Skip to content
Register Sign in Wishlist

Counterexamples in Measure and Integration


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

£ 34.99

Add to cart Add to wishlist

Other available formats:
Hardback, eBook

Looking for an inspection copy?

This title is not currently available on inspection

Product filter button
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 (

    • 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
    Read more

    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

    See more reviews

    Customer reviews

    Not yet reviewed

    Be the first to review

    Review was not posted due to profanity


    , create a review

    (If you're not , sign out)

    Please enter the right captcha value
    Please enter a star rating.
    Your review must be a minimum of 12 words.

    How do you rate this item?


    Product details

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

    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

  • Resources for

    Counterexamples in Measure and Integration

    René L. Schilling, Franziska Kühn

    General Resources

    Find resources associated with this title

    Type Name Unlocked * Format Size

    Showing of

    Back to top

    This title is supported by one or more locked resources. Access to locked resources is granted exclusively by Cambridge University Press to lecturers whose faculty status has been verified. To gain access to locked resources, lecturers should sign in to or register for a Cambridge user account.

    Please use locked resources responsibly and exercise your professional discretion when choosing how you share these materials with your students. Other lecturers may wish to use locked resources for assessment purposes and their usefulness is undermined when the source files (for example, solution manuals or test banks) are shared online or via social networks.

    Supplementary resources are subject to copyright. Lecturers are permitted to view, print or download these resources for use in their teaching, but may not change them or use them for commercial gain.

    If you are having problems accessing these resources please contact

  • 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.

Related Books

Sorry, this resource is locked

Please register or sign in to request access. If you are having problems accessing these resources please email

Register Sign in
Please note that this file is password protected. You will be asked to input your password on the next screen.

» Proceed

You are now leaving the Cambridge University Press website. Your eBook purchase and download will be completed by our partner Please see the permission section of the catalogue page for details of the print & copy limits on our eBooks.

Continue ×

Continue ×

Continue ×
warning icon

Turn stock notifications on?

You must be signed in to your Cambridge account to turn product stock notifications on or off.

Sign in Create a Cambridge account arrow icon

Find content that relates to you

Join us online

This site uses cookies to improve your experience. Read more Close

Are you sure you want to delete your account?

This cannot be undone.


Thank you for your feedback which will help us improve our service.

If you requested a response, we will make sure to get back to you shortly.

Please fill in the required fields in your feedback submission.