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Measures, Integrals and Martingales

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Measures, Integrals and Martingales

Measures, Integrals and Martingales

382 pages, parution le 19/12/2005

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
Voir tout
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Caractéristiques techniques du livre "Measures, Integrals and Martingales"

  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

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