Measures, Integrals and Martingales

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This is a concise and elementary introduction to contemporary measure and integration theory as it is needed in many parts of analysis and probability theory. Undergraduate calculus and an introductory course on rigorous analysis in R are the only essential prerequisites, making the text suitable for both lecture courses and for self-study. Numerous illustrations and exercises are included to consolidate what has already been learned and to discover variants and extensions to the main material. Hints and solutions can be found on the authors website, which can be reached at

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Editorial Reviews

From the Publisher
"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."
MAA Reviews

" 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. If it does, the author will deserve our thanks indeed."
Journal of the Royal Statistical Society

"This book will remain a good reference on the subject for years to come."
Peter Eichelsbacher, Mathematical Reviews

"The chapters contain nicely written short blocks of theory followed by good and meaningful exercises, solutions of which are available on the author's home page. This feature makes the book an attractive starting point for an undergraduate course on measure and integration theory. The book is well structured and the presentation is clear; arguments and proofs are detailed and easy to follow."
Filip Lindskog, Journal of the American Statistical Association

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Product Details

  • ISBN-13: 9780521615259
  • Publisher: Cambridge University Press
  • Publication date: 8/31/2005
  • Edition description: New Edition
  • Pages: 394
  • Product dimensions: 6.85 (w) x 9.72 (h) x 0.83 (d)

Meet the Author

Rene Schilling is a Professor of Stochastics at the University of Marburg.

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Table of Contents

Prelude; Dependence chart; Prologue; 1. The pleasures of counting; 2. sigma-algebras; 3. Measures; 4. Uniqueness of measures; 5. Existance of measures; 6. Measurable mappings; 7. Measurable functions; 8. Integration of positive functions; 9. Integrals of measurable functions and null sets; 10. Convergence theroems and their applications; 11. The function spaces; 12. Product measures and Fubini's theorem; 13. Integrals with respect to image measures; 14. Integrals of images and Jacobi's transformation rule; 15. Uniform integrability and Vitali's convergence theorem; 16. Martingales; 17. Martingale convergence theorems; 18. The Radon-Nikodym theorem and other applications of martingales; 19. Inner product spaces; 20. Hilbert space; 21. Conditional expectations in L2; 22. Conditional expectations in Lp; 23. Orthonormal systems and their convergence behaviour; Appendix A. Lim inf and lim supp; Appendix B. Some facts from point-set topology; Appendix C. The volume of a parallelepiped; Appendix D. Non-measurable sets; Appendix E. A summary of the Riemann integral; Further reading; Bibliography; Notation index; Name and subject index.

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