Measure Theory

This book presents measure and integration theory in a self­contained and step by step manner. After an informal introduction to the subject, the general extension theorem of Caratheodory is presented in Chapter 1. This is followed by the construction of Lebesgue­Stieltjes measures on the real line and Euclidean spaces, and of measures on finite and countable spaces. The presentation gives a general perspective to the subject so as to enable students to think beyond the special, albeit important, example of the Lebesgue measure on the real line. Integration theory is developed in Chapter 2 where the three basic convergence theorems and their extensions are presented. Basic aspects of the theory of Lp, Banach and Hilbert spaces are presented in chapter 3. The Lebesgue­ Radon­Nikodym theorem, signed measures and the fundamental theorem of the Lebesgue integral calculus are taken up in chapter 4. Product measures and their applications to convolutions are discussed in Chapter 5. Also included in Chapter 5 are sections on Fourier Series and Fourier transforms. The last chapter is devoted to basic aspects of probability theory including the Kolmogorov consistency theorem for the construction of stochastic processes. The appendix reviews basic set theory and advanced calculus.

Apart from providing a modern presentation of the subject, this book includes a large number of exercises that should prove very useful to the instructor and the students. This book should be valuable to M.Sc. and Ph.D. students in mathematics, statistics and related fields in India.

1. Measures and Integration: An Informal Introduction
2. Measures
3. Integration
4. L p ­Spaces
5. Differentiation
6. Product Measures, Convolutions, and Transforms
7. Probability Spaces
A.1 Elementary set theory
A.2 Real numbers, continuity, differentiability and integration
A.3 Complex numbers, exponential and trigonometric functions
A.4 Metric spaces
A.5 Problems
List of Symbols and Abbreviations
References
Subject Index