# Real Analysis and Probability

## 1st Edition

### Probability and Mathematical Statistics: A Series of Monographs and Textbooks

**Author:**Robert B. Ash

**Editors:**Z. W. Birnbaum E. Lukacs

**eBook ISBN:**9781483191423

**Imprint:**Academic Press

**Published Date:**28th March 1972

**Page Count:**494

## Description

Real Analysis and Probability provides the background in real analysis needed for the study of probability. Topics covered range from measure and integration theory to functional analysis and basic concepts of probability. The interplay between measure theory and topology is also discussed, along with conditional probability and expectation, the central limit theorem, and strong laws of large numbers with respect to martingale theory.

Comprised of eight chapters, this volume begins with an overview of the basic concepts of the theory of measure and integration, followed by a presentation of various applications of the basic integration theory. The reader is then introduced to functional analysis, with emphasis on structures that can be defined on vector spaces. Subsequent chapters focus on the connection between measure theory and topology; basic concepts of probability; and conditional probability and expectation. Strong laws of large numbers are also examined, first from the classical viewpoint, and then via martingale theory. The final chapter is devoted to the one-dimensional central limit problem, paying particular attention to the fundamental role of Prokhorov's weak compactness theorem.

This book is intended primarily for students taking a graduate course in probability.

## Table of Contents

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

## Details

- No. of pages:
- 494

- Language:
- English

- Copyright:
- © Academic Press 1972

- Published:
- 28th March 1972

- Imprint:
- Academic Press

- eBook ISBN:
- 9781483191423

## About the Author

### Robert B. Ash

## About the Editors

### Z. W. Birnbaum

### E. Lukacs

### Affiliations and Expertise

Bowling Green State University