Construction and basic properties of Gaussian measures on Fréchet spaces

Abstract

We generalize the Gross' method of the construction of a Gaussian probability measure μ in the sense of Bernstein on infinite dimensional Banach spaces to an arbitrary separable Fréchet space M. Next we describe basic properties of the constructed measure and discuss analogies between Gaussian measures in Banach and Fréchet spaces. We investigate, among other things, the Hilbert space H of admissible shifts for the measure μ and show that the closed unit ball in H is a compact subset of M. Finally, we prove a generalized version of the Brunn-Minkowski inequality for Gaussian measures in Fréchet spaces, which is the main tool in the proof of strong limit theorems for sums of Gaussian random elements in Fréchet spaces.