A Borel linear subspace of $\mathbb R^\omega$ that cannot be covered by countably many closed Haar-meager sets

Abstract

We prove that the countable product of lines contains a Haar-null Haar-meager Borel linear subspace $L$ that cannot be covered by countably many closed Haar-meager sets. This example is applied to studying the interplay between various classes of "large" sets and Kuczma-Ger classes in the topological vector spaces ${\mathbb R}^n$ for $n\le \omega$.