JournalsjemsVol. 23, No. 7pp. 2439–2466

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

  • David Preiss

    University of Warwick, Coventry, UK
  • Elena Riss

    Herzen State Pedagogical University of Russia, Saint Petersburg, Russia
  • Jaroslav Tišer

    Czech Technical University, Prague, Czechia
A set of positive Gaussian measure with uniformly zero density everywhere. cover

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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 γ\gamma in a separable Hilbert space HH for which the Density Theorem fails uniformly, i.e. there is a set MHM\subset H of positive γ\gamma-measure such that

limr0supxXγ(B(x,r)M)γB(x,r)=0.\lim_{r\searrow 0}\sup_{x\in X} \frac{\gamma(B(x,r)\cap M)}{\gamma B(x,r)}=0.

Cite this article

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

DOI 10.4171/JEMS/1058