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Real Analysis

Peter A. Loeb
Publication Date: 
Number of Pages: 
[Reviewed by
Salim Salem
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The 274 pages Loeb’s Real Analysis provide nothing short of a deep introduction to measure theory. In fact, one will find that measure is the main thread connecting the elven chapters and four appendices.

The book starts by a study of the topology of real line, where the author introduces algebras and \(\sigma\)-algebras, setting the stage for measure theory. Chapter 2 introduces measures and measurable functions. The discussion centers on Lebesgue outer-measure and measure; a non-measurable set and a null set (namely Cantor’s) are constructed. In chapter 3 the author defines measurable functions, simple functions, and limits. He gives a short new proof of Lusin’s theorem and proves Egoroff’s. The fourth chapter is a discussion of the integral of simple, bounded measurable, non-negative measurable, and measurable functions. Fatou’s Lemma is then proved and convergence in measure is defined.

In the fifth chapter the author uses local maximal functions and a simple optimal lemma to simplify the study of the relation between differentiation and integration. Chapter six is a generalization of measures to general spaces and chapter seven is an introduction to metric and normed spaces, in which one meets Baire category theory. These two chapters form a good setup for chapter eight, where Hilbert spaces are the main topic. The Radon-Nikodým theorem is proved and Fourier coefficients are calculated.

Chapter nine is concerned with general topology: neighborhood, continuous mappings, connectedness, compactness, metrization, Ascoli-Arzela and Stone-Weierstrass theorems are proved. Chapter ten discusses the construction of measure as it was done by Carathéodory, it introduces Lebesgue measure on Euclidian spaces and the product measure. The last chapter is devoted to Banach spaces: more results on \(L^p\) can be found as well as the Hahn-Banach theorem.

The last part of the book contains four appendices: the first discuss the axiom of choice and its equivalent lemmas and principles; the second is on limit and on the Besicovitch and Morse covering theorem., Non-standard analysis and measure theory is the topic of the third and the fourth is devoted to answers to some chosen problems.

This is a very well written book. Its chapters are no more than 20 pages each, which allows students to easily work through them. The proofs are sharp, lively and rigorously written. Definitions are well motivated and explained with a little dose of examples. The results discussed in this book are not only among the most beautiful but also among the most useful ones.

Each chapter ends with a set of well-chosen problems. I enjoyed solving them. They include a wealth of results and some provide proofs of theorems that appear in the text. Any interested reader will find a great pleasure in working them.

This book is one of the best reference books for a high level course on Real Analysis. It is, as well, a good introduction to measure theory on the real line, Hilbert, Banach and topological spaces. I recommend it, not only, to any student who wants to study or do research on measures and integration or who will use these notions in studying other subjects; but, also to every mathematics department’s library.

Salim Salem is Professor of Mathematics at the Saint-Joseph University of Beirut.

See the table of contents in the publisher's webpage.