Gaussian measures on Lp spaces, 1 ≤ p < ∞

Abstract

This paper contains the following three types of results: First, a 1-1 correspondence is established between Gaussian measures on L p , 1 ≤ p < ∞, and Gaussian processes with paths in L p . Second, a 0–1 law for Gaussian measures on Fréchet spaces is proved, which is subsequently applied to obtain two other 0–1 laws. In the first, it is shown that the paths of a measurable Gaussian process belong to L p with probability 0 or 1, and in the second, it is shown that a certain random series converges uniformly on any Borel subset of the real line with probability 0 or 1. Third, a Gaussian measure on L p is characterized in terms of the characteristic function; and its topological support is obtained in terms of its mean and covariance operator.