A set of positive Gaussian measure with uniformly zero density everywhere. (Q2039581)

Summary: 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 \[\lim_{r\searrow 0}\sup_{x\in X} \frac{\gamma(B(x,r)\cap M)}{\gamma B(x,r)}=0. \]