Measures, Integrals and Martingales

René Schilling

Cambridge University Press
Paperback: 0-521-61525-9

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Textbook Measures, Integrals and Martingales René Schilling (go to homepage) Description 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 will be available on the internet. Lecturers can request inspection copies of this title Features Introduction to a central mathematical topic accessible for undergraduates Easy to follow exposition with numerous illustrations and exercises included Hints and solutions are available on the internet. Text is suitable for classroom use as well as for self-study ContentsPrelude - Dependence chart - Prologue The pleasures of counting sigma-algebras Measures Uniqueness of measures Existance of measures Measurable mappings Measurable functions Integration of positive functions Integrals of measurable functions and null sets Convergence theroems and their applications The function spaces L^p Product measures and Fubini’s theorem Integrals with respect to image measures Integrals of images and Jacobi’s transformation rule Uniform integrability and Vitali’s convergence theorem Martingales Martingale convergence theorems The Radon-Nikodym theorem and other applications of martingales Inner product spaces Hilbert space Conditional expectations in L^2 Conditional expectations in L^p Orthonormal systems and their convergence behaviour Appendix A: liminf and limsup 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

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