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Essentials of Measure Theory

Carlos S. Kubrusly
Publication Date: 
Number of Pages: 
[Reviewed by
Michael Berg
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Kubrusly notes that “the … book is the outcome of attempts to meet the needs of a contemporary course in measure theory for mathematicians who [sic] will also be accessible to a wider audience of students in mathematics, statistics, economics, engineering, and physics, bearing a modest prerequisite.” Accordingly, Essentials of Measure Theory is a no-nonsense text, covering, indeed, the essentials: Kubrusly takes the reader from measurable functions in the reals and the attendant \(\sigma\)-algebras, through integration theory, all the way to decomposition and extension of measures and the very important subject of product measures; so much for Part I of the book. Thereafter Part II goes to topological spaces (and their measures), starting with more discussion of integrals and then getting to the all-important material about Borel measures, representation theorems (with Riesz properly featured), and invariant measures on topological groups: Haar measure is abundantly covered. This latter feature is exceptionally welcome, given the audience Kubrusly seeks to reach. As a number theorist who is nowadays spending a lot of time with physics (for better and for worse), this particularly resonates for me.

Kubrusly has a great deal of experience teaching this material in the world’s most beautiful city, Rio de Janeiro, at the Catholic University there, and his expertise certainly shows. His exposition is very clear and accessible; he makes wonderful connections between themes, and there is a good deal of deep material present, even given his focus on “essentials.” The exercises are wonderful: there are a lot of remarks and hints, consonant with his evident pedagogical objectives. In short, it is a very good book, meeting the objectives the author has set for himself. For readers who want to go beyond the parameters set by the text, there are ample references, and as I already indicated, Kubrusly does an excellent job in situating this centrally important mathematical subject in a proper larger context.

Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.

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