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 Borel linear subspace $L\ne {\mathbb{R}}^{\omega }$ 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 $n\le \omega$.

9 pages. arXiv admin note: text overlap with arXiv:1803.06712