Measure Theory and Probability Theory
This book arose out of two graduate courses that the authors have taught duringthepastseveralyears;the firstonebeingonmeasuretheoryfollowed by the second one on advanced probability theory. The traditional approach to a first course in measure theory, such as in Royden (1988), is to teach the Lebesgue measure on the real line, then the p differentation theorems of Lebesgue, L -spaces on R, and do general m- sure at the end of the course with one main application to the construction of product measures. This approach does have the pedagogic advantage of seeing one concrete case first before going to the general one. But this also has the disadvantage in making many students’ perspective on m- sure theory somewhat narrow. It leads them to think only in terms of the Lebesgue measure on the real line and to believe that measure theory is intimately tied to the topology of the real line. As students of statistics, probability, physics, engineering, economics, and biology know very well, there are mass distributions that are typically nonuniform, and hence it is useful to gain a general perspective. This book attempts to provide that general perspective right from the beginning. The opening chapter gives an informal introduction to measure and integration theory. It shows that the notions of σ-algebra of sets and countable additivity of a set function are dictated by certain very na- ral approximation procedures from practical applications and that they are not just some abstract ideas.
1100018647
Measure Theory and Probability Theory
This book arose out of two graduate courses that the authors have taught duringthepastseveralyears;the firstonebeingonmeasuretheoryfollowed by the second one on advanced probability theory. The traditional approach to a first course in measure theory, such as in Royden (1988), is to teach the Lebesgue measure on the real line, then the p differentation theorems of Lebesgue, L -spaces on R, and do general m- sure at the end of the course with one main application to the construction of product measures. This approach does have the pedagogic advantage of seeing one concrete case first before going to the general one. But this also has the disadvantage in making many students’ perspective on m- sure theory somewhat narrow. It leads them to think only in terms of the Lebesgue measure on the real line and to believe that measure theory is intimately tied to the topology of the real line. As students of statistics, probability, physics, engineering, economics, and biology know very well, there are mass distributions that are typically nonuniform, and hence it is useful to gain a general perspective. This book attempts to provide that general perspective right from the beginning. The opening chapter gives an informal introduction to measure and integration theory. It shows that the notions of σ-algebra of sets and countable additivity of a set function are dictated by certain very na- ral approximation procedures from practical applications and that they are not just some abstract ideas.
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Measure Theory and Probability Theory

Measure Theory and Probability Theory

Measure Theory and Probability Theory

Measure Theory and Probability Theory

Paperback(Softcover reprint of hardcover 1st ed. 2006)

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Overview

This book arose out of two graduate courses that the authors have taught duringthepastseveralyears;the firstonebeingonmeasuretheoryfollowed by the second one on advanced probability theory. The traditional approach to a first course in measure theory, such as in Royden (1988), is to teach the Lebesgue measure on the real line, then the p differentation theorems of Lebesgue, L -spaces on R, and do general m- sure at the end of the course with one main application to the construction of product measures. This approach does have the pedagogic advantage of seeing one concrete case first before going to the general one. But this also has the disadvantage in making many students’ perspective on m- sure theory somewhat narrow. It leads them to think only in terms of the Lebesgue measure on the real line and to believe that measure theory is intimately tied to the topology of the real line. As students of statistics, probability, physics, engineering, economics, and biology know very well, there are mass distributions that are typically nonuniform, and hence it is useful to gain a general perspective. This book attempts to provide that general perspective right from the beginning. The opening chapter gives an informal introduction to measure and integration theory. It shows that the notions of σ-algebra of sets and countable additivity of a set function are dictated by certain very na- ral approximation procedures from practical applications and that they are not just some abstract ideas.

Product Details

ISBN-13: 9781441921918
Publisher: Springer New York
Publication date: 11/23/2010
Series: Springer Texts in Statistics
Edition description: Softcover reprint of hardcover 1st ed. 2006
Pages: 619
Product dimensions: 6.10(w) x 9.25(h) x 0.05(d)

Table of Contents

Measures and Integration: An Informal Introduction.- Measures.- Integration.- Lp-Spaces.- Differentiation.- Product Measures, Convolutions, and Transforms.- Probability Spaces.- Independence.- Laws of Large Numbers.- Convergence in Distribution.- Characteristic Functions.- Central Limit Theorems.- Conditional Expectation and Conditional Probability.- Discrete Parameter Martingales.- Markov Chains and MCMC.- Shastic Processes.- Limit Theorems for Dependent Processes.- The Bootstrap.- Branching Processes.
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