Some analogies between Haar meager sets and Haar null sets in abelian Polish groups

Abstract

In the paper we would like to pay attention to some analogies between Haar meager sets and Haar null sets. Among others, we show that 0∈int(A−A) for each Borel non-Haar meager set A in an abelian Polish group. Moreover, we define D-measurability as a topological analog of Christensen measurability and prove that each D-measurable homomorphism is continuous. Finally, we show easy constructions of a Haar meager non-Haar null set and, conversely, a meager Haar null set which is not Haar meager in spaces of sequences. Our results refer to the papers [1,2,4].