Librería Portfolio

A concise yet elementary introduction to measure and integration theory, which are vital in many areas of mathematics, including analysis, probability, mathematical physics and finance. In this highly successful textbook, core ideas of measure and integration are explored, and martingales are used to develop the theory further. Other topics are also covered such as Jacobi´s transformation theorem, the Radon-Nikodym theorem, differentiation of measures and Hardy-Littlewood maximal functions. In this second edition, readers will find newly added chapters on Hausdorff measures, Fourier analysis, vague convergence and classical proofs of Radon-Nikodym and Riesz representation theorems. All proofs are carefully worked out to ensure full understanding of the material and its background. Requiring few prerequisites, this book is suitable for undergraduate lecture courses or self-study. Numerous illustrations and over 400 exercises help to consolidate and broaden knowledge. Full solutions to all exercises are available on the author´s webpage at www.motapa.de.

A very clear exposition which will take the ´fear´ out of measure and integration theory
Contains clearly structured proofs and numerous exercises designed to deepen understanding of the material
Full solutions to all problems are available online, making the text useful for self-study




Table of Contents

List of symbols
Prelude
Dependence chart
1. Prologue
2. The pleasures of counting
3. s-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
11. Null sets and the ´almost everywhere´
12. Convergence theorems and their applications
13. The function spaces Lp
14. Product measures and Fubini´s theorem
15. Integrals with respect to image measures
16. Jacobi´s transformation theorem
17. Dense and determining sets
18. Hausdorff measure
19. The Fourier transform
20. The Radon-Nikodym theorem
21. Riesz representation theorems
22. Uniform integrability and Vitali´s convergence theorem
23. Martingales
24. Martingale convergence theorems
25. Martingales in action
26. Abstract Hilbert spaces
27. Conditional expectations
28. Orthonormal systems and their convergence behaviour
Appendix A. Lim inf and lim sup
Appendix B. Some facts from topology
Appendix C. The volume of a parallelepiped
Appendix D. The integral of complex valued functions
Appendix E. Measurability of the continuity points of a function
Appendix F. Vitali´s covering theorem
Appendix G. Non-measurable sets
Appendix H. Regularity of measures
Appendix I. A summary of the Riemann integral
References
Name and subject index.