| Title:
|
Cardinal characteristics of the ideal of Haar null sets (English) |
| Author:
|
Banakh, T. |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
45 |
| Issue:
|
1 |
| Year:
|
2004 |
| Pages:
|
119-137 |
| . |
| Category:
|
math |
| . |
| Summary:
|
We calculate the cardinal characteristics of the $\sigma$-ideal $\Cal H\Cal N(G)$ of Haar null subsets of a Polish non-locally compact group $G$ with invariant metric and show that $\operatorname{cov}(\Cal H\Cal N(G)) \leq \frak b\leq \max \{\frak d,\operatorname{non}(\Cal N)\}\leq \operatorname{non}(\Cal H\Cal N(G))\leq \operatorname{cof}(\Cal H\Cal N(G)) \kern -0.86pt > \kern -0.86pt \min \{\frak d,\operatorname{non}(\Cal N)\}$. If $G=\prod_{n\geq 0}G_n$ is the product of abelian locally compact groups $G_n$, then $\operatorname{add}(\Cal H\Cal N(G)) \break = \operatorname{add}(\Cal N)$, $\operatorname{cov}(\Cal H\Cal N(G))=\min\{\frak b, \operatorname{cov}(\Cal N)\}$, $\operatorname{non}(\Cal H\Cal N(G))= \max \{\frak d,\operatorname{non}(\Cal N)\}$ and \linebreak $\operatorname{cof}(\Cal H\Cal N(G))\geq \operatorname{cof}(\Cal N)$, where $\Cal N$ is the ideal of Lebesgue null subsets on the real line. Martin Axiom implies that $\operatorname{cof}(\Cal H\Cal N(G))>2^{\aleph_0}$ 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 $\Cal H\Cal N(G)$ contains all $o$-bounded subsets (equivalently, subsets with the small ball property) of $G$. (English) |
| Keyword:
|
Polish group |
| Keyword:
|
Haar null set |
| Keyword:
|
Martin Axion |
| Keyword:
|
cardinal characteristics of an ideal |
| Keyword:
|
$o$-bounded set |
| Keyword:
|
the small ball property |
| MSC:
|
03E04 |
| MSC:
|
03E15 |
| MSC:
|
03E17 |
| MSC:
|
03E35 |
| MSC:
|
03E50 |
| MSC:
|
03E75 |
| MSC:
|
22A10 |
| MSC:
|
28C10 |
| MSC:
|
54A25 |
| MSC:
|
54H11 |
| idZBL:
|
Zbl 1098.03057 |
| idMR:
|
MR2076864 |
| . |
| Date available:
|
2009-05-05T16:43:43Z |
| Last updated:
|
2012-04-30 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/119441 |
| . |
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