  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
$1\in\mathrm{Int}(A^{-1}AA^{-1}A)$.
