Characteristic Functional of a Probability Measure Absolutely Continuous with Respect to a Gaussian Radon Measure.

摘要

Let mu and mu sub 1 probability measures on a locally convex Hausdorff real topological linear space E. C.R. Baker posed the problem of characterizing the absolute continuity of mu and mu sub 1 by their characteristic functionals. The aim of this paper is to give an answer to this problem in the case where mu is a Gaussian Radon measure. We shall define a Fourier transform, establish the inversion formula, and then give a necessary and sufficient condition for mu sub 1 to be absolutely continuous with respect to mu based on the characteristic functional. As applications, for the convolution mu sub 1 = mu sub v, where v is a Radon measure on E, we shall give some concrete sufficient conditions on v for mu sub v << micron. (Author)