Measures, Integrals and Martingales
Measures, Integrals and Martingales
Replaced by:
9780521615259This book, first published in 2005, introduces measure and integration theory as it is 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, 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 author's website, which can be reached from www.cambridge.org/9780521615259. This book forms a sister volume to René Schilling's other book Counterexamples in Measure and Integration (www.cambridge.org/9781009001625).
- 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 author's website, which can be reached from www.cambridge.org/9780521615259
- Text is suitable for classroom use as well as for self-study
Reviews & endorsements
'... thorough introduction to a wide variety of first year graduate level topics in analysis... accessible to anyone with a strong undergraduate background in calculus, linear algebra, and real analysis.' Zentralblatt MATH
'This is a concise and elementary introduction to measure and integration theory as need nowadays in many parts of analysis and probability theory.' L'Enseignement Mathématique
'I have not seen some of the topics that are mentioned above … treated successfully at undergraduate level before, and the book is worth having for these alone … [it] has the potential to revitalize the way that measure theory is taught.' Journal of the Royal Statistical Society
Product details
November 2005Hardback
9780521850155
394 pages
255 × 180 × 29 mm
0.944kg
15 b/w illus. 500 exercises
Replaced by 9780521615259
Table of Contents
- 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.
