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.
Explore measure and integration theory by asking 'What can go wrong if…' with this selection of over 300 counterexamples.
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 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.
'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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