Measure and Integration Theory

Measure and Integration Theory

by Heinz Bauer
     
 

ISBN-10: 3110167190

ISBN-13: 9783110167191

Pub. Date: 06/28/2001

Publisher: De Gruyter

This book gives a straightforward introduction to the field as it is nowadays required in many branches of analysis and especially in probability theory. The first three chapters (Measure Theory, Integration Theory, Product Measures) basically follow the

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Overview

This book gives a straightforward introduction to the field as it is nowadays required in many branches of analysis and especially in probability theory. The first three chapters (Measure Theory, Integration Theory, Product Measures) basically follow the clear and approved exposition given in the author's earlier book on "Probability Theory and Measure Theory". Special emphasis is laid on a complete discussion of the transformation of measures and integration with respect to the product measure, convergence theorems, parameter depending integrals, as well as the Radon-Nikodym theorem.

The final chapter, essentially new and written in a clear and concise style, deals with the theory of Radon measures on Polish or locally compact spaces. With the main results being Luzin's theorem, the Riesz representation theorem, the Portmanteau theorem, and a characterization of locally compact spaces which are Polish, this chapter is a true invitation to study topological measure theory.

The text addresses graduate students, who wish to learn the fundamentals in measure and integration theory as needed in modern analysis and probability theory. It will also be an important source for anyone teaching such a course.

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Product Details

ISBN-13:
9783110167191
Publisher:
De Gruyter
Publication date:
06/28/2001
Series:
de Gruyter Studies in Mathematics Series, #26
Edition description:
New Edition
Pages:
248
Product dimensions:
6.14(w) x 9.21(h) x 0.63(d)
Age Range:
18 Years

Table of Contents

Preface
Introduction
Notations
Ch. IMeasure Theory1
1[sigma]-algebras and their generators2
2Dynkin systems5
3Contents, premeasures, measures8
4Lebesgue premeasure14
5Extension of a premeasure to a measure18
6Lebesgue-Borel measure and measures on the number line26
7Measurable mappings and image measures34
8Mapping properties of the Lebesgue-Borel measure38
Ch. IIIntegration Theory49
9Measurable numerical functions49
10Elementary functions and their integral53
11The integral of non-negative measurable functions57
12Integrability64
13Almost everywhere prevailing properties70
14The spaces [actual symbol not reproducible][superscript p]([mu])74
15Convergence theorems79
16Applications of the convergence theorems88
17Measures with densities: the Radon-Nikodym theorem96
18Signed measures107
19Integration with respect to an image measure110
20Stochastic convergence112
21Equi-integrability121
Ch. IIIProduct Measures132
22Products of [sigma]-algebras and measures132
23Product measures and Fubini's theorem135
24Convolution of finite Borel measures147
Ch. IVMeasures on Topological Spaces152
25Borel sets, Borel and Radon measures152
26Radon measures on Polish spaces157
27Properties of locally compact spaces166
28Construction of Radon measures on locally compact spaces170
29Riesz representation theorem177
30Convergence of Radon measures188
31Vague compactness and metrizability questions204
Bibliography217
Symbol Index221
Name Index223
Subject Index225

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