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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 in a separable Hilbert space for which the Density Theorem fails uniformly, i.e. there is a set of positive -measure such that
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David Preiss, Elena Riss, Jaroslav Tišer, A set of positive Gaussian measure with uniformly zero density everywhere.. J. Eur. Math. Soc. 23 (2021), no. 7, pp. 2439–2466