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• 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)

• 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)

• Weak Convergence in Metric Spaces. The Space C. The Space D. Dependent Variables. Other Modes of Convergence. Appendix. Some Notes on the Problems. Bibliographical Notes. Bibliography. Index.
• (source: Nielsen Book Data)

A new look at weak-convergence methods in metric spaces-from a master of probability theory In this new edition, Patrick Billingsley updates his classic work Convergence of Probability Measures to reflect developments of the past thirty years. Widely known for his straightforward approach and reader-friendly style, Dr. Billingsley presents a clear, precise, up-to-date account of probability limit theory in metric spaces. He incorporates many examples and applications that illustrate the power and utility of this theory in a range of disciplines-from analysis and number theory to statistics, engineering, economics, and population biology. With an emphasis on the simplicity of the mathematics and smooth transitions between topics, the Second Edition boasts major revisions of the sections on dependent random variables as well as new sections on relative measure, on lacunary trigonometric series, and on the Poisson-Dirichlet distribution as a description of the long cycles in permutations and the large divisors of integers. Assuming only standard measure-theoretic probability and metric-space topology, Convergence of Probability Measures provides statisticians and mathematicians with basic tools of probability theory as well as a springboard to the "industrial-strength" literature available today.
(source: Nielsen Book Data)

• Probability-- Measure-- Integration-- Random Variables and Expected Values-- Convergence of Distributions-- Derivatives and Conditional Probability-- Stochastic Processes-- Appendix-- Notes of the Problems-- Bibliography-- List of Symbols-- Index.
• (source: Nielsen Book Data)

Borel's normal number theorem, proved by calculus alone, followed by short sections that establish the existence and fundamental properties of probability measures, presenting lebesque measure on the unit interval. Coverage includes key topics in measure, integration, random variables and expected values, convergence of distributions, derivatives and conditional probability and stochastic processes.
(source: Nielsen Book Data)
PROBABILITY AND MEASURE Third Edition Now in its new third edition, Probability and Measure offers advanced students, scientists, and engineers an integrated introduction to measure theory and probability. Retaining the unique approach of the previous editions, this text interweaves material on probability and measure, so that probability problems generate an interest in measure theory and measure theory is then developed and applied to probability. Probability and Measure provides thorough coverage of probability, measure, integration, random variables and expected values, convergence of distributions, derivatives and conditional probability, and stochastic processes. The Third Edition features an improved treatment of Brownian motion and the replacement of queuing theory with ergodic theory. Like the previous editions, this new edition will be well received by students of mathematics, statistics, economics, and a wide variety of disciplines that require a solid understanding of probability theory.
(source: Nielsen Book Data)

• Probability-- Measure-- Integration-- Random Variables and Expected Values-- Convergence of Distributions-- Derivatives and Conditional Probability-- Stochastic Processes-- Appendix-- Notes of the Problems-- Bibliography-- List of Symbols-- Index.
• (source: Nielsen Book Data)

Borel's normal number theorem, proved by calculus alone, followed by short sections that establish the existence and fundamental properties of probability measures, presenting lebesque measure on the unit interval. Coverage includes key topics in measure, integration, random variables and expected values, convergence of distributions, derivatives and conditional probability and stochastic processes.
(source: Nielsen Book Data)

• Weak convergence-- Random elements and convergence in distribution-- Prokhorov's theorem-- The Space C-- A maximal Inequality-- Tightness-- Limit theorems.
• (source: Nielsen Book Data)

A treatment of the convergence of probability measures from the foundations to applications in limit theory for dependent random variables. Mapping theorems are proved via Skorokhod's representation theorem; Prokhorov's theorem is proved by construction of a content. The limit theorems at the conclusion are proved under a new set of conditions that apply fairly broadly, but at the same time make possible relatively simple proofs.
(source: Nielsen Book Data)