Integration and probability
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Bibliographic Information
Integration and probability
（Graduate texts in mathematics, 157）
Springer-Verlag, c1995
- Other Title
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Intégration et probabilités : analyse de Fourier et analyse spectrale
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Note
Includes bibliographical references and index
Description and Table of Contents
Description
An introduction to analysis with the right mix of abstract theories and concrete problems. Starting with general measure theory, the book goes on to treat Borel and Radon measures and introduces the reader to Fourier analysis in Euclidean spaces with a treatment of Sobolev spaces, distributions, and the corresponding Fourier analysis. 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 gives a taste of the fact that analysis is not a collection of independent theories, but can be treated as a whole.
Table of Contents
- I Measurable Spaces and Integrable Functions.- 1 ?-algebras.- 1.1 Sub-?-algebras. Intersection of ?-algebras.- 1.2 ?-algebra generated by a family of sets.- 1.3 Limit of a monotone sequence of sets.- 1.4 Theorem (Boolean algebras and monotone classes).- 1.5 Product ?-algebras.- 2 Measurable Spaces.- 2.1 Inverse image of a ?-algebra.- 2.2 Closure under inverse images of the generated ?-algebra.- 2.3 Measurable spaces and measurable mappings.- 2.4 Borel algebras. Measurability and continuity. Operations on measurable functions.- 2.5 Pointwise convergence of measurable mappings.- 2.6 Supremum of a sequence of measurable functions.- 3 Measures and Measure Spaces.- 3.1 Convexity inequality.- 3.2 Measure of limits of monotone sequences.- 3.3 Countable convexity inequality.- 4 Negligible Sets and Classes of Measurable Mappings.- 4.1 Negligible sets.- 4.2 Complete measure spaces.- 4.3 The space M((X, A)
- (X?,A?)).- 5 Convergence in M ((X,A)
- (Y,BY)).- 5.1 Convergence almost everywhere.- 5.2 Convergence in measure.- 6 The Space of Integrable Functions.- 6.1 Simple measurable functions.- 6.2 Finite ?-algebras.- 6.3 Simple functions and indicator functions.- 6.4 Approximation by simple functions.- 6.5 Integrable simple functions.- 6.6 Some spaces of bounded measurable functions.- 6.7 The truncation operator.- 6.8 Construction of L1.- 7 Theorems on Passage to the Limit under the Integral Sign.- 7.1 Fatou-Beppo Levi theorem.- 7.2 Lebesgue's theorem on series.- 7.3 Theorem (truncation operator a contraction).- 7.4 Integrability criteria.- 7.5 Definition of the integral on a measurable set.- 7.6 Lebesgue's dominated convergence theorem.- 7.7 Fatou's lemma.- 7.8 Applications of the dominated convergence theorem to integrals which depend on a parameter.- 8 Product Measures and the Fubini-Lebesgue Theorem.- 8.1 Definition of the product measure.- 8.2 Proposition (uniqueness of the product measure).- 8.3 Lemma (measurability of sections).- 8.4 Construction of the product measure.- 8.5 The Fubini-Lebesgue theorem.- 9 The Lp Spaces.- 9.0 Integration of complex-valued functions.- 9.1 Definition of the Lp spaces.- 9.2 Convexity inequalities.- 9.3 Completeness theorem.- 9.4 Notions of duality.- 9.5 The space L?.- 9.6 Theorem (containment relations between Lp spaces if (X) < ?).- II Borel Measures and Radon Measures.- 1 Locally Compact Spaces and Partitions of Unity.- 1.0 Definition of locally compact spaces which are countable at infinity.- 1.1 Urysohn's lemma.- 1.2 Support of a function.- 1.3 Subordinate covers.- 1.4 Partitions of unity.- 2 Positive Linear Functionals onCK(X) and Positive Radon Measures.- 2.1 Borel measures.- 2.2 Radon-Riesz theorem.- 2.3 Proof of uniqueness of the Riesz representation.- 2.4 Proof of existence of the Riesz representation.- 3 Regularity of Borel Measures and Lusin's Theorem.- 3.1 Proposition (Borel measures and Radon measures).- 3.2 Theorem (regularity of Radon measures).- 3.3 Theorem (regularity of locally finite Borel measures).- 3.4 The classes G?(X) and F?(X).- 3.5 Theorem (density of CK in Lp).- 4 The Lebesgue Integral on R and on Rn.- 4.1 Definition of the Lebesgue integral on R.- 4.2 Properties of the Lebesgue integral.- 4.3 Lebesgue measure on Rn.- 4.4 Change of variables in the Lebesgue integral on Rn.- 5 Linear Functionals on CK(X) and Signed Radon Measures.- 5.1 Continuous linear functionals on C(X) (X compact).- 5.2 Decomposition theorem.- 5.3 Signed Borel measures.- 5.4 Dirac measures and discrete measures.- 5.5 Support of a signed Radon measure.- 6 Measures and Duality with Respect to Spaces of Continuous Functions on a Locally Compact Space.- 6.1 Definitions.- 6.2 Proposition (relationships among CbCK, and C0)..- 6.3 The Alexandroff compactification.- 6.4 Proposition.- 6.5 The space M1(X).- 6.6 Theorem (M1(X) the dual of C0(X)).- 6.7 Defining convergence by duality.- 6.8 Theorem (relationships among types of convergence).- 6.9 Theorem (narrow density of Md,f1in M1).- III Fourier Analysis.- 1 Convolutions and Spectral Analysis on Locally Compact Abelian Groups.- 1.1 Notation.- 1.2 Examples.- 1.3 The group algebra.- 1.4 The dual group. The Fourier transform on M1.- 1.5 Invariant measures. The space L1.- 1.6 The space L1(G).- 1.7 The translation operator.- 1.8 Extensions of the convolution product.- 1.9 Convergence theorem.- 2 Spectral Synthesis on Tn and Rn.- 2.1 The character groups of Rn and Tn.- 2.2 Spectral synthesis on T.- 2.3 Extension of the results to Tn.- 2.4 Spectral synthesis on R.- 2.5 Spectral synthesis on Rn.- 2.6 Parseval's lemma.- 3 Vector Differentiation and Sobolev Spaces.- 3.1 Differentiation in the vector sense. The spaces Wsp.- 3.2 The space D(Rn).- 3.3 Weak differentiation.- 3.4 Action of D on Wsp. The space Ws,locp.- 3.5 Sobolev spaces.- 4 Fourier Transform of Tempered Distributions.- 4.1 The space S(Rn).- 4.2 Isomorphism of S(Rn) under the Fourier transform.- 4.3 The Fourier transform in spaces of distributions.- 5 Pseudo-differential Operators.- 5.1 Symbol of a differential operator.- 5.2 Definition of a pseudo-differential operator on D(E).- 5.3 Extension of pseudo-differential operators to Sobolev spaces.- 5.4 Calderon's symbolic pseudo-calculus.- 5.5 Elliptic regularity.- IV Hilbert Space Methods and Limit Theorems in Probability Theory.- 1 Foundations of Probability Theory.- 1.1 Introductory remarks on the mathematical representation of a physical system.- 1.2 Axiomatic definition of abstract Boolean algebras.- 1.3 Representation of a Boolean algebra.- 1.4 Probability spaces.- 1.5 Morphisms of probability spaces.- 1.6 Random variables and distributions of random variables.- 1.7 Mathematical expectation and distributions.- 1.8 Various notions of convergence in probability theory.- 2 Conditional Expectation.- 2.0 Phenomenological meaning.- 2.1 Conditional expectation as a projection operator on L2.- 2.2 Conditional expectation and positivity.- 2.3 Extension of conditional expectation to L1.- 2.4 Calculating EBwhen B is a finite ?-algebra.- 2.5 Approximation by finite ?-algebras.- 2.6 Conditional expectation and Lp spaces.- 3 Independence and Orthogonality.- 3.0 Independence of two sub-?-algebras.- 3.1 Independence of random variables and of ?-algebras.- 3.2 Expectation of a product of independent r. v..- 3.3 Conditional expectation and independence.- 3.4 Independence and distributions (case of two random variables).- 3.5 A function space on the ?-algebra generated by two ?-algebras.- 3.6 Independence and distributions (case of n random variables).- 4 Characteristic Functions and Theorems on Convergence in Distribution.- 4.1 The characteristic function of a random variable.- 4.2 Characteristic function of a sum of independent r. v..- 4.3 Laplace's theorem and Gaussian distributions.- 5 Theorems on Convergence of Martingales.- 5.1 Martingales.- 5.2 Energy equality.- 5.3 Theory of L2 martingales.- 5.4 Stopping times and the maximal inequality.- 5.5 Convergence of regular martingales.- 5.6 L1 martingales.- 5.7 Uniformly integrable sets.- 5.8 Regularity criterion.- 6 Theory of Differentiation.- 6.0 Separability.- 6.1 Separability and approximation by finite ?-algebras.- 6.2 The Radon-Nikodym theorem.- 6.3 Duality of the Lp spaces.- 6.4 Isomorphisms of separable probability spaces.- 6.5 Conditional probabilities.- 6.6 Product of a countably infinite set of probability spaces.- V Gaussian Sobolev Spaces and Stochastic Calculus of Variations.- 1 Gaussian Probability Spaces.- 1.1 Definition (Gaussian random variables).- 1.2 Definition (Gaussian spaces).- 1.3 Hermite polynomials.- 1.4 Hermite series expansion.- 1.5 The Ornstein-Uhlenbeck operator on R.- 1.6 Canonical basis for the L2 space of a Gaussian probability space.- 1.7 Isomorphism theorem.- 1.8 The Cameron-Martin theorem on (RN,B?,v): quasi-invariance under the action of ?2.- 2 Gaussian Sobolev Spaces.- 2.1 Finite-dimensional spaces.- 2.2 Using Hermite series to characterize Ds2(R) in the Gaussian L2 space.- 2.3 The spaces Ds2(Rk) (k?1).- 2.4 Approximation of Lp(RN,v) by Lp(Rn, v).- 2.5 The spaces Drp(RN).- 3 Absolute Continuity of Distributions.- 3.1 The Gaussian Space on R.- 3.2 The Gaussian space on RN.- 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.- Appendix II. 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.
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