Real Analysis and Probability

Preface

Summary of Notation


1 Fundamentals of Measure and Integration Theory


1.1 Introduction


1.2 Fields, σ-Fields, and Measures


1.3 Extension of Measures


1.4 Lebesgue—Stieltjes Measures and Distribution Functions


1.5 Measurable Functions and Integration


1.6 Basic Integration Theorems


1.7 Comparison of Lebesgue and Riemann Integrals


2 Further Results in Measure and Integration Theory


2.1 Introduction


2.2 Radon—Nikodym Theorem and Related Results


2.3 Applications to Real Analysis


2.4 Lp Spaces


2.5 Convergence Of Sequences of Measurable Functions


2.6 Product Measures and Fubini’s Theorem


2.7 Measures on Infinite Product Spaces


2.8 References


3 Introduction to Functional Analysis


3.1 Introduction


3.2 Basic Properties of Hilbert Spaces


3.3 Linear Operators on Normed Linear Spaces


3.4 Basic Theorems of Functional Analysis


3.5 Some Properties of Topological Vector Spaces


3.6 References


4 The Interplay between Measure Theory and Topology


4.1 Introduction


4.2 The Daniell Integral


4.3 Measures on Topological Spaces


4.4 Measures on Uncountably Infinite Product Spaces


4.5 Weak Convergence of Measures


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


5.9 Some Examples from Basic Probability


5.10 Expectation


5.11 Infinite Sequences of Random Variables


5.12 References


6 Conditional Probability and Expectation


6.1 Introduction


6.2 Applications


6.3 The General Concept of Conditional Probability and Expectation


6.4 Conditional Expectation, Given a σ-Field


6.5 Properties Of Conditional Expectation


6.6 Regular Conditional Probabilities


6.7 References


7 Strong Laws of Large Numbers and Martingale Theory


7.1 Introduction


7.2 Convergence Theorems


7.3 Martingales


7.4 Martingale Convergence Theorems


7.5 Uniform Integrability


7.6 Uniform Integrability and Martingale Theory


7.7 Optional Sampling Theorems


7.8 Applications of Martingale Theory


7.9 Applications To Markov Chains


7.10 References


8 The Central Limit Theorem


8.1 Introduction


8.2 The Fundamental Weak Compactness Theorem


8.3 Convergence to a Normal Distribution


8.4 Stable Distributions


8.5 Infinitely Divisible Distributions


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

A2 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

Bibliography

Solutions to Problems

Subject Index