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Measure and Integration Theory

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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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Editorial Reviews

From The Critics
In 1990, Bauer (mathematics, U. Erlangen-Nürnberg, Germany) split off the first three introductory chapters of his textbook on probability theory, originally published over 30 years ago, to form a separate volume providing the measure-theoretic background for the rest of the text and a self-contained introduction to measure and integration. Burckel (mathematics, Kansas State U.) has taken the opportunity of translating the 1992 second German edition of to revise and update the treatment. Annotation c. Book News, Inc., Portland, OR (booknews.com)
From the Publisher
"It is a pleasure to see Heinz Bauer's famous monograph Maß- und Intergrationstheorie [...] published in English. [...] The translation by Robert B. Burckel is careful and close to the German original. He has replaced references to German textbooks by references to English textbooks, and he has also added several interesting exercises, comments, and references. the typographical appearance of the book is excellent."Klaus D. Schmidt in: Zentralblatt Math, 10/2000 "Like the German original, the present textbook is a very readable and concise introduction to measure theory and integration.[...] In any case, it definitely is a valuable resource for both students and teachers."G. Teschl in: Internationale Mathematische Nachrichten, Wien, 189/2002
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Product Details

  • ISBN-13: 9783110167191
  • Publisher: De Gruyter
  • Publication date: 6/28/2001
  • 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)

Meet the Author

Professor Heinz Bauer (1928--2002) was Professor at the Mathematical Institute of the Friedrich-Alexander-University Erlangen-Nürnberg, Erlangen, Germany.
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Table of Contents

Preface
Introduction
Notations
Ch. I Measure Theory 1
1 [sigma]-algebras and their generators 2
2 Dynkin systems 5
3 Contents, premeasures, measures 8
4 Lebesgue premeasure 14
5 Extension of a premeasure to a measure 18
6 Lebesgue-Borel measure and measures on the number line 26
7 Measurable mappings and image measures 34
8 Mapping properties of the Lebesgue-Borel measure 38
Ch. II Integration Theory 49
9 Measurable numerical functions 49
10 Elementary functions and their integral 53
11 The integral of non-negative measurable functions 57
12 Integrability 64
13 Almost everywhere prevailing properties 70
14 The spaces [actual symbol not reproducible][superscript p]([mu]) 74
15 Convergence theorems 79
16 Applications of the convergence theorems 88
17 Measures with densities: the Radon-Nikodym theorem 96
18 Signed measures 107
19 Integration with respect to an image measure 110
20 Stochastic convergence 112
21 Equi-integrability 121
Ch. III Product Measures 132
22 Products of [sigma]-algebras and measures 132
23 Product measures and Fubini's theorem 135
24 Convolution of finite Borel measures 147
Ch. IV Measures on Topological Spaces 152
25 Borel sets, Borel and Radon measures 152
26 Radon measures on Polish spaces 157
27 Properties of locally compact spaces 166
28 Construction of Radon measures on locally compact spaces 170
29 Riesz representation theorem 177
30 Convergence of Radon measures 188
31 Vague compactness and metrizability questions 204
Bibliography 217
Symbol Index 221
Name Index 223
Subject Index 225
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