The descriptive set theory of the Lebesgue density theorem

Abstract

Given an equivalence class $[A]$ in the measure algebra of the Cantor space, let $\hatΦ([A])$ be the set of points having density 1 in $A$. Sets of the form $\hatΦ([A])$ are called $\mathcal{T}$-regular. We establish several results about $\mathcal{T}$-regular sets. Among these, we show that $\mathcal{T}$-regular sets can have any complexity within $Π^{0}_{3}$ (=$ \mathbf{F}_{σδ}$), that is for any $Π^{0}_{3}$ subset $X$ of the Cantor space there is a $\mathcal{T}$-regular set that has the same topological complexity of $X$. Nevertheless, the generic $\mathcal{T}$-regular set is $Π^{0}_{3}$-complete, meaning that the classes $[A]$ such that $\hatΦ([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 $\mathcal{T}$-regular sets with empty interior turn out to be $Π^{0}_{3}$-complete. Finally we show that the generic $[A]$ does not contain a $Δ^{0}_{2}$ set, i.e., a set which is in $\mathbf{F}_σ\cap\mathbf{G}_δ$