The Steinhaus property and Haar-null sets

TL;DR

This paper investigates properties of universally measurable sets in uncountable Polish groups, showing that certain measure-theoretic conditions imply the set of measures null on all translates is co-meager, with implications for Haar-null sets.

Contribution

It establishes a link between the Steinhaus property and Haar-null sets in uncountable Polish groups, providing new conditions for when sets are non-Haar-null.

Findings

If $A^{-1}A$ is meager, then the set of measures null on all translates of $A$ is co-meager.

Analytic sets not left Haar-null have interior points in their product sets.

The results connect measure-theoretic properties with topological group structures.

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