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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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