# Lectures on Measure and Integration

###### Harold Widom
Publisher:
Dover Publications
Publication Date:
2017
Number of Pages:
164
Format:
Paperback
Price:
14.95
ISBN:
9780486810287
Category:
Monograph
[Reviewed by
Allen Stenger
, on
05/8/2017
]

This is a concise treatment of measure and integration in general measure spaces. The present book is a 2016 Dover corrected reprint of the 1969 Van Nostrand edition. It is a polished set of lecture notes prepared by David Drasin and Anthony J. Tromba.

The book starts out in general measure spaces; the Lebesgue integral on the real line is one of many examples and does not get any special treatment itself. The book covers the important consequences of the Lebesgue theory, such as the convergence theorems, the improved Fundamental Theorem of Calculus, the Fubini theorems, the $L^p$ spaces, and the application of the theory to Fourier series. Most books on advanced analysis also cover all these topics, along with much else. For example, the coverage and level of detail are about the same as in Chapters 1–9 of Rudin’s Real and Complex Analysis. So the question arises: How does the present book compare?

One reason the present book is useful is that exposition is especially clear (which Rudin often is not). Another reason is that the presentation is especially streamlined (you can think of it as the Named Theorems version: if the theorem is not important enough to have a name, we don’t treat it), and it is attractive if you want to understand the theory without learning everything there is to know about it. In some ways it is too streamlined: there are no exercises, although it does have a good collection of examples. It is also the “pure math” version of the subject, and does not give any applications, even to other areas of math, except for the chapter on Fourier series. Very Bad Feature: no index.

The book is not as detailed or as general as more specialized tomes such as Halmos’s Measure Theory or Berberian’s Measure and Integration. These latter two books also have a lot on measures on topological spaces and topological groups, which are largely omitted from Widom’s book (except for a discussion of $L^p$ spaces over normal topological spaces and normed linear spaces).

Bottom line: a streamlined and easy-to-follow exposition of measure and integration, but not for beginners: readers should already understand the Lebesgue theory on the real line before they tackle this book.

Allen Stenger is a math hobbyist and retired software developer. He is an editor of the Missouri Journal of Mathematical Sciences. His personal web page is allenstenger.com. His mathematical interests are number theory and classical analysis.

I. Measures

II. Integration

III. The Theorems of Fubini

IV. Representations of Measures

V.The Lebesgue Spaces

VI. Differentiation

VII. Fourier Series