• 1. Measure theory--
• 2. Laws of large numbers--
• 3. Central limit theorems--
• 4. Martingales--
• 5. Markov chains--
• 6. Ergodic theorems--
• 7. Brownian motion--
• 8. Applications to random walk--
• 9. Multidimensional Brownian motion-- Appendix. Measure theory details.
• (source: Nielsen Book Data)

This lively introduction to measure-theoretic probability theory covers laws of large numbers, central limit theorems, random walks, martingales, Markov chains, ergodic theorems, and Brownian motion. Concentrating on results that are the most useful for applications, this comprehensive treatment is a rigorous graduate text and reference. Operating under the philosophy that the best way to learn probability is to see it in action, the book contains extended examples that apply the theory to concrete applications. This fifth edition contains a new chapter on multidimensional Brownian motion and its relationship to partial differential equations (PDEs), an advanced topic that is finding new applications. Setting the foundation for this expansion, Chapter 7 now features a proof of Ito's formula. Key exercises that previously were simply proofs left to the reader have been directly inserted into the text as lemmas. The new edition re-instates discussion about the central limit theorem for martingales and stationary sequences.
(source: Nielsen Book Data)

• chapter 1 Foundations; Set Theory / Richard M. Dudley
• chapter 2 General Topology / Richard M. Dudley
• chapter 3 Measures / Richard M. Dudley
• chapter 4 Integration / Richard M. Dudley
• chapter 5 Lp Spaces; Introduction to Functional Analysis / Richard M. Dudley
• chapter 6 Convex Sets and Duality of Normed Spaces / Richard M. Dudley
• chapter 7 Measure, Topology, and Differentiation / Richard M. Dudley
• chapter 8 Introduction to Probability Theory / Richard M. Dudley
• chapter 9 Convergence of Laws and Central Limit Theorems / Richard M. Dudley
• chapter 10 Conditional Expectations and Martingales / Richard M. Dudley
• chapter 11 Convergence of Laws on Separable Metric Spaces / Richard M. Dudley
• chapter 12 Stochastic Processes / Richard M. Dudley
• chapter 13 Measurability: Borel Isomorphism and Analytic Sets / Richard M. Dudley

Written by one of the best-known probabilists in the world this text offers a clear and modern presentation of modern probability theory and an exposition of the interplay between the properties of metric spaces and those of probability measures. This text is the first at this level to include discussions of the subadditive ergodic theorems, metrics for convergence in laws and the Borel isomorphism theory. The proofs for the theorems are consistently brief and clear and each chapter concludes with a set of historical notes and references. This book should be of interest to students taking degree courses in real analysis and/or probability theory.
(source: Nielsen Book Data)

• FOREWORD xi PREFACE xiii Patrick Billingsley 1925
• 2011 xv Chapter1 PROBABILITY 1
• 1. BOREL S NORMAL NUMBER THEOREM, 1 The Unit Interval The Weak Law of Large Numbers The Strong Law of Large Numbers Strong Law Versus Weak Length The Measure Theory of Diophantine Approximation*
• 2. PROBABILITY MEASURES, 18 Spaces Assigning Probabilities Classes of Sets Probability Measures Lebesgue Measure on the Unit Interval Sequence Space* Constructing s-Fields*
• 3. EXISTENCE AND EXTENSION, 39 Construction of the Extension Uniqueness and the p ? Theorem Monotone Classes Lebesgue Measure on the Unit Interval Completeness Nonmeasurable Sets Two Impossibility Theorems*
• 4. DENUMERABLE PROBABILITIES, 53 General Formulas Limit Sets Independent Events Subfields The Borel-Cantelli Lemmas The Zero-One Law
• 5. SIMPLE RANDOM VARIABLES, 72 Definition Convergence of Random Variables Independence Existence of Independent Sequences Expected Value Inequalities
• 6. THE LAW OF LARGE NUMBERS, 90 The Strong Law The Weak Law Bernstein's Theorem A Refinement of the Second Borel-Cantelli Lemma
• 7. GAMBLING SYSTEMS, 98 Gambler's Ruin Selection Systems Gambling Policies Bold Play* Timid Play*
• 8. MARKOV CHAINS, 117 Definitions Higher-Order Transitions An Existence Theorem Transience and Persistence Another Criterion for Persistence Stationary Distributions Exponential Convergence* Optimal Stopping*
• 9. LARGE DEVIATIONS AND THE LAW OF THE ITERATED LOGARITHM, 154 Moment Generating Functions Large Deviations Chernoff's Theorem* The Law of the Iterated Logarithm Chapter2 MEASURE 167
• 10. GENERAL MEASURES, 167 Classes of Sets Conventions Involving 8 Measures Uniqueness
• 11. OUTER MEASURE, 174 Outer Measure Extension An Approximation Theorem
• 12. MEASURES IN EUCLIDEAN SPACE, 181 Lebesgue Measure Regularity Specifying Measures on the Line Specifying Measures in Rk Strange Euclidean Sets*
• 13. MEASURABLE FUNCTIONS AND MAPPINGS, 192 Measurable Mappings Mappings into Rk Limits and Measurability Transformations of Measures
• 14. DISTRIBUTION FUNCTIONS, 198 Distribution Functions Exponential Distributions Weak Convergence Convergence of Types* Extremal Distributions* Chapter3 INTEGRATION 211
• 15. THE INTEGRAL, 211 Definition Nonnegative Functions Uniqueness
• 16. PROPERTIES OF THE INTEGRAL, 218 Equalities and Inequalities Integration to the Limit Integration over Sets Densities Change of Variable Uniform Integrability Complex Functions
• 17. THE INTEGRAL WITH RESPECT TO LEBESGUE MEASURE, 234 The Lebesgue Integral on the Line The Riemann Integral The Fundamental Theorem of Calculus Change of Variable The Lebesgue Integral in Rk Stieltjes Integrals
• 18. PRODUCT MEASURE AND FUBINI S THEOREM, 245 Product Spaces Product Measure Fubini's Theorem Integration by Parts Products of Higher Order
• 19. THE Lp SPACES*, 256 Definitions Completeness and Separability Conjugate Spaces Weak Compactness Some Decision Theory The Space L2 An Estimation Problem Chapter4 RANDOM VARIABLES AND EXPECTED VALUES 271
• 20. RANDOM VARIABLES AND DISTRIBUTIONS, 271 Random Variables and Vectors Subfields Distributions Multidimensional Distributions Independence Sequences of Random Variables Convolution Convergence in Probability The Glivenko-Cantelli Theorem*
• 21. EXPECTED VALUES, 291 Expected Value as Integral Expected Values and Limits Expected Values and Distributions Moments Inequalities Joint Integrals Independence and Expected Value Moment Generating Functions
• 22. SUMS OF INDEPENDENT RANDOM VARIABLES, 300 The Strong Law of Large Numbers The Weak Law and Moment Generating Functions Kolmogorov's Zero-One Law Maximal Inequalities Convergence of Random Series Random Taylor Series*
• 23. THE POISSON PROCESS, 316 Characterization of the Exponential Distribution The Poisson Process The Poisson Approximation Other Characterizations of the Poisson Process Stochastic Processes
• 24. THE ERGODIC THEOREM*, 330 Measure-Preserving Transformations Ergodicity Ergodicity of Rotations Proof of the Ergodic Theorem The Continued-Fraction Transformation Diophantine Approximation Chapter5 CONVERGENCE OF DISTRIBUTIONS 349
• 25. WEAK CONVERGENCE, 349 Definitions Uniform Distribution Modulo 1* Convergence in Distribution Convergence in Probability Fundamental Theorems Helly's Theorem Integration to the Limit
• 26. CHARACTERISTIC FUNCTIONS, 365 Definition Moments and Derivatives Independence Inversion and the Uniqueness Theorem The Continuity Theorem Fourier Series*
• 27. THE CENTRAL LIMIT THEOREM, 380 Identically Distributed Summands The Lindeberg and Lyapounov Theorems Dependent Variables*
• 28. INFINITELY DIVISIBLE DISTRIBUTIONS*, 394 Vague Convergence The Possible Limits Characterizing the Limit
• 29. LIMIT THEOREMS IN Rk, 402 The Basic Theorems Characteristic Functions Normal Distributions in Rk The Central Limit Theorem
• 30. THE METHOD OF MOMENTS*, 412 The Moment Problem Moment Generating Functions Central Limit Theorem by Moments Application to Sampling Theory Application to Number Theory Chapter6 DERIVATIVES AND CONDITIONAL PROBABILITY 425
• 31. DERIVATIVES ON THE LINE*, 425 The Fundamental Theorem of Calculus Derivatives of Integrals Singular Functions Integrals of Derivatives Functions of Bounded Variation
• 32. THE RADON NIKODYM THEOREM, 446 Additive Set Functions The Hahn Decomposition Absolute Continuity and Singularity The Main Theorem
• 33. CONDITIONAL PROBABILITY, 454 The Discrete Case The General Case Properties of Conditional Probability Difficulties and Curiosities Conditional Probability Distributions
• 34. CONDITIONAL EXPECTATION, 472 Definition Properties of Conditional Expectation Conditional Distributions and Expectations Sufficient Subfields* Minimum-Variance Estimation*
• 35. MARTINGALES, 487 Definition Submartingales Gambling Functions of Martingales Stopping Times Inequalities Convergence Theorems Applications: Derivatives Likelihood Ratios Reversed Martingales Applications: de Finetti's Theorem Bayes Estimation A Central Limit Theorem* Chapter7 STOCHASTIC PROCESSES 513
• 36. KOLMOGOROV'S EXISTENCE THEOREM, 513 Stochastic Processes Finite-Dimensional Distributions Product Spaces Kolmogorov's Existence Theorem The Inadequacy of RT A Return to Ergodic Theory The Hewitt Savage Theorem*
• 37. BROWNIAN MOTION, 530 Definition Continuity of Paths Measurable Processes Irregularity of Brownian Motion Paths The Strong Markov Property The Reflection Principle Skorohod Embedding Invariance*
• 38. NONDENUMERABLE PROBABILITIES, 558 Introduction Definitions Existence Theorems Consequences of Separability*
• APPENDIX 571 NOTES ON THE PROBLEMS 587 BIBLIOGRAPHY 617 INDEX 619.
• (source: Nielsen Book Data)

Praise for the Third Edition "It is, as far as I'm concerned, among the best books in math ever written...if you are a mathematician and want to have the top reference in probability, this is it." (Amazon.com, January 2006) A complete and comprehensive classic in probability and measure theory Probability and Measure, Anniversary Edition by Patrick Billingsley celebrates the achievements and advancements that have made this book a classic in its field for the past 35 years. Now re-issued in a new style and format, but with the reliable content that the third edition was revered for, this Anniversary Edition builds on its strong foundation of measure theory and probability with Billingsley's unique writing style. In recognition of 35 years of publication, impacting tens of thousands of readers, this Anniversary Edition has been completely redesigned in a new, open and user-friendly way in order to appeal to university-level students. This book adds a new foreward by Steve Lally of the Statistics Department at The University of Chicago in order to underscore the many years of successful publication and world-wide popularity and emphasize the educational value of this book. The Anniversary Edition contains features including: * An improved treatment of Brownian motion * Replacement of queuing theory with ergodic theory * Theory and applications used to illustrate real-life situations * Over 300 problems with corresponding, intensive notes and solutions * Updated bibliography * An extensive supplement of additional notes on the problems and chapter commentaries Patrick Billingsley was a first-class, world-renowned authority in probability and measure theory at a leading U.S. institution of higher education. He continued to be an influential probability theorist until his unfortunate death in 2011. Billingsley earned his Bachelor's Degree in Engineering from the U.S. Naval Academy where he served as an officer. he went on to receive his Master's Degree and doctorate in Mathematics from Princeton University.Among his many professional awards was the Mathematical Association of America's Lester R. Ford Award for mathematical exposition. His achievements through his long and esteemed career have solidified Patrick Billingsley's place as a leading authority in the field and been a large reason for his books being regarded as classics. This Anniversary Edition of Probability and Measure offers advanced students, scientists, and engineers an integrated introduction to measure theory and probability. Like the previous editions, this Anniversary Edition is a key resource for students of mathematics, statistics, economics, and a wide variety of disciplines that require a solid understanding of probability theory.
(source: Nielsen Book Data)

• 1. Measure theory--
• 2. Laws of large numbers--
• 3. Central limit theorems--
• 4. Random walks--
• 5. Martingales--
• 6. Markov chains--
• 7. Ergodic theorems--
• 8. Brownian motion-- Appendix A. Measure theory details.
• (source: Nielsen Book Data)

This classic introduction to probability theory for beginning graduate students covers laws of large numbers, central limit theorems, random walks, martingales, Markov chains, ergodic theorems, and Brownian motion. It is a comprehensive treatment concentrating on the results that are the most useful for applications. Its philosophy is that the best way to learn probability is to see it in action, so there are 200 examples and 450 problems. The new edition begins with a short chapter on measure theory to orient readers new to the subject.
(source: Nielsen Book Data)

• 1. Foundations: set theory--
• 2. General topology--
• 3. Measures--
• 4. Integration--
• 5. Lp spaces: introduction to functional analysis--
• 6. Convex sets and duality of normed spaces--
• 7. Measure, topology, and differentiation--
• 8. Introduction to probability theory--
• 9. Convergence of laws and central limit theorems--
• 10. Conditional expectations and martingales--
• 11. Convergence of laws on separable metric spaces--
• 12. Stochastic processes--
• 13. Measurability: Borel isomorphism and analytic sets-- Appendixes: A. Axiomatic set theory-- B. Complex numbers, vector spaces, and Taylor's theorem with remainder-- C. The problem of measure-- D. Rearranging sums of nonnegative terms-- E. Pathologies of compact nonmetric spaces-- Indices.
• (source: Nielsen Book Data)

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.
(source: Nielsen Book Data)

• 1. Foundations: set theory--
• 2. General topology--
• 3. Measures--
• 4. Integration--
• 5. Lp spaces: introduction to functional analysis--
• 6. Convex sets and duality of normed spaces--
• 7. Measure, topology, and differentiation--
• 8. Introduction to probability theory--
• 9. Convergence of laws and central limit theorems--
• 10. Conditional expectations and martingales--
• 11. Convergence of laws on separable metric spaces--
• 12. Stochastic processes--
• 13. Measurability: Borel isomorphism and analytic sets-- Appendixes: A. Axiomatic set theory-- B. Complex numbers, vector spaces, and Taylor's theorem with remainder-- C. The problem of measure-- D. Rearranging sums of nonnegative terms-- E. Pathologies of compact nonmetric spaces-- Indices.
• (source: Nielsen Book Data)

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.
(source: Nielsen Book Data)

• 1. Foundations: set theory--
• 2. General topology--
• 3. Measures--
• 4. Integration--
• 5. Lp spaces: introduction to functional analysis--
• 6. Convex sets and duality of normed spaces--
• 7. Measure, topology, and differentiation--
• 8. Introduction to probability theory--
• 9. Convergence of laws and central limit theorems--
• 10. Conditional expectations and martingales--
• 11. Convergence of laws on separable metric spaces--
• 12. Stochastic processes--
• 13. Measurability: Borel isomorphism and analytic sets-- Appendixes: A. Axiomatic set theory-- B. Complex numbers, vector spaces, and Taylor's theorem with remainder-- C. The problem of measure-- D. Rearranging sums of nonnegative terms-- E. Pathologies of compact nonmetric spaces-- Indices.
• (source: Nielsen Book Data)

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.
(source: Nielsen Book Data)

• 1. A branching-process example-- Part I. Foundations:
• 2. Measure spaces--
• 3. Events--
• 4. Random variables--
• 5. Independence--
• 6. Integration--
• 7. Expectation--
• 8. An easy strong law: product measure-- Part II. Martingale Theory:
• 9. Conditional expectation--
• 10. Martingales--
• 11. The convergence theorem--
• 12. Martingales bounded in L2--
• 13. Uniform integrability--
• 14. UI martingales--
• 15. Applications-- Part III. Characteristic Functions:
• 16. Basic properties of CFs--
• 17. Weak convergence--
• 18. The central limit theorem-- Appendices-- Exercises.
• (source: Nielsen Book Data)

Probability theory is nowadays applied in a huge variety of fields including physics, engineering, biology, economics and the social sciences. This book is a modern, lively and rigorous account which has Doob's theory of martingales in discrete time as its main theme. It proves important results such as Kolmogorov's Strong Law of Large Numbers and the Three-Series Theorem by martingale techniques, and the Central Limit Theorem via the use of characteristic functions. A distinguishing feature is its determination to keep the probability flowing at a nice tempo. It achieves this by being selective rather than encyclopaedic, presenting only what is essential to understand the fundamentals; and it assumes certain key results from measure theory in the main text. These measure-theoretic results are proved in full in appendices, so that the book is completely self-contained. The book is written for students, not for researchers, and has evolved through several years of class testing. Exercises play a vital role. Interesting and challenging problems, some with hints, consolidate what has already been learnt, and provide motivation to discover more of the subject than can be covered in a single introduction.
(source: Nielsen Book Data)