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Measures, Integrals and Martingales

2nd Edition

$49.99 (X)

  • Date Published: April 2017
  • availability: Available
  • format: Paperback
  • isbn: 9781316620243

$ 49.99 (X)

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About the Authors
  • A concise yet elementary introduction to measure and integration theory, which are vital in many areas of mathematics, including analysis, probability, mathematical physics and finance. In this highly successful textbook, core ideas of measure and integration are explored, and martingales are used to develop the theory further. Other topics are also covered such as Jacobi's transformation theorem, the Radon–Nikodym theorem, differentiation of measures and Hardy–Littlewood maximal functions. In this second edition, readers will find newly added chapters on Hausdorff measures, Fourier analysis, vague convergence and classical proofs of Radon–Nikodym and Riesz representation theorems. All proofs are carefully worked out to ensure full understanding of the material and its background. Requiring few prerequisites, this book is suitable for undergraduate lecture courses or self-study. Numerous illustrations and over 400 exercises help to consolidate and broaden knowledge. Full solutions to all exercises are available on the author's webpage at This book forms a sister volume to René Schilling's other book Counterexamples in Measure and Integration (

    • A very clear exposition which will take the 'fear' out of measure and integration theory
    • Contains clearly structured proofs and numerous exercises designed to deepen understanding of the material
    • Full solutions to all problems are available online, making the text useful for self-study
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    Reviews & endorsements

    Review of previous edition: ‘… thorough introduction to a wide variety of first-year graduate-level topics in analysis … accessible to anyone with a strong undergraduate background in calculus, linear algebra and real analysis.' Zentralblatt MATH

    Review of previous edition: ‘The author truly covers a wide range of topics … Proofs are written in a very organized and detailed manner … I believe this to be a great book for self-study as well as for course use. The book is ideal for future probabilists as well as statisticians, and can serve as a good introduction for mathematicians interested in measure theory.' Ita Cirovic Donev, MAA Reviews

    Review of previous edition: ‘… succeeds in handling the technicalities of measure theory, which is traditionally regarded as dry and inaccessible to students (and, I think, the most difficult material that I have taught at undergraduate level) with a light touch. The book is eminently suitable for a course (or two) for good final-year or first-year post-graduate students and has the potential to revitalize the way that measure theory is taught.' N. H. Bingham, Journal of the Royal Statistical Society

    Review of previous edition: ‘This book will remain a good reference on the subject for years to come.' Peter Eichelsbacher, Mathematical Reviews

    Review of previous edition: ‘… this well-written and carefully structured book is an excellent choice for an undergraduate course on measure and integration theory. Most good books on measure and integration are graduate books and, therefore, are not optimal for undergraduate courses … This book is aimed at both (future) analysts and (future) probabilists, and is therefore suitable for students from both these groups.' Filip Lindskog, Journal of the American Statistical Association

    'This book is an admirable counterpart, both to the first author's well-known text Measures, Integrals and Martingales (CUP, 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.' Professor N. H. Bingham, Imperial College London

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    Product details

    • Edition: 2nd Edition
    • Date Published: April 2017
    • format: Paperback
    • isbn: 9781316620243
    • length: 490 pages
    • dimensions: 247 x 174 x 22 mm
    • weight: 0.97kg
    • contains: 40 b/w illus. 420 exercises
    • availability: Available
  • Table of Contents

    List of symbols
    Dependence chart
    1. Prologue
    2. The pleasures of counting
    3. σ-algebras
    4. Measures
    5. Uniqueness of measures
    6. Existence of measures
    7. Measurable mappings
    8. Measurable functions
    9. Integration of positive functions
    10. Integrals of measurable functions
    11. Null sets and the 'almost everywhere'
    12. Convergence theorems and their applications
    13. The function spaces Lp
    14. Product measures and Fubini's theorem
    15. Integrals with respect to image measures
    16. Jacobi's transformation theorem
    17. Dense and determining sets
    18. Hausdorff measure
    19. The Fourier transform
    20. The Radon–Nikodym theorem
    21. Riesz representation theorems
    22. Uniform integrability and Vitali's convergence theorem
    23. Martingales
    24. Martingale convergence theorems
    25. Martingales in action
    26. Abstract Hilbert spaces
    27. Conditional expectations
    28. Orthonormal systems and their convergence behaviour
    Appendix A. Lim inf and lim sup
    Appendix B. Some facts from topology
    Appendix C. The volume of a parallelepiped
    Appendix D. The integral of complex valued functions
    Appendix E. Measurability of the continuity points of a function
    Appendix F. Vitali's covering theorem
    Appendix G. Non-measurable sets
    Appendix H. Regularity of measures
    Appendix I. A summary of the Riemann integral
    Name and subject index.

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    Measures, Integrals and Martingales

    René L. Schilling

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  • Author

    René L. Schilling, Technische Universität, Dresden
    René L. Schilling is a Professor of Mathematics at Technische Universität, Dresden. His main research area is stochastic analysis and stochastic processes.

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