Exercises and Solutions Manual for Integration and Probability: By Paul MalliavinThis book is designed to be an introduction to analysis with the proper mix of abstract theories and concrete problems. It starts with general measure theory, treats Borel and Radon measures (with particular attention paid to Lebesgue measure) and introduces the reader to Fourier analysis in Euclidean spaces with a treatment of Sobolev spaces, distributions, and the Fourier analysis of such. 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 to the existing literature gives the reader a taste of the fact that analysis is not a collection of independent theories but can be treated as a whole. |
Contents
Measurable Spaces and Integrable Functions | 1 |
Borel Measures and Randon Measures | 17 |
Fourier Analysis | 49 |
Hilbert Space Methods and Limit Theorems in Probability Theory | 79 |
Gaussian Sobolev Spaces and Stochastic Calculus of Variations | 133 |
Common terms and phrases
Applying Borel Borel algebra bounded C₁ change of variable compute conclude consider continuous function convergence theorem converges narrowly convex convex function defined denote density Dirac measure Euclidean space Euclidean space Rd exists finite or countable follows formula Fourier transform Fourier-Plancherel transform Fubini function f g₁ Gaussian gives Hence Hilbert transform implies independent random variables integral L¹(R L²(R large numbers law of large Lebesgue measure lemma Let f let ƒ limn linear mapping martingale METHOD monotone convergence monotone convergence theorem nonnegative norm o-algebra open set polynomials positive measure probability measure probability space Problem III-1 Problem IV-11 Prove Radon measures real numbers real random variables REMARK result Schwarz's inequality sequence of independent SOLUTION subsets uniqueness unit sphere V₁ Y₁ µ(dx μ₁