Measure and Probability  book cover
1st Edition

Measure and Probability

ISBN 9781138114180
Published February 7, 2019 by CRC Press
232 Pages

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Book 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.

Table of Contents

Probabilities and Measures
σ-algebras as events
Algebras, monotone classes, etc.
Preliminaries on measures
Outer measures and Caratheodory extension
Lebesgue measure
Bernoulli trials

Measurable functions
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
Classification of states
Strong Markov property
Stationary distribution
Limit theorems

Some Analysis
Complex measures
Lp spaces
Radon–Nikodym theorem
Change of variables
The Riesz representation theorem

Metric spaces
Topological spaces
The Stone–Weierstrass theorem




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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.


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