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基本説明
A concise and elementary introduction. Undergraduate calculus and an introductory course on rigorous analysis in R are the only essential prerequisites. Making this text suitable for both lecture courses and for self-study.
Full Description
This 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.
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.
