Haar Null Sets and the Consistent Reflection of Non-meagreness

Elekes, Márton ; Steprāns, Juris

Canadian journal of mathematics, Tome 66 (2014) no. 2, pp. 303-322

Résumé

A subset $X$ of a Polish group $G$ is called Haar null if there exist a Borel set $B\,\supset \,X$ and Borel probability measure $\mu$ on $G$ such that $\mu \left( g\,Bh \right)\,=\,0$ for every $g,\,h\,\in \,G$ . We prove that there exist a set $X\,\subset \,\text{R}$ that is not Lebesgue null and a Borel probability measure $\mu$ such that $\mu \left( X\,+\,t \right)\,=\,0$ for every $t\,\in \,\text{R}$ . 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 $X\,\subset \,\text{G}$ there exists a $t\in \text{G}$ such that $C\,\cap \,\left( X\,+\,t \right)$ is relatively non-meagre in $C$ . This essentially generalizes results of Bartoszyński and Burke–Miller.

Détail

DOI : 10.4153/CJM-2012-058-5

Mots-clés : 28C10, 03E35, 03E17, 22C05, 28A78, Haar null, Christensen, non-locally compact Polish group, packing dimension, ProblemFC on Fremlin's list, forcing, generic real