Characteristic functional of a probability measure absolutely continuous with respect to a Gaussian Radon measure

Abstract

Let μ and μ 1 be probability measures on a locally convex Hausdorff real topological linear space E. C. R. Baker (Lecture Notes in Mathematics No. 109, pp. 33–44, Springer-Verlag, Berlin/New York, 1979) posed the problem of characterizing the absolute continuity of μ and μ 1 by their characteristic functionals. The aim of this paper is to give an answer to this problem in the case where μ is a Gaussian Radon measure. A Fourier transform shall be defined, the inversion formula established, and then a necessary and sufficient condition given for μ 1 to be absolutely continuous with respect to μ based on the characteristic functional. As applications, for the convolution μ 1 = μ∗v , where v is a Radon measure on E, we shall give some concrete sufficient conditions on v for μ∗v ⪡ μ.