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Counterexamples in Measure and Integration

$44.99 (P)

  • Date Published: August 2021
  • availability: In stock
  • format: Paperback
  • isbn: 9781009001625

$ 44.99 (P)
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About the Authors
  • 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

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    Product details

    • Date Published: August 2021
    • format: Paperback
    • isbn: 9781009001625
    • length: 420 pages
    • dimensions: 244 x 170 x 23 mm
    • weight: 0.74kg
    • availability: In stock
  • 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.

  • Resources for

    Counterexamples in Measure and Integration

    René L. Schilling, Franziska Kühn

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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.

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