Densities 282
9.2 Convergence of Random Variables 287
9.3 Convergence of Laws 291
9.4 Characteristic Functions 298
9.5 UniquenessofCharacteristicFunctions
and a Central Limit Theorem 303
9.6 Triangular Arrays and Lindeberg’s Theorem 315
9.7 Sums of Independent Real Random Variables 320
Contents vii
*9.8 TheLe´vyContinuityTheorem;InfinitelyDivisible
and Stable Laws 325
10 Conditional Expectations and Martingales 336
10.1 Conditional Expectations 336
10.2 RegularConditionalProbabilitiesandJensen’s
Inequality 341
10.3 Martingales 353
10.4 Optional Stopping and Uniform Integrability 358
10.5 Convergence of Martingales and Submartingales 364
*10.6 Reversed Martingales and Submartingales 370
*10.7 Subadditive and Superadditive Ergodic Theorems 374
11 Convergence of Laws on Separable Metric Spaces 385
11.1 Laws and Their Convergence 385
11.2 Lipschitz Functions 390
11.3 Metrics for Convergence of Laws 393
11.4 Convergence of Empirical Measures 399
11.5 Tightness and Uniform Tightness 402
*11.6 Strassen’sTheorem:NearbyVariables
with Nearby Laws 406
*11.7 AUniformityforLawsandAlmostSurelyConverging
Realizations of Converging Laws 413
*11.8 Kantorovich-Rubinstein Theorems 420
*11.9 U-Statistics 426
12 Stochastic Processes 439
12.1 Existence of Processes and Brownian Motion 439
12.2 The Strong Markov Property of Brownian Motion 450
12.3 ReflectionPrinciples,TheBrownianBridge,
and Laws of Suprema 459
12.4 LawsofBrownianMotionatMarkovTimes:
Skorohod Imbedding 469
12.5 Laws of the Iterated Logarithm 476
13 Measurability: Borel Isomorphism and Analytic Sets 487
*13.1 Borel Isomorphism 487
*13.2 Analytic Sets 493
Appendix A Axiomatic Set Theory 503
A.1 Mathematical Logic 503
A.2 Axioms for Set Theory 505
viii Contents
A.3 Ordinals and Cardinals 510
A.4 From Sets to Numbers 515
AppendixB ComplexNumbers,VectorSpaces,
and Taylor’s Theorem with Remainder 521
Appendix C The Problem of Measure 526
Appendix D Rearranging Sums of Nonnegative Terms 528
Appendix E Pathologies of Compact Nonmetric Spaces 530
Author Index 541
Subject Index 546
Notation Index 554
Preface to the Cambridge Edition
This is a text at the beginning graduate level. Some study of intermediate
analysis in Euclidean spaces will provide helpful background, but in this
editionsuchbackgroundisnotaformalprerequisite.Effortstomakethebook
moreself-containedincludeinsertingmaterialontherealnumbersysteminto
Chapter1,addingatreatmentoftheStone-Weierstrasstheorem,andgenerally
eliminating references for proofs to other books except at very few points,
suchassomecomplexvariabletheoryinAppendixB.
Chapters1through5provideaone-semestercourseinrealanalysis.Fol-
lowingthat,aone-semestercourseonprobabilitycanbebasedonChapters
8 through 10 and parts of 11 and 12. Starred paragraphs and sections, such
asthosefoundinChapter6andmostofChapter7,arecalledonrarely,ifat
all,laterinthebook.Theycanbeskipped,atleastonfirstreading,oruntil
needed.
Relativelyfewproofsoflessvitalfactshavebeenlefttothereader.Iwould
beverygladtoknowofanysubstantialunintentionalgapsorerrors.Although
I have worked and checked all the problems and hints, experience suggests
thatmistakesinproblems,andhintsthatmaymislead,arelessobviousthan
errorsinthetext.Sotakehintswithagrainofsaltandperhapsmakeafirst
tryattheproblemswithoutusingthehints.
Ilookedforthebestandshortestavailableproofsforthetheorems.Short
proofsthathaveappearedinjournalarticles,butinfewifanyothertextbooks,
aregivenforthecompletionofmetricspaces,thestronglawoflargenumbers,
the ergodic theorem, the martingale convergence theorem, the subadditive
ergodictheorem,andtheHartman-Wintnerlawoftheiteratedlogarithm.
Around1950,whenHalmos’classicMeasureTheoryappeared,themore
advanced parts of the subject headed toward measures on locally compact
spaces, as in, for example, §7.3 of this book. Since then, much of the re-
searchinprobabilitytheoryhasmovedmoreinthedirectionofmetricspaces.
Chapter11givessomefactsconnectingmetricsandprobabilitieswhichfol-
lowthenewertrend.AppendixEindicateswhatcangowrongwithmeasures
ix
