Overview

This book is intended to be an introductory, yet sophisticated, treatment of measure theory. It should provide an in-depth reference for the practicing mathematician. It is hoped that advanced students as well as instructors will find it useful. The first part of the book should prove useful to both analysts and probabilists. One may treat the second and third parts as an introduction to the theory of probability, or use the fourth part as an introduction to analysis.The treatment is for the most part self-contained. Other than familiarity with general topology, some functional analysis and a certain degree of mathematical sophistication, little is required for profitable reading of this text. At the end of each chapter, exercises are provided which are designed to present some additional material and examples.

Full Product Details

Author:   Michel Simonnet ,  C.-M. Marle
Publisher:   Springer-Verlag New York Inc.
Imprint:   Springer-Verlag New York Inc.
Edition:   Softcover reprint of the original 1st ed. 1996
Dimensions:   Width: 15.50cm , Height: 2.70cm , Length: 23.50cm
Weight:   1.620kg
ISBN:  

9780387946443

ISBN 10:   0387946446
Pages:   510
Publication Date:   06 June 1996
Audience:   College/higher education ,  Professional and scholarly ,  Postgraduate, Research & Scholarly ,  Professional & Vocational
Format:   Paperback
Publisher's Status:   Active
Availability:   In Print  
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Table of Contents

I Integration Relative to Daniell Measures.- 1 Riesz Spaces.- 1.1 Ordered Groups.- 1.2 Riesz Spaces.- 1.3 Order Dual of a Riesz Space.- 1.4 Daniell Measures.- 2 Measures on Semirings.- 2.1 Semirings, Rings, and ?-Rings.- 2.2 Measures on Semirings.- 2.3 Lebesgue Measure on an Interval.- 3 Integrable and Measurable Functions.- 3.1 Upper Integral of a Positive Function.- 3.2 Convergence Theorems.- 3.3 Integrable Sets.- 3.4 ?-Measurable Spaces.- 3.5 Measurable Mappings.- 3.6 Essentially Integrable Mappings.- 3.7 Upper and Lower Integrals.- 3.8 Atoms.- 3.9 Prolongations of ?.- 4 Lebesgue Measure on R.- 4.1 Base-b Expansions of a Real Number.- 4.2 The Cantor Singular Function.- 4.3 Example of a Nonmeasurable Set.- 5 Lp Spaces.- 5.1 Definition of Lp Spaces.- 5.2 Convergence Theorems.- 5.3 Convergence in Measure.- 5.4 Uniformly Integrable Sets.- 6 Integrable Functions for Measures on Semirings.- 6.1 Measurability.- 6.2 Complements on the Lp Spaces.- 6.3 Measures Defined by Masses.- 6.4 Prolongations of a Measure.- 7 Radon Measures.- 7.1 Locally Compact Spaces.- 7.2 Radon Measures.- 7.3 Regularity of Radon Measures.- 7.4 Lusin Measurable Mappings.- 7.5 Atomic Radon Measures.- 7.6 The Riemann Integral.- 7.7 Weak Convergence.- 7.8 Tight Sequences.- 8 Regularity.- 8.1 Regular Measures.- II Operations on Measures Defined on Semirings.- 9 Induced Measures and Product Measures.- 9.1 Measure Induced on a Measurable Set.- 9.2 Fubini's Theorem.- 9.3 Lebesgue Measure on Rk.- 10 Radon-Nikodym Derivatives.- 10.1 Sums of Measures.- 10.2 Locally Integrable Functions.- 10.3 The Radon-Nikodym Theorem.- 10.4 Combination of Operations on Measures.- 10.5 Duality of Lp Spaces.- 10.6 The Yosida-Hewitt Decomposition Theorem.- 11 Images of Measures.- 11.1 ?-Suited Pairs.- 11.2 Infinite Product of Measures.- 11.3 Change of Variable.- 11.4 Elements of Ergodic Theory.- 12 Change of Variables.- 12.1 Differentiation in Rk.- 12.2 The Modulus of an Automorphism.- 12.3 Change of Variables.- 12.4 Polar Coordinates.- 13 Stieltjes Integral.- 13.1 Functions of Bounded Variation.- 13.2 Stieltjes Measures.- 13.3 Line Integrals.- 13.4 The Lebesgue Decomposition of a Function.- 13.5 Upper and Lower Derivatives.- 14 The Fourier Transform in Rk.- 14.1 Measures in Rk.- 14.2 Distribution Functions.- 14.3 Covariance Matrix.- 14.4 The Fourier Transform.- 14.5 Normal Laws in Rn.- III Convergence of Random Variables; Conditional Expectation.- 15 The Strong Law of Large Numbers.- 15.1 Convergence in Probability.- 15.2 Independence of Random Variables.- 15.3 An Example of Independent Random Variables.- 15.4 The One-Sided Shift Transformation.- 15.5 Borel's Normal Number Theorem.- 16 The Central Limit Theorem.- 16.1 Convergence in Law.- 16.2 The Lindeberg Theorem.- 16.3 The Central Limit Theorem.- 17 Order Statistics.- 17.1 Definition of the Order Statistics.- 17.2 Convergence of the Empirical Median.- 18 Conditional Probability.- 18.1 Conditional Expectation.- 18.2 The Converse of the Mean-Value Theorem.- 18.3 Jensen's Inequality.- 18.4 Conditional Expected Value Given a Random Variable.- 18.5 Conditional Law of Y Given X.- 18.6 Computation of Conditional Laws.- 18.7 Existence of Conditional Laws when G = Rk.- IV Operations on Radon Measures.- 19 ?-Adequate Family of Measures.- 19.1 Induced Radon Measure.- 19.2 ?-Dense Families of Compact Sets.- 19.3 Sums of Radon Measures.- 19.4 ?-Adequate Families.- 19.5 ?-Adapted Pairs.- 20 Radon Measures Defined by Densities.- 20.1 Integration with Respect to Induced Measures.- 20.2 Radon Measures with Base ?.- 20.3 The Radon-Nikodym Theorem.- 20.4 Duality of Lp Spaces.- 21 Images of Radon Measures and Product Measures.- 21.1 Images of Radon Measures.- 21.2 Decomposition of a Measure in Slices.- 21.3 Product of Radon Measures.- 22 Operations on Regular Measures.- 22.1 Some Operations on Regular Measures.- 22.2 Baire Sets.- 22.3 Product of Regular Measures.- 22.4 Change of Variable Formula.- 23 Haar Measures.- 23.1 Invariant Measures.- 23.2 Existence and Uniqueness of Left Haar Measure.- 23.3 Modular Function on G.- 23.4 Relatively Invariant Measures on a Group.- 23.5 Homogeneous Spaces.- 23.6 Integration with Respect to ?#.- 23.7 Reconstitution of ?#/?.- 23.8 Quasi-Invariant Measures on Homogeneous Spaces.- 23.9 Relatively Invariant Measures on G/H.- 23.10 Haar Measure on SO(n + 1,R).- 23.11 Haar Measure on SH(n,R).- 24 Convolution of Measures.- 24.1 Convolvable Measures.- 24.2 Convolution of a Measure and a Function.- 24.3 Convolution of a Measure and a Continuous Function.- 24.4 Convolution of ? ? M(G, C) and f ? $$\overline {{\mathcal{L}^{\text{p}}}}$$(?).- 24.5 Convolution and Transposition.- 24.6 Convolution of Functions on a Group.- 24.7 Regularization.- 24.8 Definition of Gelfand Pair.- Symbol Index.

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