Measures, Integrals and Martingales
by René L. Schilling
2nd Edition
Publisher: Cambridge University Press
Book Details
| Print ISBN | 9781316620243 |
| eText ISBN | 9781108161275 |
| Publisher | Cambridge University Press |
| Publishing Year | 2017 |
| Edition | 2nd Edition |
| Language | English |
| Pages | 496 |
Measures, Integrals and Martingales, 2nd Edition, is a textbook by René L. Schilling published by Cambridge University Press. The volume provides a concise, elementary introduction to measure and integration theory designed to require few prerequisites as the theory is developed quickly and simply.
Coverage introduces core principles of measure and integration alongside their connections to stochastic analysis. Key topics include σ-algebras, the existence and uniqueness of measures, measurable functions, null sets, convergence theorems, and Lp spaces, leading to the study of conditional expectations and martingales.
A dependence chart outlines the relationships among topics, and an external author website provides additional material. Publisher solution resources are restricted exclusively to adopting instructors.
Table of Contents
Chapter List of symbols: List of symbols
Chapter Prelude: Prelude
Chapter Dependence chart: Dependence chart
Chapter 1: Prologue
Chapter 2: The pleasures of counting
Chapter 3: σ-algebras
Chapter 4: Measures
Chapter 5: Uniqueness of measures
Chapter 6: Existence of measures
Chapter 7: Measurable mappings
Chapter 8: Measurable functions
Chapter 9: Integration of positive functions
Chapter 10: Integrals of measurable functions
Chapter 11: Null sets and the 'almost everywhere'
Chapter 12: Convergence theorems and their applications
Chapter 13: The function spaces Lp
Chapter 14: Product measures and Fubini's theorem
Chapter 15: Integrals with respect to image measures
Chapter 16: Jacobi's transformation theorem
Chapter 17: Dense and determining sets
Chapter 18: Hausdorff measure
Chapter 19: The Fourier transform
Chapter 20: The Radon–Nikodym theorem
Chapter 21: Riesz representation theorems
Chapter 22: Uniform integrability and Vitali's convergence theorem
Chapter 23: Martingales
Chapter 24: Martingale convergence theorems
Chapter 25: Martingales in action
Chapter 26: Abstract Hilbert spaces
Chapter 27: Conditional expectations
Chapter 28: Orthonormal systems and their convergence behaviour
Chapter Appendix A: Lim inf and lim sup
Chapter Appendix B: Some facts from topology
Chapter Appendix C: The volume of a parallelepiped
Chapter Appendix D: The integral of complex valued functions
Chapter Appendix E: Measurability of the continuity points of a function
Chapter Appendix F: Vitali's covering theorem
Chapter Appendix G: Non-measurable sets
Chapter Appendix H: Regularity of measures
Chapter Appendix I: A summary of the Riemann integral
Chapter References: References
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