Abstract view
Haar Null Sets and the Consistent Reflection of Nonmeagreness


Published:20130206
Printed: Apr 2014
Márton Elekes,
Rényi Alfréd Institute, Reáltanoda u. 1315, 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 nonmeagre set $X \subset G$ there exists a $t
\in G$ such that $C \cap (X + t)$ is relatively nonmeagre in $C$. This
essentially generalises results of Bartoszyński and BurkeMiller.
Keywords: 
Haar null, Christensen, nonlocally compact Polish group, packing dimension, Problem FC on Fremlin's list, forcing, generic real
Haar null, Christensen, nonlocally 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
