Measure and Integration Theory

by Bauer, Heinz

• ISBN13: 9783110167191
• ISBN10: 3110167190
• Format: Hardcover
• Copyright: 2001-12-01
• Publisher: De Gruyter

Summary

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.

Table of Contents

Preface	p. vii
Introduction	p. ix
Notations	p. xi
Measure Theory	p. 1
[sigma]-algebras and their generators	p. 2
Dynkin systems	p. 5
Contents, premeasures, measures	p. 8
Lebesgue premeasure	p. 14
Extension of a premeasure to a measure	p. 18
Lebesgue-Borel measure and measures on the number line	p. 26
Measurable mappings and image measures	p. 34
Mapping properties of the Lebesgue-Borel measure	p. 38
Integration Theory	p. 49
Measurable numerical functions	p. 49
Elementary functions and their integral	p. 53
The integral of non-negative measurable functions	p. 57
Integrability	p. 64
Almost everywhere prevailing properties	p. 70
The spaces L[superscript p]([mu])	p. 74
Convergence theorems	p. 79
Applications of the convergence theorems	p. 88
Measures with densities: the Radon-Nikodym theorem	p. 96
Signed measures	p. 107
Integration with respect to an image measure	p. 110
Stochastic convergence	p. 112
Equi-integrability	p. 121
Product Measures	p. 132
Products of [sigma]-algebras and measures	p. 132
Product measures and Fubini's theorem	p. 135
Convolution of finite Borel measures	p. 147
Measures on Topological Spaces	p. 152
Borel sets, Borel and Radon measures	p. 152
Radon measures on Polish spaces	p. 157
Properties of locally compact spaces	p. 166
Construction of Radon measures on locally compact spaces	p. 170
Riesz representation theorem	p. 177
Convergence of Radon measures	p. 188
Vague compactness and metrizability questions	p. 204
Bibliography	p. 217
Symbol Index	p. 221
Name Index	p. 223
Subject Index	p. 225
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