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 (www.cambridge.org/9781316620243).
- More than 300 examples and counterexamples illustrating the (im)possibilities of measure and integration theory
- Concise non-technical overview of the main points of measure and integration
- Companion volume to the popular textbook Measures, Integrals and Martingales, now in its second edition
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Reviews & endorsements
'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, 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, 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, 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, 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, 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, The Mathematical Gazette
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Product details
- Date Published: June 2021
- format: Hardback
- isbn: 9781316519134
- length: 330 pages
- dimensions: 251 x 174 x 27 mm
- weight: 0.87kg
- availability: Available
Table of Contents
Preface
User's guide
List of topics and phenomena
1. A panorama of Lebesgue integration
2. A refresher of topology and ordinal numbers
3. Riemann is not enough
4. Families of sets
5. Set functions and measures
6. Range and support of a measure
7. Measurable and non-measurable sets
8. Measurable maps and functions
9. Inner and outer measure
10. Integrable functions
11. Modes of convergence
12. Convergence theorems
13. Continuity and a.e. continuity
14. Integration and differentiation
15. Measurability on product spaces
16. Product measures
17. Radon–Nikodým and related results
18. Function spaces
19. Convergence of measures
References
Index.
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Authors
René L. Schilling, Technische Universität, Dresden
René L. Schilling is Professor of Probability Theory at Technische Universität Dresden. His research focuses on stochastic analysis and the theory of stochastic processes.
Franziska Kühn, Technische Universität, Dresden
Franziska Kühn is Research Assistant at Technische Universität Dresden, where she finished her Ph.D. in 2016. She is interested in the interplay of probability theory and analysis, with a focus on jump processes and non-local operators.