## Haar null and non-dominating sets

### Volume 170 / 2001

Fundamenta Mathematicae 170 (2001), 197-217
MSC: 28C10, 03E15, 43A07.
DOI: 10.4064/fm170-1-11

#### Abstract

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.