Counterexamples in Measure and Integration

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

René L. Schilling (Author), Franziska Kühn (Author)

9781009001625, Cambridge University Press

Paperback / softback, published 17 June 2021

420 pages
24.4 x 17 x 2.3 cm, 0.74 kg

'… 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

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).

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.

Subject Areas: Mensuration & systems of measurement [PDDM], Probability & statistics [PBT], Calculus & mathematical analysis [PBK]