Gaussian Measures in the Sense of Bernstein: Factorization, Supports, Zero-One Law

Abstract

Let $X$ be a second countable locally compact Abelian group, and let $\mu$ be a Gaussian measure in the sense of Bernstein on $X$. Under the assumption that the connected component of zero of $X$ contains a finite number of elements of order 2, we prove that $\mu$ is a convolution of a Gaussian measure, the Haar distribution of a compact subgroup of $X$, and a signed measure supported in the subgroup of $X$ generated by elements of order 2. We describe the support of $\mu$ on an arbitrary group $X$. We prove that if the connected component of zero of $X$ has finite dimension, then the zero-one law holds for $\mu$ under the assumption that $\mu$ has no idempotent factors.