The Steinhaus property and Haar-null sets

Abstract

It is shown that if $G$ is an uncountable Polish group and $A\subseteq G$ is a universally measurable set such that $A^{-1}A$ is meager, then the set $T_l(A)=\{\mu \in P(G):\mu (gA)=0\text{for all}g\in G\}$ is co-meager. In particular, if $A$ is analytic and not left Haar-null, then $1\in \mathrm{Int}(A^{-1}A{A}^{-1}A)$.

9 pages, no figures

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