Measure and Probability

This book covers the fundamentals of measure theory and probability theory. It begins with the construction of Lebesgue measure via Caratheodory’s outer measure approach and goes on to discuss integration and standard convergence theorems and contains an entire chapter devoted to complex measures, Lp spaces, Radon–Nikodym theorem, and the Riesz representation theorem. It presents the elements of probability theory, the law of large numbers, and central limit theorem. The book then discusses discrete time Markov chains, stationary distributions and limit theorems. The appendix covers many basic topics such as metric spaces, topological spaces and the Stone–Weierstrass theorem.

Table of Contents

Probabilities and Measures

Introduction

σ-algebras as events

Algebras, monotone classes, etc.

Preliminaries on measures

Outer measures and Caratheodory extension

Lebesgue measure

Regularity

Bernoulli trials

Integration

Measurable functions

Integration

a.e. considerations

Random Variables

Distribution and expectation

Independent events and tail σ-algebra

Some distributions

Conditional expectation

Probability Measures on Product Spaces

Product measures

Joint distribution and independence

Probability measures on infinite product spaces

Kolmogorov consistency theorem

Characteristics and Convergences

Characteristic functions

Modes of convergence

Central limit theorem

Law of large numbers

Markov Chains

Discrete time MC

Examples

Classification of states

Strong Markov property

Stationary distribution

Limit theorems

Some Analysis

Complex measures

Lp spaces

Radon–Nikodym theorem

Change of variables

Differentiation

The Riesz representation theorem

Appendix

Metric spaces

Topological spaces

Compactness

The Stone–Weierstrass theorem

Tables

References

Index