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Essentials of Integration Theory for Analysis

Daniel W. Stroock
Publication Date: 
Number of Pages: 
Graduate Texts in Mathematics 262
[Reviewed by
Peter Rabinovitch
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Essentials of Integration Theory for Analysis is, as the title states, a textbook on measure theory for analysis. It does not, as the author writes, embed measure theory in topics in which measure theory plays a central role, such as probability or Fourier analysis. However, there are examples from both in the book.

Essentials is a significantly expanded version of the author’s A Concise Introduction to the Theory of Integration from 1990. I see that according to the Springer web site Concise was updated in 1998 to 262 pages (not so concise anymore), but I don’t have a copy of that edition handy. Compared to the 1990 edition that I used when I first learned measure theory, Essentials has about 90 more pages than Concise, with new sections on the rate of convergence of Riemann approximations, Steiner symmetrization, Fourier series and transforms, and Daniell integration. There are also more exercises.

The target audience is students who have completed a good analysis course (baby Rudin, say).

Stroock writes beautifully and is full of insight and advice. For example he states that “the essence of any theory of integration is a divide and conquer strategy” and then goes on to describe what would be needed in “a reasonable notion of measure.” It is my belief that these messages to the student learning the material are very beneficial, and help place the material in context.

On the negative side, some notation is used without definition. For example, the notation for the interior of a set is used on page 1 without being explained.

The difference between the two books is exactly what you would expect from their titles — one is a very concise introduction, the other a more detailed, fleshed out text. Each is extremely well suited to its stated purpose.

Peter Rabinovitch is a Systems Architect at Research in Motion, and a PhD student in probability. He s currently writing about applications of Mallows permutations.