We show that every non-Haar-null analytic subset of Z ω contains a non-Haar-null closed subset. Moreover, we also prove that the codes of Haar-null analytic subsets, and, consequently, closed Haar-null sets in the Effros Borel space of Z ω form a ∆ 1 2 set.It is not hard to see that non-locally-compact Polish groups do not admit a Haar measure (that is, an invariant σ-finite Borel measure). However, Christensen [4] (and later, independently, Hunt-Sauer-Yorke [8]) generalized the ideal of Haar measure zero sets to every Polish group as follows: Definition 0.1. Let (G, •) be a Polish group and S ⊂ G. We say that S is Haar-null, (in symbols, S ∈ HN ) if there exists a universally measurable set U ⊃ S (that is, a set measurable with respect to every Borel probability measure) and a Borel probability measure µ on G such that for every g, h ∈ G we have µ(gUh) = 0. Such a measure µ is called a witness measure for S. This notion has found wide application in diverse areas such as functional analysis, dynamical systems, group theory, geometric measure theory, and analysis (see, e.g., [12,2,14,13,5,1]). It provides a well-behaved notion of "almost every" (or "prevalent") element of a Polish group. It is natural to investigate the regularity properties of Haar-null sets. In particular, one might wonder whether "small sets are contained in nice small sets" and whether "large sets contain nice large sets". Concerning the first question, Solecki [13] has shown a positive statement, namely, that every analytic Haar-null set is contained in a Borel Haar-null set. On the negative side, the first and the third author [6] proved that, unlike the situation in locally-compact groups,