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Measure Theory and Integration

American Mathematical Society
Number of Pages: 

Can one write a good 249-page textbook (plus nine short appendices) that covers measure and integration theory and includes introductions to important topics such as: ergodic theory, probability spaces, Wiener measure and Brownian motion, martingales and Hausdorff's r-dimensional measures? Well, it depends on your criteria.

The author accurately summarizes his goals on his website as follows: Quickly get to the construction of Lebesgue measure on the real line. Then quickly get to the basic results on the Lebesgue integral. Follow this with further constructions of measures, on Euclidean space, manifolds, etc. Then include some more advanced topics.

Taylor's treatment throughout is elegant and very efficient. The core chapters on the basics of measure and integration, Chapters 1-9, are written as if the only purpose in studying this material is to get past it and go on to something else. What is missing is the author's obviously great wisdom and deep perspective. No motivation. No historical context in the core chapters. I didn't see the words "interesting" or "beautiful," or "tedious" for that matter, anywhere.

The author leaves out many details and references. Beginners will have trouble deciding which reasons are obvious and which require extra work or a reference. Moreover, as one might expect in such a short book, a lot of results are shunted to the exercises.

For the unprepared student, Chapter 7 about integration on manifolds will be a shock. In fact, this chapter and related Appendices B, E, F and G would make a semester course for many students.

Taylor's book has many fine features. But how well will it serve students? I found some of the proofs too slick and unmotivated, for example, Egoroff's Theorem 3.9, the Radon-Nikodym Theorem 4.10, and the Law of the Iterated Logarithm 17.14.

The book is extraordinarily clean. I found no mistakes, and no misprints until page 63. Still, it retains features of class notes. Some notation is used without explanation or before it is defined. When Hilbert space is defined, the scalar field of complex numbers isn't even mentioned, but it's clear from the proofs that these are complex Hilbert spaces.

I found reading the text very enjoyable, but I usually knew where details or references are needed. Very good students who work slowly and carefully through the book, and work many of the exercises, should enjoy the book. But many students will find the relentless pace discouraging. If I were teaching this material, I would not assign this text, but I would have a copy on my desk.

Kenneth A. Ross ( taught at the University of Oregon from 1965 to 2000. He was President of the MAA during 1995-1996. Before that he served as AMS Associate Secretary, MAA Secretary, and MAA Associate Secretary. His research area of interest was commutative harmonic analysis, especially where it has a probabilistic flavor. He is the author of the book Elementary Analysis: The Theory of Calculus (1980, now in 14th printing), co-author of Discrete Mathematics (with Charles R.B. Wright, 2003, fifth edition), and, as Ken Ross, the author of A Mathematician at the Ballpark: Odds and Probabilities for Baseball Fans (2004).

Date Received: 
Monday, August 14, 2006
Include In BLL Rating: 
Michael E. Taylor
Graduate Studies in Mathematics 76
Publication Date: 
Kenneth A. Ross
Publish Book: 
Modify Date: 
Saturday, May 1, 2010