Haar-positive closed subsets of Haar-positive analytic sets

Abstract

We show that every non-Haar-null analytic subset of ${\mathbb{Z}}^{\omega }$ contains a non-Haar-null closed subset. Moreover, we also prove that the codes of Haar-null analytic subsets, and, consequently, closed Haar-null sets in the Effros Borel space of ${\mathbb{Z}}^{\omega }$ form a ${\mathbf{\Delta }}_2^1$ set.