Real Analysis and Probability: Probability and Mathematical Statistics: a Series of Monographs and Textbooks

This book, the first of a projected two volume series, is designed for a graduate course in modern probability. The first four chapters, along with the Appendix: On General Topology, provide the background in analysis needed for the study of probability. This material is available as a separate book called" Measure, Integration, and Functional Analysis." Preface Summary of Notation 1 Sets 2 Real Numbers 3 Functions 4 Topology S Vector Spaces 6 Zorn's Lemma 1 Fundamentals of Measure and Integration Theory 1.1 Introduction Problems 1.2 Fields, O"-Fields, and Measures Problems 1.3 Extension of Measures Problems 1.4 Lebesgue-Stieltjes Measures and Distribution Functions Problems 1.5 Measurable Functions and Integration Problems 1.6 Basic Integration Theorems Problems 1.7 Comparison of Lebesgue and Riemann Integrals Problems 2 Further Results in Measure and Integration Theory 2.1 Introduction Problems 2.2 Radon-Nikodym Theorem and Related Results Problems 2.3 Applications to Real Analysis Problems 2.4 L^p Spaces Problems 2.5 Convergence of Sequences of Measurable Functions Problems 2.6 Product Measures and Fubini's Theorem Problems 2.7 Measures on Infinite Product Spaces Problems 2.8 References 3 Introduction to Functional Analysis 3.1 Introduction 3.2 Basic Properties of Hilbert Spaces Problems 3.3 Linear Operators on Normed Linear Spaces Problems 3.4 Basic Theorems of Functional Analysis Problems 3.5 Some Properties of Topological Vector Spaces Problems 3.6 References 4 The Interplay between Measure Theory and Topology 4.1 Introduction 4.2 The Daniell Integral Problems 4.3 Measures on Topological Spaces Problems 4.4 Measures on Uncountably Infinite Product Spaces Problems 4.5 Weak Convergence of Measures Problems 4.6 References 5 Basic Concepts of Probability 5.1 Introduction 5.2 Discrete Probability Spaces 5.3 Independence 5.4 Bernoulli Trials 5.5 Conditional Probability 5.6 Random Variables 5.7 Random Vectors 5.8 Independent Random Variables Problems 5.9 Some Examples from Basic Probability Problems 5.10 Expectation Problems 5.11 Infinite Sequences of Random Variables Problems 5.12 References 6 Conditional Probability and Expectation 6.1 Introduction 6.2 Applications 6.3 The General Concept of Conditional Probability and Expectation Problems 6.4 Conditional Expectation Given a \sigma-field Problem 6.5 Properties of Conditional Expectation Problems 6.6 Regular Conditional Probabilities Problems 6.7 References 7 Strong Laws of Large Numbers and Martingale Theory 7.1 Introduction Problems 7.2 Convergence Theorems Problems 7.3 Martingales Problems 7.4 Martingale Convergence Theorems Problems 7.5 Uniform Integrability Problems 7.6 Uniform Integrability and Martingale Theory Problems 7.7 Optional Sampling Theorems Problems 7.8 Applications of Martingale Theory Problems 7.9 Applications to Markov Chains 7.10 References 8 The Central Liinit Theorem 8.1 Introduction Problems 8.2 The Fundamental Weak Compactness Theorem Problems 8.3 Convergence to a Normal Distribution Problems 8.4 Stable Distributions Problem 8.5 Infinitely Divisible Distributions Problems 8.6 Uniform Convergence in the Central Limit Theorem 8.7 Proof of the Inversion Formula (8.1.4) 8.8 Completion of the Proof of Theorem 8.3.2 8.9 Proof of the Convergence of Types Theorem (8.3.4) 8.10 References Appendix on General Topology A1 Introduction Al Convergence A3 Product and Quotient Topologies A4 Separation Properties and Other Ways of Classifying Topological Spaces A5 Compactness A6 Semicontinuous Functions A7 The Stone-Weierstrass Theorem A8 Topologies on Function Spaces A9 Complete Metric Spaces and Category Theorems A10 Uniform Spaces Solutions to Problems Chapter 1 Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Chapter 7 Chapter 8 Subject Index