Abstract view
Haar Null Sets and the Consistent Reflection of Non-meagreness
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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
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
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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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