Naively Haar null sets in Polish groups

Abstract

Let $(G,\cdot )$ be a Polish group. We say that a set $X\subset G$ is Haar null if there exists a universally measurable set $U\supset X$ and a Borel probability measure $\mu$ such that for every $g,h\in G$ we have $\mu (gUh)=0$. We call a set $X$ naively Haar null if there exists a Borel probability measure $\mu$ such that for every $g,h\in G$ we have $\mu (gXh)=0$. Generalizing a result of Elekes and Steprāns, which answers the first part of Problem FC from Fremlin's list, we prove that in every abelian Polish group there exists a naively Haar null set that is not Haar null.