Measure and Probability: 1st Edition (Paperback) book cover

Measure and Probability

1st Edition

By Siva Athreya, V. S. Sunder

CRC Press

232 pages

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Description

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.

Reviews

This textbook is suitable for a one-semester course on measure theory and probability for beginning graduate students in mathematics, probability and statistics. It can also be used as a textbook for advanced undergraduate students in mathematics … The topics are well selected to meet the needs of students who are interested in graduate studies in areas related to analysis, probability, stochastic processes and statistics … This makes the book student-friendly. A motivated student can use it by him- or herself to learn the topics well.

—Yimin Xiao, Mathematical Reviews, 2010

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

About the Authors

Siva Athreya is a professor at the Indian Academy of Sciences, Bangalore, and V.S. Sunder is a professor at the Institute of Mathematical Sciences, Chennai.

Subject Categories

BISAC Subject Codes/Headings:
MAT000000
MATHEMATICS / General
MAT029010
MATHEMATICS / Probability & Statistics / Bayesian Analysis
MAT037000
MATHEMATICS / Functional Analysis