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Robert B. Ash, "Measure, Integration and Functional Analysis"

The subject matter of this book is fundamental to all areas of mathematical

analysis and should be accessible to students who are in the early stages of

their professional training. An undergraduate course in real variables is a

prerequisite, and an acquaintance with complex analysis is desirable. How-

ever, since no use is made of the Cauchy theory, the exposure to complex

variables need not be extensive. To fully appreciate Chapters 3 and 4 some

background in elementary point-set topology is essential. The intended

audience thus consists of mathematics majors, most likely seniors or beginning

graduate students, plus students of engineering and physics who use measure

theory or functional analysis in their work.

The book has been arranged so that it may be used in several ways. The

first two chapters present the fundamentals of measure and integration theory,

and can serve as the text for a short course in this subject. If time permits,

material from Chapter 4, on the interplay between measure theory and

topology, may be added. Chapter 4 is almost independent of Chapter 3; the

only dependence occurs in Theorems 4.3.13 and 4.3.15. If the particular

group of students has some background in measure theory, Chapter 3 may

be used as a text for an introductory course in functional analysis. There is an

Appendix on General Topology, in which those areas of topology that occur

in the book are treated. Selections from the appendix and from Chapter 4

can be blended into a course covering aspects of topology that are of interest

in analysis. Of course, the entire book may be covered, possibly in a leisurely

one year course. Problems are given at the end of each section. Fairly detailed solutions

are given to many problems, and instructors may obtain solutions to those

problems not worked out in the text by writing the publisher.

" Measure, Integration and Functional Analysis" is actually the first half

of another book. The author's " Real Analysis and Probability" consists of

the present text and four additional chapters on probability. Thus I have been

careful to include those results that are of particular importance in probability.

For example, the generalized product measure theorem (Section 2.6) is

developed in such a way that it is immediately applicable to compound experi-

ments in probability, where the probability of an event associated with step n

of the experiment depends on the result of the first n - I steps. Also, measures

on arbitrary product spaces are discussed, and the Kolmogorov extension

theorem is proved in full generality.

Although the book has a probabilistic flavor and indeed its main purpose

is to prepare probability students for later work, nevertheless considerable

care has been taken to make the presentation suitable for the analysis student

who is not necessarily a probability specialist. The basic training needed for

work in probability theory is quite similar to the background required for the

study of other areas of modern analysis. There are only a few sections in the

book which can be regarded as specialized, and these sections may be omitted

without loss of continuity. Specifically, Section 1.4 (Parts 6-10), Section 2.7,

and Sections 4.4 and 4.5 may be skipped, although even the nonprobabilist

may encounter this material in his later work.

It is a pleasure to thank Professors Melvin Gardner and Samuel Saslaw,

who used the manuscript in their classes and made many helpful suggestions,

Mrs. Dee Keel for her expert typing, and the staff at Academic for their

encouragement and cooperation.

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