Counterexamples in Measure and Integration

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René L. Schilling (Technische Universität, Dresden) Franziska Kühn (Technische Universität, Dresden)

Counterexamples in Measure and Integration

René L. Schilling (Technische Universität, Dresden), Franziska Kühn (Technische Universität, Dresden)

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English

Cambridge University Press
17 June 2021

Calculus & mathematical analysis; Probability & statistics; Mensuration & systems of measurement

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).

By: René L. Schilling (Technische Universität Dresden), Franziska Kühn (Technische Universität, Dresden)
Imprint: Cambridge University Press
Country of Publication: United Kingdom
Dimensions: Height: 244mm, Width: 170mm, Spine: 23mm
Weight: 740g
ISBN: 9781009001625
ISBN 10: 1009001620
Pages: 420
Publication Date: 17 June 2021
Audience: Professional and scholarly , Undergraduate
Format: Paperback
Publisher's Status: Active

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.

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.

Reviews for Counterexamples in Measure and Integration

'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