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Introduction to Measure Theory and Functional Analysis

Piermarco Cannarsa and Teresa D'Aprile
Publication Date: 
Number of Pages: 
[Reviewed by
Ittay Weiss
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This book, written by leading experts, is a well-crafted textbook covering a medley of relevant topics in measure theory and functional analysis in a rather get-to-the-point-quickly fashion, yet resulting in a very readable and enjoyable journey. The material covered is largely classical, heavily influenced by the good old textbooks in the field (i.e, Rudin, Royden, and Yosida), but the authors’ own agenda and point-of-view are tangibly present and give the book a unique feel.

The book manages to cover quite a lot of ground in a relatively short number of pages strewn with plenty of exercises. The core topics are presented nicely, with more technical results, or results that would otherwise interrupt the didactic flow, secluded in an appendix.

Some rather minor imperfections should be noted too, if merely to alert the reader and sharpen one’s expectations. Firstly, the final chapter in the book is devoted to set-valued analysis. A good treatment of this would have been a wonderful supplement to the book, as many books completely neglect this increasingly important topic. I find the chapter to be quite lacking, however, both in content and in perspective. It covers very little and fails to provide the reader with a potentially fascinating view of a landscape of current research; an unfortunate missed opportunity.

Another point to take note of are the bibliographies at the end of each chapter, a requirement for books in the UNITEXT series. Almost without exception, the references are to old and classical textbooks, and occasionally to a source written in French. The bibliography is thus rather useless for the intended readership. It would have been much more useful if the mandatory bibliography was used to provide the reader with current and most relevant suggestions for further, or supplementary, reading, or to indicate accessible research level material; another missed opportunity.

All-in-all, the student reading this book stands to gain much knowledge, develop and hone her skills, and enjoy the trip. With only a few pitfalls, it’s probably going to be a smooth ride.

Ittay Weiss is Lecturer of Mathematics at the School of Computing, Information and Mathematical Sciences of the University of the South Pacific in Suva, Fiji.