Author | Malliavin, Paul. author |
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Title | Integration and Probability [electronic resource] / by Paul Malliavin |
Imprint | New York, NY : Springer New York, 1995 |
Connect to | http://dx.doi.org/10.1007/978-1-4612-4202-4 |
Descript | XXII, 326 p. online resource |
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 Mยต ((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 Lusinโs 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 Bochnerโs 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