A set of positive Gaussian measure with uniformly zero density everywhere.

Abstract

Existing negative results on invalidity of analogues of classical Density and Differentiation Theorems in infinite-dimensional spaces are considerably strengthened by a construction of a Gaussian measure $\gamma$ in a separable Hilbert space $H$ for which the Density Theorem fails uniformly, i.e. there is a set $M\subset H$ of positive $\gamma$-measure such that