Measures, Integrals and Martingales
eBook - PDF

Measures, Integrals and Martingales

  1. English
  2. PDF
  3. Available on iOS & Android
eBook - PDF

Measures, Integrals and Martingales

About this book

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 www.motapa.de. This book forms a sister volume to René Schilling's other book Counterexamples in Measure and Integration (www.cambridge.org/9781009001625).

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Yes, you can access Measures, Integrals and Martingales by René L. Schilling in PDF and/or ePUB format, as well as other popular books in Mathematics & Probability & Statistics. We have over one million books available in our catalogue for you to explore.

Information

Publisher
Cambridge University Press
Year
2017
Print ISBN
9781316620243
eBook ISBN
9781108162395
Edition
2
Topic
Mathematics
Subtopic
Probability & Statistics
Index
Mathematics

Table of contents

  1. Cover
  2. Half-Title
  3. Title Page
  4. Imprints Page
  5. Contents
  6. List of Symbols
  7. Prelude
  8. Dependence Chart
  9. 1 Prologue
  10. 2 The Pleasures of Counting
  11. 3 σ-Algebras
  12. 4 Measures
  13. 5 Uniqueness of Measures
  14. 6 Existence of Measures
  15. 7 Measurable Mappings
  16. 8 Measurable Functions
  17. 9 Integration of Positive Functions
  18. 10 Integrals of Measurable Functions
  19. 11 Null Sets and the 'Almost Everywhere'
  20. 12 Convergence Theorems and Their Applications
  21. 13 The Function Spaces L[sup(p)]
  22. 14 Product Measures and Fubini's Theorem
  23. 15 Integrals with Respect to Image Measures
  24. 16 Jacobi's Transformation Theorem
  25. 17 Dense and Determining Sets
  26. 18 Hausdorff Measure
  27. 19 The Fourier Transform
  28. 20 The Radon–Nikodm Theorem
  29. 21 Riesz Representation Theorems
  30. 22 Uniform Integrability and Vitali's Convergence Theorem
  31. 23 Martingales
  32. 24 Martingale Convergence Theorems
  33. 25 Martingales in Action
  34. 26 Abstract Hilbert Spaces
  35. 27 Conditional Expectations
  36. 28 Orthonormal Systems and Their Convergence Behaviour
  37. Appendix A: lim inf and lim sup
  38. Appendix B: Some Facts from Topology
  39. Appendix C: The Volume of a Parallelepiped
  40. Appendix D: The Integral of Complex-Valued Functions
  41. Appendix E: Measurability of the Continuity Points of a Function
  42. Appendix F: Vitali's Covering Theorem
  43. Appendix G: Non-measurable Sets
  44. Appendix H: Regularity of Measures
  45. Appendix I: A Summary of the Riemann Integral
  46. References
  47. Index