Characteristic Functional of a Probability Measure Absolutely Continuous with Respect to a Gaussian Radon Measure
NORTH CAROLINA UNIV AT CHAPEL HILL DEPT OF STATISTICS
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Let mu and mu sub 1 be 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 v, where v is a Radon measure on E, we shall give some concrete sufficient conditions on v for mu v mu.
- Statistics and Probability