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Counterexamples in Measure and Integration

René L. Schilling and Franziska Kühn
Cambridge University Press
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The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by
John Ross
, on
In Counterexamples in Measure and Integration, Schilling and Kühn create an excellent reference text and companion reader for anyone interested in deepening their understanding of measure theory. As we all know, and as the text explicitly states, students of mathematics will “gain a better understanding of a theorem or theory if [they know] its limitations – which may be expressed in the form of counterexamples.” Indeed, it is often more informative to learn “’what can go wrong’ and to understand ‘why a result fails’ than to plod through yet another piece of theory.” This philosophy motivates and undergirds the text. While the result likely would not serve as a standalone textbook on measure theory, it would undoubtedly provide valuable support to any standard text on the subject.
This textbook contains 19 chapters with the first two serving as an informal overview of important subjects. Chapter 1 gives us an overview of Lebesgue Integration, refreshing the reader on the details of (and motivation behind) many major definitions in measure theory. Chapter 2 provides details on important topics in topology, which will help us construct many of our counterexamples in the chapters to come. The bulk of the text – from Chapter 3 onwards – is dedicated to providing counterexamples in measure theory. There are over 300 counterexamples in total, and the remaining chapters serve to separate overarching topics in a narratively appropriate manner. (Which means, loosely, starting with Riemann integration and moving through measurable sets, sigma-algebras, measurable functions, integration, and function spaces.) The counterexamples are each self-contained and discussed in relatively short expositions (most are covered in less than two pages).
The text is not meant to be read linearly, but it has a very precise organization that makes it an outstanding reference text. Every counterexample provided is given a number of the form [M.N] (referring to counterexample N found in chapter M), and so specific counterexamples can be referenced easily throughout the text. This allows readers to skip around the book with ease, following one example by reading about a related counterexample. This cross-referencing is robust throughout the text, including in the introductory chapters. Indeed, those first few chapters help motivate their definitions by “pointing” to future counterexamples in the text. While theorems in the introductory chapters rarely include full proofs, references are provided for the interested reader. Finally, the text contains a thorough table of contents, as well as a “List of Topics and Phenomena” that are covered, each carefully referenced. As a result, specific counterexamples and general ideas are each easily looked up in this
It is worth noting that the text is most naturally set to be a companion to Measures, Integrals, and Martingales. This is true in part because the texts share an author (Schilling), but also because many stated theorems in Counterexamples point to specific chapters/pages in MIMS to supply a proof.
However, in this reviewer’s opinion, the text would be useful as a companion to any standard measure theory textbook, and is not solely bound to MIMS. With its conversational tone, long list of examples and its precise organization, Counterexamples in Measure and Integration makes a strong case to appear on the bookshelf of any student of measure theory.


John Ross is an assistant professor of mathematics at Southwestern University