The descriptive set theory of the Lebesgue Density Theorem.

Given an equivalence class [A] in the measure algebra of the Cantor space, let View the MathML source be the set of points having density 1 in A. Sets of the form View the MathML source 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 View the MathML source (View the MathML source), that is for any View the MathML source 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 View the MathML source-complete, meaning that the classes [A] such that View the MathML source is View the MathML source-complete form a comeager 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 View the MathML source-complete. Finally we show that the generic [A] does not contain a View the MathML source set, i.e., a set which is in View the MathML source.