Measures, Integrals and Martingales

René L. Schilling

382 pages, parution le 19/12/2005

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Résumé

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.

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

Sommaire

Prelude
Dependence chart
Prologue
The pleasures of counting
σ-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
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
Conditional expectations in
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

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Caractéristiques techniques

	PAPIER
Éditeur(s)	Cambridge University Press
Auteur(s)	René L. Schilling
Parution	19/12/2005
Nb. de pages	382
Format	17,5 x 24,5
Couverture	Broché
Poids	778g
Intérieur	Noir et Blanc
EAN13	9780521615259
ISBN13	978-0-521-61525-9