A set of positive Gaussian measure with uniformly zero density everywhere.
David Preiss, Elena Riss, Jaroslav Tišer · Journal of the European Mathematical Society · 2021
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.