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Given an equivalence class [A] in the measure algebra of the Cantor space, letΦ([A]) be the set of points having density 1 in A. Sets of the form Φ([A]) are called T -regular. We establish several results about T -regular sets. Among these, we show that T -regular sets can have any complexity within Π 0 3 (=F σδ ), that is for any Π 0 3 subset X of the Cantor space there is a T -regular set that has the same topological complexity of X. Nevertheless, the generic T -regular set is Π 0 3 -complete, meaning that the classes [A] such thatΦ([A]) is Π 0 3 -complete form a comeagre subset of the measure algebra. We prove that this set is also dense in the sense of forcing, as T -regular sets with empty interior turn out to be Π 0 3 -complete. Finally we show that the generic [A] does not contain a ∆ 0 2 set, i.e., a set which is in F σ ∩ G δ .

Abstract. Two sets of reals are Borel equivalent if one is the Borel pre-image of the other, and a Borel-Wadge degree is a collection of pairwise Borel equivalent subsets of R. In this note we investigate the structure of Borel-Wadge degrees under the assumption of the Axiom of Determinacy.1. Introduction and statements of the results. Let X be a Polish (i.e., separable, completely metrizable) space, and let A, B ⊆ X. We say that A is Borel reducible to B, in symbols The work of Wadge and others has shown that the Axiom of Determinacy, AD from now on, imposes a rich and detailed structure on the Wadge degrees. In this paper it is shown that the Borel-Wadge degrees exhibit a similar behavior, namely: the relation ≤ B is well-founded; self-dual degrees and non-self-dual pairs alternate, with a self-dual degree at limit levels of countable cofinality and non-self-dual pairs at limit levels of uncountable co-

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