Integration and Probability

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.