We survey results about Haar null subsets of (not necessarily locally compact) Polish groups. The aim of this paper is to collect the fundamental properties of the various possible definitions of Haar null sets, and also to review the techniques that may enable the reader to prove results in this area. We also present several recently introduced ideas, including the notion of Haar meager sets, which are closely analogous to Haar null sets. We prove some results in a more general setting than that of the papers where they were originally proved and prove some results for Haar meager sets which were already known for Haar null sets.
Following Darji, we say that a Borel subset B of an abelian Polish group G is Haar meager if there is a compact metric space K and a continuous function f : K → G such that the preimage of the translate,The main open problem in this area is Darji's question asking whether these two notions are the same. Even though there have been several partial results suggesting a positive answer, in this paper we construct a counterexample. More specifically, we construct a G δ set in Z ω that is Haar meager but not strongly Haar meager. We also show that no Fσ counterexample exists, hence our result is optimal.
We consider a real-valued function f defined on the set of infinite branches X of a countably branching pruned tree T. The function f is said to be a limsup function if there is a function u : T → R such that f (x) = lim sup t→∞ u(x 0 , . . . , x t ) for each x ∈ X. We study a game characterization of limsup functions, as well as a novel game characterization of functions of Baire class 1.
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