Search results
Search: MSC category 28C10
( Set functions and measures on topological groups or semigroups, Haar measures, invariant measures [See also 22Axx, 43A05] )
1. CJM 2013 (vol 66 pp. 303)
 Elekes, Márton; Steprāns, Juris

Haar Null Sets and the Consistent Reflection of Nonmeagreness
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 BartoszyÅski and BurkeMiller.
Keywords:Haar null, Christensen, nonlocally compact Polish group, packing dimension, Problem FC on Fremlin's list, forcing, generic real Categories:28C10, 03E35, 03E17, , , , , 22C05, 28A78 

2. CJM 2009 (vol 61 pp. 124)
 Dijkstra, Jan J.; Mill, Jan van

Characterizing Complete Erd\H os Space
The space now known as {\em complete Erd\H os
space\/} $\cerdos$ was introduced by Paul Erd\H os in 1940 as the
closed subspace of the Hilbert space $\ell^2$ consisting of all
vectors such that every coordinate is in the convergent sequence
$\{0\}\cup\{1/n:n\in\N\}$. In a solution to a problem posed by Lex G.
Oversteegen we present simple and useful topological
characterizations of $\cerdos$.
As an application we determine the class
of factors of $\cerdos$. In another application we determine
precisely which of the spaces that can be constructed in the Banach
spaces $\ell^p$ according to the `Erd\H os method' are homeomorphic
to $\cerdos$. A novel application states that if $I$ is a
Polishable $F_\sigma$ideal on $\omega$, then $I$ with the Polish
topology is homeomorphic to either $\Z$, the Cantor set $2^\omega$,
$\Z\times2^\omega$, or $\cerdos$. This last result answers a
question that was asked
by Stevo Todor{\v{c}}evi{\'c}.
Keywords:Complete Erd\H os space, Lelek fan, almost zerodimensional, nowhere zerodimensional, Polishable ideals, submeasures on $\omega$, $\R$trees, linefree groups in Banach spaces Categories:28C10, 46B20, 54F65 
