Integration and Probability

It is a distinct pleasure to have the opportunity to introduce Professor Malliavin's book to the English-speaking mathematical world. In recent years there has been a noticeable retreat from the level of ab straction at which graduate-level courses in analysis were previously taught in the United States and elsewhere. In contrast to the practices used in the 1950s and 1960s, when great emphasis was placed on the most general context for integration and operator theory, we have recently witnessed an increased emphasis on detailed discussion of integration over Euclidean space and related problems in probability theory, harmonic analysis, and partial differential equations. Professor Malliavin is uniquely qualified to introduce the student to anal ysis with the proper mix of abstract theories and concrete problems. His mathematical career includes many notable contributions to harmonic anal ysis, complex analysis, and related problems in probability theory and par tial differential equations. Rather than developed as a thing-in-itself, the abstract approach serves as a context into which special models can be couched. For example, the general theory of integration is developed at an abstract level, and only then specialized to discuss the Lebesgue measure and integral on the real line. Another important area is the entire theory of probability, where we prefer to have the abstract model in mind, with no other specialization than total unit mass. Generally, we learn to work at an abstract level so that we can specialize when appropriate.

Zusammenfassung
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.

Inhalt
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.