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.

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Information

Publisher

Academic Press

Year

2014

ISBN

9781483191423

Topic

Mathematics

Subtopic

Probability & Statistics

Index

Mathematics

Front Cover
Real Analysis and Probability
Copyright Page
Table of Contents
Preface
Summary of Notation
Chapter 1. Fundamentals of Measure and Integration Theory
Chapter 2. Further Results in Measure and Integration Theory
Chapter 3. Introduction to Functional Analysis
Chapter 4. The Interplay between Measure Theory and Topology
Chapter 5. Basic Concepts of Probability
Chapter 6. Conditional Probability and Expectation
Chapter 7. Strong Laws of Large Numbers and Martingale Theory
Chapter 8. The Central Limit Theorem
Appendix on General Topology
Bibliography
Solutions to Problems
Subject Index