Haar null sets and the consistent reflection of non-meagreness

Details Haar null sets and the consistent reflection of non-meagreness Haar null sets and the consistent reflection of non-meagreness

2011-09-28

arxiv.org/abs/1109.6164v1

Mathematics

Classical Analysis and ODEs

Scientific paper

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 Lebesgue null set $X \su \RR$ such that for every Borel probability measure $\mu$ there exists $t \in \RR$ such that $\mu(X + t) = 0$. 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 $C \su G$ such that for every non-meagre set $X \su G$ 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.

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