The descriptive set theory of the Lebesgue Density Theorem

Given an equivalence class [A][A] in the measure algebra of the Cantor space, let Φˆ([A]) be the set of points having density 11 in AA. Sets of the form Φˆ([A]) are called TT-regular. We establish several results about TT-regular sets. Among these, we show that TT-regular sets can have any complexity within Π30 (=Fσδ), that is for any Π30 subset XX of the Cantor space there is a TT-regular set that has the same topological complexity of XX. Nevertheless, the generic TT-regular set is Π30-complete, meaning that the classes [A][A] such that Φˆ([A]) is Π30-complete form a comeager subset of the measure algebra. We prove that this set is also dense in the sense of forcing, as TT-regular sets with empty interior turn out to be Π30-complete. Finally we show that the generic [A][A] does not contain a Δ20 set, i.e., a set which is in Fσ∩Gδ.