Measures, Integrals and Martingales

 Introduction to a central mathematical topic accessible for undergraduates ? Easy to follow exposition with numerous illustrations and exercises included. Hints and solutions can be found on the authors website, which can be reached from www.cambridge.org/9780521615259 ? Text is suitable for classroom use as well as for self-study

This is a concise and elementary introduction to measure and integration theory as it is nowadays needed in many parts of analysis and probability theory. The basic theory - measures, integrals, convergence theorems, Lp-spaces and multiple integrals - is explored in the first part of the book. The second part then uses the notion of martingales to develop the theory further, covering topics such as Jacobi's generalized transformation Theorem, the Radon-Nikodym theorem, differentiation of measures, Hardy-Littlewood maximal functions or general Fourier series. Undergraduate calculus and an introductory course on rigorous analysis are the only essential prerequisites, making this text suitable for both lecture courses and for self-study. Numerous illustrations and exercises are included and these are not merely drill problems but are there to consolidate what has already been learnt and to discover variants, sideways and extensions to the main material. Hints and solutions can be found on the authors website, which can be reached from www.cambridge.org/9780521615259. Now in its third printing, the book has been recently updated with corrections from the author.

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.