The descriptive set theory of the Lebesgue density theorem

TL;DR

This paper explores the complexity of sets with Lebesgue density properties in the measure algebra of the Cantor space, revealing their topological complexity and generic characteristics within descriptive set theory.

Contribution

It establishes the possible complexities of a-regular sets, shows the generic a-regular set is a-3-complete, and analyzes their topological and forcing properties.

Findings

a-regular sets can have any a-3 complexity within the Cantor space.

The generic a-regular set is a-3-complete and comeagre in the measure algebra.

a-regular sets with empty interior are a-3-complete.

Abstract

Given an equivalence class $[A]$ in the measure algebra of the Cantor space, let $\hat\Phi([A])$ be the set of points having density 1 in $A$. Sets of the form $\hat\Phi([A])$ 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 $\Pi^{0}_{3}$ (=$ \mathbf{F}_{\sigma\delta}$), that is for any $\Pi^{0}_{3}$ 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 $\Pi^{0}_{3}$-complete, meaning that the classes $[A]$ such that $\hat{\Phi}([A]) $ is $\Pi^{0}_{3}$-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 $\Pi^{0}_{3}$-complete. Finally we show that the generic $[A]$ does not contain a $\Delta^{0}_{2}$ set, i.e., a set which is in $\mathbf{F}_\sigma\cap\mathbf{G}_\delta$