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 will show that 0 ∈ int (A− A) for each Borel set A, which is not Haar meager in an abelian Polish group. Moreover, we will give an example of a Borel non-Haar meager set A ⊂ c 0 such that int (A + A) = ∅. Finally, we will define D-measurability as a topological analog of Christensen measurability, and apply our generalization of Piccard's theorem to prove that each D-measurable homomorphism is continuous. Our results refer to the papers [1], [2] and [4].

Haar meager sets revisited

“…Theorem 2]; see also [5]) and Haar meager (see [10]), and, as expected, the reason is the same in both cases: If A is a Borel subset of an abelian Polish group which is non-Haar null or non-Haar meager, then A − A contains an open neighborhood of 0. (To complete the argument, it suffices to observe that if A is compact, so is A − A.…”

Haar meager sets, their hulls, and relationship to compact sets

“…These results clearly correlate to the analogous well known results concerning Haar null sets proved by Christensen in [3]. Other similarities of the σ-ideals of Haar meager sets and of Haar null sets were investigated in [4] and [2]. We should note that it is not known whether every (naively) Haar meager set is (naively) strongly Haar meager.…”

“…Then we show if G is either the symmetric group S ∞ or any non-locally compact Polish group with a translation invariant metric then all compact subsets of G satisfy this sufficient condition, and thus they are strongly Haar meager. This improves the result by Jab lońska who proved that every compact subset of a non-locally compact abelian Polish group is Haar meager [4].…”

Beyond Lebesgue and Baire IV: Density topologies and a converse Steinhaus–Weil theorem

“…Analogously to the left-Haar-null sets, define in G the family of left-Haarmeagre sets, HM(G), to comprise the sets M coverable by a universally Baire set B for which there are a compact Hausdorff space K and a continuous f : K → G with f −1 (gB) meagre in K for all g ∈ G. These were introduced, in the abelian Polish group setting with K metrizable, by Darji [Dar], cf. [Jab1], and shown there to form a σ-ideal of meagre sets (co-extensive with the meagre sets for G locally compact); as HM(G)⊆ B 0 (G), the family is not studied here. 9.…”