Haar null and non-dominating sets

S/lawomir Solecki¹

¹ Department of Mathematics Indiana University Bloomington, IN 47405, U.S.A.

Fundamenta Mathematicae, Tome 170 (2001) no. 1, pp. 197-217

Résumé

We study the $\sigma $-ideal of Haar null sets on Polish groups. It is shown that on a non-locally compact Polish group with an invariant metric this $\sigma $-ideal is closely related, in a precise sense, to the $\sigma $-ideal of non-dominating subsets of $\omega ^\omega $. Among other consequences, this result implies that the family of closed Haar null sets on a Polish group with an invariant metric is Borel in the Effros Borel structure if, and only if, the group is locally compact. This answers a question of Kechris. We also obtain results connecting Haar null sets on countable products of locally compact Polish groups with amenability of the factor groups.

Détail

DOI : 10.4064/fm170-1-11

Keywords: study sigma ideal haar null sets polish groups shown non locally compact polish group invariant metric sigma ideal closely related precise sense sigma ideal non dominating subsets omega omega among other consequences result implies family closed haar null sets polish group invariant metric borel effros borel structure only group locally compact answers question kechris obtain results connecting haar null sets countable products locally compact polish groups amenability factor groups