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Measure and Integration Theory
     

Measure and Integration Theory

by Heinz Bauer, Robert B. Burckel
 

ISBN-10: 3110167190

ISBN-13: 9783110167191

Pub. Date: 06/21/2001

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 clear and approved exposition given in the author's earlier book on "Probability Theory and Measure Theory".

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.

Product Details

ISBN-13:
9783110167191
Publisher:
De Gruyter
Publication date:
06/21/2001
Series:
De Gruyter Studies in Mathematics Series , #26
Edition description:
New Edition
Pages:
246
Product dimensions:
6.10(w) x 9.06(h) x 0.02(d)
Age Range:
18 Years

Table of Contents

Prefacevii
Introductionix
Notationsxi
Chapter IMeasure Theory1
1.[sigma]-algebras and their generators2
2.Dynkin systems5
3.Contents, premeasures, measures8
4.Lebesgue premeasure14
5.Extension of a premeasure to a measure18
6.Lebesgue-Borel measure and measures on the number line26
7.Measurable mappings and image measures34
8.Mapping properties of the Lebesgue-Borel measure38
Chapter IIIntegration Theory49
9.Measurable numerical functions49
10.Elementary functions and their integral53
11.The integral of non-negative measurable functions57
12.Integrability64
13.Almost everywhere prevailing properties70
14.The spaces L[superscript p]([mu])74
15.Convergence theorems79
16.Applications of the convergence theorems88
17.Measures with densities: the Radon-Nikodym theorem96
18.Signed measures107
19.Integration with respect to an image measure110
20.Stochastic convergence112
21.Equi-integrability121
Chapter IIIProduct Measures132
22.Products of [sigma]-algebras and measures132
23.Product measures and Fubini's theorem135
24.Convolution of finite Borel measures147
Chapter IVMeasures on Topological Spaces152
25.Borel sets, Borel and Radon measures152
26.Radon measures on Polish spaces157
27.Properties of locally compact spaces166
28.Construction of Radon measures on locally compact spaces170
29.Riesz representation theorem177
30.Convergence of Radon measures188
31.Vague compactness and metrizability questions204
Bibliography217
Symbol Index221
Name Index223
Subject Index225

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