Overview

This book is designed to be an introduction to analysis with the proper mix of abstract theories and concrete problems. It starts with general measure theory, treats Borel and Radon measures (with particular attention paid to Lebesgue measure) and introduces the reader to Fourier analysis in Euclidean spaces with a treatment of Sobolev spaces, distributions, and the Fourier analysis of such. It continues with a Hilbertian treatment of the basic laws of probability including Doob's martingale convergence theorem and finishes with Malliavin's ""stochastic calculus of variations"" developed in the context of Gaussian measure spaces. This invaluable contribution to the existing literature gives the reader a taste of the fact that analysis is not a collection of independent theories but can be treated as a whole.

Full Product Details

Author:   H. Airault ,  Paul Malliavin ,  L. Kay ,  L. Kay
Publisher:   Springer-Verlag New York Inc.
Imprint:   Springer-Verlag New York Inc.
Edition:   1995 ed.
Volume:   157
Dimensions:   Width: 15.60cm , Height: 2.00cm , Length: 23.40cm
Weight:   1.480kg
ISBN:  

9780387944098

ISBN 10:   0387944095
Pages:   326
Publication Date:   13 June 1995
Audience:   College/higher education ,  Professional and scholarly ,  Postgraduate, Research & Scholarly ,  Professional & Vocational
Format:   Hardback
Publisher's Status:   Active
Availability:   In Print  
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Table of Contents

I Measurable Spaces and Integrable Functions.- 1 ?-algebras.- 2 Measurable Spaces.- 3 Measures and Measure Spaces.- 4 Negligible Sets and Classes of Measurable Mappings.- 5 Convergence in Mµ ((X,A);(Y,BY)).- 6 The Space of Integrable Functions.- 7 Theorems on Passage to the Limit under the Integral Sign.- 8 Product Measures and the Fubini-Lebesgue Theorem.- 9 The Lp Spaces.- II Borel Measures and Radon Measures.- 1 Locally Compact Spaces and Partitions of Unity.- 2 Positive Linear Functionals onCK(X) and Positive Radon Measures.- 3 Regularity of Borel Measures and Lusin’s Theorem.- 4 The Lebesgue Integral on R and on Rn.- 5 Linear Functionals on CK(X) and Signed Radon Measures.- 6 Measures and Duality with Respect to Spaces of Continuous Functions on a Locally Compact Space.- III Fourier Analysis.- 1 Convolutions and Spectral Analysis on Locally Compact Abelian Groups.- 2 Spectral Synthesis on Tn and Rn.- 3 Vector Differentiation and Sobolev Spaces.- 4 Fourier Transform of Tempered Distributions.- 5 Pseudo-differential Operators.- IV Hilbert Space Methods and Limit Theorems in Probability Theory.- 1 Foundations of Probability Theory.- 2 Conditional Expectation.- 3 Independence and Orthogonality.- 4 Characteristic Functions and Theorems on Convergence in Distribution.- 5 Theorems on Convergence of Martingales.- 6 Theory of Differentiation.- V Gaussian Sobolev Spaces and Stochastic Calculus of Variations.- 1 Gaussian Probability Spaces.- 2 Gaussian Sobolev Spaces.- 3 Absolute Continuity of Distributions.- Appendix I. Hilbert Spectral Analysis.- 1 Functions of Positive Type.- 2 Bochner’s Theorem.- 3 Spectral Measures for a Unitary Operator.- 4 Spectral Decomposition Associated with a Unitary Operator.- 5 Spectral Decomposition for Several Unitary Operators.- AppendixII. Infinitesimal and Integrated Forms of the Change-of-Variables Formula.- 1 Notation.- 2 Velocity Fields and Densities.- Exercises for Chapter I.- Exercises for Chapter II.- Exercises for Chapter III.- Exercises for Chapter IV.- Exercises for Chapter V.

Reviews

If I were asked to recommend texts to research students who need a grounding in integration theory this book would be on the list...The book is excellent - C. Barnett, Imperial College of Science, Technology and Medicine, London

""If I were asked to recommend texts to research students who need a grounding in integration theory this book would be on the list...The book is excellent"" - C. Barnett, Imperial College of Science, Technology and Medicine, London

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