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

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

Published $2005$, Cambridge University Press.

### 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