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Cambridge University Press
Online publication date:
May 2021
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Book description

Often it is more instructive to know 'what can go wrong' and to understand 'why a result fails' than to plod through yet another piece of theory. In this text, the authors gather more than 300 counterexamples - some of them both surprising and amusing - showing the limitations, hidden traps and pitfalls of measure and integration. Many examples are put into context, explaining relevant parts of the theory, and pointing out further reading. The text starts with a self-contained, non-technical overview on the fundamentals of measure and integration. A companion to the successful undergraduate textbook Measures, Integrals and Martingales, it is accessible to advanced undergraduate students, requiring only modest prerequisites. More specialized concepts are summarized at the beginning of each chapter, allowing for self-study as well as supplementary reading for any course covering measures and integrals. For researchers, it provides ample examples and warnings as to the limitations of general measure theory. This book forms a sister volume to René Schilling's other book Measures, Integrals and Martingales (


'This book is an admirable counterpart, both to the first author's well-known text Measures, Integrals and Martingales (Cambridge, 2005/2017), and to the books on counter-examples in analysis (Gelbaum and Olmsted), topology (Steen and Seebach) and probability (Stoyanov). To paraphrase the authors' preface: in a good theory, it is valuable and instructive to probe the limits of what can be said by investigating what cannot be said. The task is thus well-conceived, and the execution is up to the standards one would expect from the books of the first author and of their papers. I recommend it warmly.'

N. H. Bingham - Imperial College

‘… an excellent reference text and companion reader for anyone interested in deepening their understanding of measure theory.’

John Ross Source: MAA Reviews

‘… the unique nature of the book makes it an essential acquisition for any university with a doctoral program in pure mathematics … Essential.’

M. Bona Source: Choice Connect

‘The book is well written, the demonstrations are clear and the bibliographic references are competent. We appreciate this work as extremely useful for those interested in measure theory and integration, starting with beginners and extending even to advanced researchers in the field.’

Liviu Constantin Florescu Source: Mathematical Reviews/MathSciNet

‘Counterexamples in Measure and Integration is an ideal companion to help better understand canonically problematic examples in analysis … This collection of counterexamples is an excellent resource to researchers who rely on measure and integration theory. It would be helpful for students studying for their analysis qualifying exam as it draws on common misconceptions and enables readers to build intuition about why a given counterexample works and how conditions can be changed to make a particular statement hold.’

Katelynn Kochalski Source: Notices of the AMS

‘This is a remarkable book covering Measure and Integration, perhaps one of the most important parts of Mathematics. It is written in a master style by following the best traditions in writing this kind of books. The authors are passionate about the topic. Look at the great care with which each of the counterexamples is presented. It is done in a way to help maximally the reader. The names of the counterexamples are chosen very carefully. Any name can be considered as a 'door' behind which is a treasure!’

Jordan M. Stoyanov Source: zbMATH

‘… compendia of counterexamples remain a useful and thought-provoking resource, and this new text is a high-quality example in an analytic direction.’

Dominic Yeo Source: The Mathematical Gazette

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