This classic textbook offers a clear exposition 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 functional 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's convergence. A chapter on stochastic processes introduces Brownian motion and the Brownian bridge. The edition has been made even more self-contained than before; it now includes 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 comprehensive historical notes have been further amplified. A number of new exercises have been added, together with hints for solution.

zlibzh/no-category/（美）达德利（Dudley，R.M.）著, (美)R.M.Dudley著, Dley Du, R. M Dudley/实分析和概率论 第2版_31133008.pdf

Real Analysis and Probability (Cambridge Studies in Advanced Mathematics, Series Number 74)

Real analysis and probability = 实分析和概率论 / monograph

Richard M. Dudley

Cambridge University Press (Virtual Publishing)

China Machine Press

Cambridge studies in advanced mathematics ;, 74, Cambridge, New York, England, 2002

Cambridge studies in advanced mathematics, 74, 2. ed., reprint, Cambridge, 2008

Cambridge studies in advanced mathematics, 74, 2nd ed, Cambridge, 2005

Jing dian yuan ban shu ku, Ying yin ban, Beijing, 2006

Cambridge University Press, Cambridge, 2002

United Kingdom and Ireland, United Kingdom

China, People's Republic, China

2, 2002

Bookmarks: p1 (p1): 1 Foundations;Set Theory
p1-1 (p1): 1.1 Definitions for Set Theory and the Real Number System
p1-2 (p9): 1.2 Relations and Orderings
p1-3 (p12): 1.3 Transfinite Induction and Recursion
p1-4 (p16): 1.4 Cardinality
p1-5 (p18): 1.5 The Axiom of Choice and Its Equivalents
p2 (p24): 2 General Topology
p2-1 (p24): 2.1 Topologies,Metrics,and Continuity
p2-2 (p34): 2.2 Compactness and Product Topologies
p2-3 (p44): 2.3 Complete and Compact Metric Spaces
p2-4 (p48): 2.4 Some Metrics for Function Spaces
p2-5 (p58): 2.5 Completion and Completeness of Metric Spaces
p2-6 (p63): 2.6 Extension of Continuous Functions
p2-7 (p67): 2.7 Uniformities and Uniform Spaces
p2-8 (p71): 2.8 Compactification
p3 (p85): 3 Measures
p3-1 (p85): 3.1 Introduction to Measures
p3-2 (p94): 3.2 Semirings and Rings
p3-3 (p101): 3.3 Completion of Measures
p3-4 (p105): 3.4 Lebesgue Measure and Nonmeasurable Sets
p3-5 (p109): 3.5 Atomic and Nonatomic Measures
p4 (p114): 4 Integration
p4-1 (p114): 4.1 Simple Functions
p4-2 (p123): 4.2 Measurability
p4-3 (p130): 4.3 Convergence Theorems for Integrals
p4-4 (p134): 4.4 Product Measures
p4-5 (p142): 4.5 Daniell-Stone Integrals
p5 (p152): 5 Lp Spaces;Introduction to Functional Analysis
p5-1 (p152): 5.1 Inequalities for Integrals
p5-2 (p158): 5.2 Norms and Completeness of Lp
p5-3 (p160): 5.3 Hilbert Spaces
p5-4 (p165): 5.4Orthonormal Sets and Bases
p5-5 (p173): 5.5 LinearForms on Hilbert Spaces,Inclusions of Lp Spaces,and Relations Between Two Measures
p5-6 (p178): 5.6 Signed Measures
p6 (p188): 6 Convex Sets and Duality of Normed Spaces
p6-1 (p188): 6.1 Lipschitz,Continuous,and Bounded Functionals
p6-2 (p195): 6.2 Convex Sets and Their Separation
p6-3 (p203): 6.3 Convex Functions
p6-4 (p208): 6.4 Duality of Lp Spaces
p6-5 (p211): 6.5 Uniform Boundedness and Closed Graphs
p6-6 (p215): 6.6 The Brunn-Minkowski Inequality
p7 (p222): 7 Measure,Topology,and Differentiation
p7-1 (p222): 7.1 Baire and Borel σ-Algebras and Regularity of Measures
p7-2 (p228): 7.2 Lebesgue's Differentiation Theorems
p7-3 (p235): 7.3 The Regularity Extension
p7-4 (p239): 7.4 The Dual of C(K)and Fourier Series
p7-5 (p243): 7.5 Almost Uniform Convergence and Lusin's Theorem
p8 (p250): 8 Introduction to Probability Theory
p8-1 (p251): 8.1 Basic Definitions
p8-2 (p255): 8.2 Infinite Products of Probability Spaces
p8-3 (p260): 8.3 Laws of Large Numbers
p8-4 (p267): 8.4 Ergodic Theorems
p9 (p282): 9 Convergence of Laws and Central Limit Theorems
p9-1 (p282): 9.1 Distribution Functions and Densities
p9-2 (p287): 9.2 Convergence of Random Variables
p9-3 (p291): 9.3 Convergence of Laws
p9-4 (p298): 9.4 Characteristic Functions
p9-5 (p303): 9.5 Uniqueness of Characteristic Functions and a Central Limit Theorem
p9-6 (p315): 9.6 Triangular Arrays and Lindeberg's Theorem
p9-7 (p320): 9.7 Sums of Independent Real Random Variables
p9-8 (p325): 9.8 The Lévy Continuity Theorem;Infinitely Divisible and Stable Laws
p10 (p336): 10 Conditional Expectations and Martingales
p10-1 (p336): 10.1 Conditional Expectations
p10-2 (p341): 10.2 Regular Conditional Probabilities and Jensen's Inequality
p10-3 (p353): 10.3 Martingales
p10-4 (p358): 10.4 Optional Stopping and Uniform Integrability
p10-5 (p364): 10.5 Convergence of Martingales and Submartingales
p10-6 (p370): 10.6 Reversed Martingales and Submartingales
p10-7 (p374): 10.7 Subadditive and Superadditive Ergodic Theorems
p11 (p385): 11 Convergence of Laws on Separable Metric Spaces
p11-1 (p385): 11.1 Laws and Their Convergence
p11-2 (p390): 11.2 Lipschitz Functions
p11-3 (p393): 11.3 Metrics for Convergence of Laws
p11-4 (p399): 11.4 Convergence of Empirical Measures
p11-5 (p402): 11.5 Tightness and Uniform Tightness
p11-6 (p406): 11.6 Strassen's Theorem:Nearby Variables with Nearby Laws
p11-7 (p413): 11.7 A Uniformity for Laws and Almost Surely Converging Realizations of Converging Laws
p11-8 (p420): 11.8 Kantorovich-Rubinstein Theorems
p11-9 (p426): 11.9 U-Statistics
p12 (p439): 12 Stochastic Processes
p12-1 (p439): 12.1 Existence of Processes and Brownian Motion
p12-2 (p450): 12.2 The Strong Markov Property of Brownian Motion
p12-3 (p459): 12.3 Reflection Principles,The Brownian Bridge,and Laws of Suprema
p12-4 (p469): 12.4 Laws of Brownian Motion at Markov Times:Skorohod Imbedding
p12-5 (p476): 12.5 Laws of the Iterated Logarithm
p13 (p487): 13 Measurability:Borel Isomorphism and Analytic Sets
p13-1 (p487): 13.1 Borel Isomorphism
p13-2 (p493): 13.2 Analytic Sets
p14 (p503): Appendix A Axiomatic Set Theory
p14-1 (p503): A.1 Mathematical Logic
p14-2 (p505): A.2 Axioms for Set Theory
p14-3 (p510): A.3 Ordinals and Cardinals
p14-4 (p515): A.4 From Sets to Numbers
p15 (p521): Appendix B Complex Numbers,Vector Spaces,and Taylor's Theorem with Remainder
p16 (p526): Appendix C The Problem of Measure
p17 (p528): Appendix D Rearranging Sums of Nonnegative Terms
p18 (p530): Appendix E Pathologies of Compact Nonmetric Spaces
p19 (p541): Author Index
p20 (p546): Subject Index
p21 (p554): Notation Index

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Includes bibliographical references and index.

This classic textbook, now reissued, offers a clear exposition of modern probability theory and of the interplay between the properties of metric spaces and probability measures. The new edition has been made even more self-contained than before; it now includes 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 comprehensive historical notes have been further amplified. A number of new exercises have been added, together with hints for solution.

2024-06-13