Cardinal characteristics of the ideal of Haar null sets

Banakh, Taras O.. "Cardinal characteristics of the ideal of Haar null sets." Commentationes Mathematicae Universitatis Carolinae 45.1 (2004): 119-137. <http://eudml.org/doc/249366>.

@article{Banakh2004,
abstract = {We calculate the cardinal characteristics of the $\sigma $-ideal $\mathcal \{H\}\mathcal \{N\}(G)$ of Haar null subsets of a Polish non-locally compact group $G$ with invariant metric and show that $\operatorname\{cov\}(\mathcal \{H\}\mathcal \{N\}(G)) \le \mathfrak \{b\}\le \max \lbrace \mathfrak \{d\},\operatorname\{non\}(\mathcal \{N\})\rbrace \le \operatorname\{non\}(\mathcal \{H\}\mathcal \{N\}(G))\le \operatorname\{cof\}(\mathcal \{H\}\mathcal \{N\}(G)) \hspace\{-0.86pt\}> \hspace\{-0.86pt\}\min \lbrace \mathfrak \{d\},\operatorname\{non\}(\mathcal \{N\})\rbrace $. If $G=\prod _\{n\ge 0\}G_n$ is the product of abelian locally compact groups $G_n$, then $\operatorname\{add\}(\mathcal \{H\}\mathcal \{N\}(G)) = \operatorname\{add\}(\mathcal \{N\})$, $\operatorname\{cov\}(\mathcal \{H\}\mathcal \{N\}(G))=\min \lbrace \mathfrak \{b\}, \operatorname\{cov\}(\mathcal \{N\})\rbrace $, $\operatorname\{non\}(\mathcal \{H\}\mathcal \{N\}(G))= \max \lbrace \mathfrak \{d\},\operatorname\{non\}(\mathcal \{N\})\rbrace $ and $\operatorname\{cof\}(\mathcal \{H\}\mathcal \{N\}(G))\ge \operatorname\{cof\}(\mathcal \{N\})$, where $\mathcal \{N\}$ is the ideal of Lebesgue null subsets on the real line. Martin Axiom implies that $\operatorname\{cof\}(\mathcal \{H\}\mathcal \{N\}(G))>2^\{\aleph _0\}$ and hence $G$ contains a Haar null subset that cannot be enlarged to a Borel or projective Haar null subset of $G$. This gives a negative (consistent) answer to a question of S. Solecki. To obtain these estimates we show that for a Polish non-locally compact group $G$ with invariant metric the ideal $\mathcal \{H\}\mathcal \{N\}(G)$ contains all $o$-bounded subsets (equivalently, subsets with the small ball property) of $G$.},
author = {Banakh, Taras O.},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {Polish group; Haar null set; Martin Axion; cardinal characteristics of an ideal; $o$-bounded set; the small ball property; Polish group; Haar null set; cardinal characteristics of an ideal},
language = {eng},
number = {1},
pages = {119-137},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Cardinal characteristics of the ideal of Haar null sets},
url = {http://eudml.org/doc/249366},
volume = {45},
year = {2004},
}

TY - JOUR
AU - Banakh, Taras O.
TI - Cardinal characteristics of the ideal of Haar null sets
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2004
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 45
IS - 1
SP - 119
EP - 137
AB - We calculate the cardinal characteristics of the $\sigma $-ideal $\mathcal {H}\mathcal {N}(G)$ of Haar null subsets of a Polish non-locally compact group $G$ with invariant metric and show that $\operatorname{cov}(\mathcal {H}\mathcal {N}(G)) \le \mathfrak {b}\le \max \lbrace \mathfrak {d},\operatorname{non}(\mathcal {N})\rbrace \le \operatorname{non}(\mathcal {H}\mathcal {N}(G))\le \operatorname{cof}(\mathcal {H}\mathcal {N}(G)) \hspace{-0.86pt}> \hspace{-0.86pt}\min \lbrace \mathfrak {d},\operatorname{non}(\mathcal {N})\rbrace $. If $G=\prod _{n\ge 0}G_n$ is the product of abelian locally compact groups $G_n$, then $\operatorname{add}(\mathcal {H}\mathcal {N}(G)) = \operatorname{add}(\mathcal {N})$, $\operatorname{cov}(\mathcal {H}\mathcal {N}(G))=\min \lbrace \mathfrak {b}, \operatorname{cov}(\mathcal {N})\rbrace $, $\operatorname{non}(\mathcal {H}\mathcal {N}(G))= \max \lbrace \mathfrak {d},\operatorname{non}(\mathcal {N})\rbrace $ and $\operatorname{cof}(\mathcal {H}\mathcal {N}(G))\ge \operatorname{cof}(\mathcal {N})$, where $\mathcal {N}$ is the ideal of Lebesgue null subsets on the real line. Martin Axiom implies that $\operatorname{cof}(\mathcal {H}\mathcal {N}(G))>2^{\aleph _0}$ and hence $G$ contains a Haar null subset that cannot be enlarged to a Borel or projective Haar null subset of $G$. This gives a negative (consistent) answer to a question of S. Solecki. To obtain these estimates we show that for a Polish non-locally compact group $G$ with invariant metric the ideal $\mathcal {H}\mathcal {N}(G)$ contains all $o$-bounded subsets (equivalently, subsets with the small ball property) of $G$.
LA - eng
KW - Polish group; Haar null set; Martin Axion; cardinal characteristics of an ideal; $o$-bounded set; the small ball property; Polish group; Haar null set; cardinal characteristics of an ideal
UR - http://eudml.org/doc/249366
ER -