Measures, integrals and martingales

Abstract

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.