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Haar Null Sets and the Consistent Reflection of Non-meagreness

  Published:2013-02-06
 Printed: Apr 2014
  • Márton Elekes,
    Rényi Alfréd Institute, Reáltanoda u. 13-15, Budapest 1053, Hungary
  • Juris Steprāns,
    Department of Mathematics, York University, Toronto, ON M3J 1P3
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Abstract

A subset $X$ of a Polish group $G$ is called 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 exist 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 Bartoszyński and Burke-Miller.
Keywords: Haar null, Christensen, non-locally compact Polish group, packing dimension, Problem FC on Fremlin's list, forcing, generic real Haar null, Christensen, non-locally compact Polish group, packing dimension, Problem FC on Fremlin's list, forcing, generic real
MSC Classifications: 28C10, 03E35, 03E17, 22C05, 28A78 show english descriptions Set functions and measures on topological groups or semigroups, Haar measures, invariant measures [See also 22Axx, 43A05]
Consistency and independence results
Cardinal characteristics of the continuum
Compact groups
Hausdorff and packing measures
28C10 - Set functions and measures on topological groups or semigroups, Haar measures, invariant measures [See also 22Axx, 43A05]
03E35 - Consistency and independence results
03E17 - Cardinal characteristics of the continuum
22C05 - Compact groups
28A78 - Hausdorff and packing measures
 

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