# Book:René L. Schilling/Measures, Integrals and Martingales

## René L. Schilling: *Measures, Integrals and Martingales*

Published $2005$, **Cambridge University Press**.

### Subject Matter

### Contents

- Prelude
- Dependence chart
- 1: Prologue
- 2: The pleasures of counting
- 3: $\sigma$-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 and null sets
- 11: Convergence theorems and their applications
- 12: The function spaces $\mathcal{L}^p$, $1 \le p \le \infty$
- 13: Product measures and Fubini's theorem
- 14: Integrals with respect to image measures
- 15: Integrals of images and Jacobi's transformation rule
- 16: Uniform integrability and Vitali's convergence theorem
- 17: Martingales
- 18: Martingale convergence theorems
- 19: The Radon-Nikodým theorem and other applications of martingales
- 20: Inner product spaces
- 21: Hilbert space $\mathfrak H$
- 22: Conditional expectations in $L^2$
- 23: Conditional expectations in $L^p$
- 24: Orthonormal systems and their convergence behaviour
- Appendix A: lim inf and lim sup
- 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
- References
- Notation index
- Name and subject index