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Haar Null Sets and the Consistent Reflection of Non-meagreness

Published online by Cambridge University Press:  20 November 2018

Márton Elekes  and
Juris Steprāns
Márton Elekes
Affiliation:
Rényi Alfréd Institute, Reáltanoda u. 13-15, Budapest 1053, Hungary Institute of Mathematics, Eötvös Loránd University, Pázmány Péter s. 1/c, Budapest 1117, Hungary e-mail: emarci@renyi.hu
Juris Steprāns
Affiliation:
Department of Mathematics, York University, TorontoON M3J 1P3. e-mail: steprans@yorku.ca
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Abstract

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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.

Keywords

28C1003E3503E1722C0528A78Haar nullChristensennon-locally compact Polish grouppacking dimensionProblem FC on Fremlin's listforcinggeneric real
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 66 , Issue 2 , 01 April 2014 , pp. 303 - 322
Copyright
Copyright © Canadian Mathematical Society 2014

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