This is a “self-contained introduction to real analysis” at the graduate level, comparable to H. L. Royden's Real Analysis or Walter Rudin's Real and Complex Analysis. The style of DiBenedetto’s book is closer to Rudin than to Royden. For example, Royden first develops the idea of a Lebesgue measure and then later generalizes to the idea of a general measure. DiBenedetto and Rudin on the other hand, develop the general theory of measures and then present Lebesgue, Lebesgue-Stieltjes, and Hausdorff measures as specific examples of the general theory. Like Rudin's, DiBenedetto’s book is quite dense, containing a lot of information in very few pages.
The book begins with a standard preliminary chapter covering topics such as countable sets, cardinality, the axiom of choice, and well-ordering. In addition, chapter one covers topics in topology and metric spaces including Urysohn’s lemma, the Tietze extension theorem, bases, and topological vector spaces. Thus, all of the prerequisite material needed for the entire book has been covered by chapter two. Chapters three and four give a thorough presentation of the Lebesgue integral, covering measurable functions, the Radon-Nikodym theorem, decomposition of measures, derivatives, the Stone-Weierstrass theorem, and the Ascoli-Arzela theorem. Chapters five and six offer an equally thorough presentation of Lp and Banach spaces. The remaining chapters cover distributions, advanced topics on integrable functions, and embeddings.
DiBenedetto's writing style is quite concise. There is generally not a lot of introductory or motivational material. The theorems are well marked, and the material is well organized. I find the book to be an excellent reference book for anyone already trained in real analysis, but I would hesitate to utilize it as the textbook for an introductory course in real analysis, even at the graduate level, for all the reasons previously mentioned.
There are several aspects of DiBenedetto’s book that I did not like. First, on page xxiv DiBenedetto states that his view is that every theorem needs motivation, but I find his motivations weak or nonexistent. I believe that this would make the material very difficult for most students in a graduate level real analysis course. Second, the density of the material, which I liked as a practitioner, tends to make for laborious reading for students. Third, the problems for each chapter are found in a section entitled Problems and Complements. The problems were not identified as such, and thus I found it difficult to distinguish between the problems and the complementary information. Finally, the index is not alphabetized. For example, “connected spaces” occur before “complex Hahn-Banach theorem” (page 475). This problem seems to be epidemic.
To summarize, I found DiBenedetto’s book to be on par with Rudin's Real and Complex Analysis book. In my estimation, this means that it would serve as an outstanding reference book for someone already familiar with the topic, but would not be the best textbook for the typical introductory graduate course in real analysis.
Andrew Siefker is assistant professor of mathematics at Angelo State University. After earning Bachelor and Master of Science degrees in electrical engineering from the Georgia Institute of Technology, he worked for several years in industry before attending Arizona State University, where he earned a Ph.D. in mathematics. He has publications in mathematics, electrical engineering, and most recently in mathematical biology.