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

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 γ on a separable Hilbert space H for which the Density Theorem fails uniformly, i.e., there is a set M ⊂ H of positive γ-measure such that lim rց0 sup x∈H γ(B(x, r) ∩M) γB(x, r) = 0.