Instytut Matematyczny Polskiej Akademii Nauk / pl / Wydawnictwa / Czasopisma IMPAN / Fundamenta Mathematicae / Wszystkie zeszyty

Streszczenie

We show that for every $2\le \xi \lt \omega _1$ there exists a Haar null set in $\mathbb {Z}^\omega$ that is the difference of two $\mathbf {\Pi }^0_\xi$ sets but is not contained in any $\mathbf {\Pi }^0_\xi$ Haar null set. In particular, there exists a Haar null set in $\mathbb {Z}^\omega$ that is the difference of two $G_\delta$ sets but is not contained in any $G_\delta$ Haar null set. This partially answers a question of M. Elekes and Z. Vidnyánszky. To prove this, we also prove a theorem which characterizes the Haar null subsets of $\mathbb {Z}^\omega$.