Haar null sets and the consistent reflection of non-meagreness

Abstract

A subset $X$ of a Polish group $G$ is called \emph{Haar null} if there exists a Borel set $B \supset X$ and Borel probability measure $\mu$ on $G$ such that $\mu(gBh)=0$ for every $g,h \in G$. We prove that there exists a set $X \subset \mathbb{R}$ that is not Lebesgue null and a Borel probability measure $\mu$ such that $\mu(X + t) = 0$ for every $t \in \mathbb{R}$. This answers a question from David Fremlin's problem list by showing that one cannot simplify the definition of a Haar null set by leaving out the Borel set $B$. (The answer was already known assuming the Continuum Hypothesis.) This result motivates the following Baire category analogue. It is consistent with $ZFC$ that there exist an abelian Polish group $G$ and a Cantor set $C \subset G$ such that for every non-meagre set $X \subset G$ there exists a $t \in G$ such that $C \cap (X + t)$ is relatively non-meagre in $C$. This essentially generalises results of Bartoszy\'nski and Burke-Miller.