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REAL ANALYSIS AND PROBABILITY
This much admired textbook, now reissued in paperback, offers a clear expo-
sition of modern probability theory and of the interplay between the properties
of metric spaces and probability measures.
The first half of the book gives an exposition of real analysis: basic set
theory, general topology, measure theory, integration, an introduction to func-
tional analysis in Banach and Hilbert spaces, convex sets and functions,
and measure on topological spaces. The second half introduces probability
based on measure theory, including laws of large numbers, ergodic theorems,
the central limit theorem, conditional expectations, and martingale conver-
gence. A chapter on stochastic processes introduces Brownian motion and the
Brownian bridge.
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
the extensive historical notes have been further amplified. A number of new
exercises, and hints for solution of old and new ones, have been added.
R. M. Dudley is Professor of Mathematics at the Massachusetts Institute of
Technology in Cambridge, Massachusetts.
CAMBRIDGE STUDIES IN ADVANCED MATHEMATICS
Editorial Board:
B. Bollobas, W. Fulton, A. Katok, F. Kirwan, P. Sarnak
Already published
17 W. Dicks & M. Dunwoody Groups acting on graphs
18 L.J. Corwin & F.P. Greenleaf Representations of nilpotent Lie groups and their
applications
19 R. Fritsch & R. Piccinini Cellular structures in topology
20 H. Klingen Introductory lectures on Siegel modular forms
21 P. Koosis The logarithmic integral II
22 M.J. Collins Representations and characters of finite groups
24 H. Kunita Stochastic flows and stochastic differential equations
25 P. Wojtaszczyk Banach spaces for analysts
26 J.E. Gilbert & M.A.M. Murray Clifford algebras and Dirac operators in
harmonic analysis
27 A. Frohlich & M.J. Taylor Algebraic number theory
28 K. Goebel & W.A. Kirk Topics in metric fixed point theory
29 J.F. Humphreys Reflection groups and Coxeter groups
30 D.J. Benson Representations and cohomology I
31 D.J. Benson Representations and cohomology II
32 C. Allday & V. Puppe Cohomological methods in transformation groups
33 C. Soule et al. Lectures on Arakelov geometry
34 A. Ambrosetti & G. Prodi A primer of nonlinear analysis
35 J. Palis & F. Takens Hyperbolicity, stability and chaos at homoclinic bifurcations
37 Y. Meyer Wavelets and operators 1
38 C. Weibel An introduction to homological algebra
39 W. Bruns & J. Herzog Cohen-Macaulay rings
40 V. Snaith Explicit Brauer induction
41 G. Laumon Cohomology of Drinfeld modular varieties I
42 E.B. Davies Spectral theory and differential operators
43 J. Diestel, H. Jarchow, & A. Tonge Absolutely summing operators
44 P. Mattila Geometry of sets and measures in Euclidean spaces
45 R. Pinsky Positive harmonic functions and diffusion
46 G. Tenenbaum Introduction to analytic and probabilistic number theory
47 C. Peskine An algebraic introduction to complex projective geometry
48 Y. Meyer & R. Coifman Wavelets
49 R. Stanley Enumerative combinatorics I
50 I. Porteous Clifford algebras and the classical groups
51 M. Audin Spinning tops
52 V. Jurdjevic Geometric control theory
53 H. Volklein Groups as Galois groups
54 J. Le Potier Lectures on vector bundles
55 D. Bump Automorphic forms and representations
56 G. Laumon Cohomology of Drinfeld modular varieties II
57 D.M. Clark & B.A. Davey Natural dualities for the working algebraist
58 J. McCleary A user’s guide to spectral sequences II
59 P. Taylor Practical foundations of mathematics
60 M.P. Brodmann & R.Y. Sharp Local cohomology
61 J.D. Dixon et al. Analytic pro-P groups
62 R. Stanley Enumerative combinatorics II
63 R.M. Dudley Uniform central limit theorems
64 J. Jost & X. Li-Jost Calculus of variations
65 A.J. Berrick & M.E. Keating An introduction to rings and modules
66 S. Morosawa Holomorphic dynamics
67 A.J. Berrick & M.E. Keating Categories and modules with K-theory in view
68 K. Sato Levy processes and infinitely divisible distributions
69 H. Hida Modular forms and Galois cohomology
70 R. Iorio & V. Iorio Fourier analysis and partial differential equations
71 R. Blei Analysis in integer and fractional dimensions
72 F. Borceaux & G. Janelidze Galois theories
73 B. Bollobas Random graphs
REAL ANALYSIS AND
PROBABILITY
R. M. DUDLEY
Massachusetts Institute of Technology
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
ISBN 0-521-80972-X hardback
ISBN 0-521-00754-2 paperback
(netLibrary)
Contents
Preface to the Cambridge Edition
page ix
1 Foundations; Set Theory
1.1 Definitions for Set Theory and the Real Number System
1.2 Relations and Orderings
*1.3 Transfinite Induction and Recursion
1.4 Cardinality
1.5 The Axiom of Choice and Its Equivalents
2 General Topology
2.1 Topologies, Metrics, and Continuity
2.2 Compactness and Product Topologies
2.3 Complete and Compact Metric Spaces
2.4 Some Metrics for Function Spaces
2.5 Completion and Completeness of Metric Spaces
*2.6 Extension of Continuous Functions
*2.7 Uniformities and Uniform Spaces
*2.8 Compactification
3 Measures
3.1 Introduction to Measures
3.2 Semirings and Rings
3.3 Completion of Measures
3.4 Lebesgue Measure and Nonmeasurable Sets
*3.5 Atomic and Nonatomic Measures
4 Integration
4.1 Simple Functions
*4.2 Measurability
4.3 Convergence Theorems for Integrals
v
1
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vi
Contents
4.4 Product Measures
*4.5 Daniell-Stone Integrals
5 L p Spaces; Introduction to Functional Analysis
5.1 Inequalities for Integrals
5.2 Norms and Completeness of Lp
5.3 Hilbert Spaces
5.4 Orthonormal Sets and Bases
5.5 Linear Forms on Hilbert Spaces, Inclusions of Lp Spaces,
and Relations Between Two Measures
5.6 Signed Measures
6 Convex Sets and Duality of Normed Spaces
6.1 Lipschitz, Continuous, and Bounded Functionals
6.2 Convex Sets and Their Separation
6.3 Convex Functions
*6.4 Duality of L p Spaces
6.5 Uniform Boundedness and Closed Graphs
*6.6 The Brunn-Minkowski Inequality
7 Measure, Topology, and Differentiation
7.1 Baire and Borel σ-Algebras and Regularity of Measures
*7.2 Lebesgue’s Differentiation Theorems
*7.3 The Regularity Extension
*7.4 The Dual of C(K) and Fourier Series
*7.5 Almost Uniform Convergence and Lusin’s Theorem
8 Introduction to Probability Theory
8.1 Basic Definitions
8.2 Infinite Products of Probability Spaces
8.3 Laws of Large Numbers
*8.4 Ergodic Theorems
9 Convergence of Laws and Central Limit Theorems
9.1 Distribution Functions and Densities
9.2 Convergence of Random Variables
9.3 Convergence of Laws
9.4 Characteristic Functions
9.5 Uniqueness of Characteristic Functions
and a Central Limit Theorem
9.6 Triangular Arrays and Lindeberg’s Theorem
9.7 Sums of Independent Real Random Variables
134
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Contents
*9.8 The L´evy Continuity Theorem; Infinitely Divisible
and Stable Laws
10 Conditional Expectations and Martingales
10.1 Conditional Expectations
10.2 Regular Conditional Probabilities and Jensen’s
Inequality
10.3 Martingales
10.4 Optional Stopping and Uniform Integrability
10.5 Convergence of Martingales and Submartingales
*10.6 Reversed Martingales and Submartingales
*10.7 Subadditive and Superadditive Ergodic Theorems
11 Convergence of Laws on Separable Metric Spaces
11.1 Laws and Their Convergence
11.2 Lipschitz Functions
11.3 Metrics for Convergence of Laws
11.4 Convergence of Empirical Measures
11.5 Tightness and Uniform Tightness
*11.6 Strassen’s Theorem: Nearby Variables
with Nearby Laws
*11.7 A Uniformity for Laws and Almost Surely Converging
Realizations of Converging Laws
*11.8 Kantorovich-Rubinstein Theorems
*11.9 U-Statistics
12 Stochastic Processes
12.1 Existence of Processes and Brownian Motion
12.2 The Strong Markov Property of Brownian Motion
12.3 Reflection Principles, The Brownian Bridge,
and Laws of Suprema
12.4 Laws of Brownian Motion at Markov Times:
Skorohod Imbedding
12.5 Laws of the Iterated Logarithm
13 Measurability: Borel Isomorphism and Analytic Sets
*13.1 Borel Isomorphism
*13.2 Analytic Sets
Appendix A Axiomatic Set Theory
A.1 Mathematical Logic
A.2 Axioms for Set Theory
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