Triangular transformations of measures

Abstract

We obtain a new identity for the entropy of a nonlinear image of a measure on R n , which yields the wellknown inequality of Talagrand.We study triangular mappings on R n and on R ∞ , i.e., the mappings T such that the ith coordinate function T i depends only on the variables x 1 , . . ., x i .With the help of such mappings we give a positive solution to the well-known open problem on the representability of every probability measure ν that is absolutely continuous with respect to a Gaussian measure γ on an infinite dimensional space as the image of γ under a mapping of the form T (x) = x + F (x), where F takes on values in the Cameron-Martin space of the measure γ.As an application we also prove a generalized logarithmic Sobolev inequality.