The Reverse Mathematics of Analytic Measurability

TL;DR

This paper analyzes the reverse mathematical strength of Lusin's theorem on analytic sets' measurability, linking it to subsystems of second-order arithmetic and employing class forcing techniques.

Contribution

It establishes the equivalence of analytic sets being Lebesgue-regular with $ ext{Sigma}_1^1$-$ ext{IND}$ over $ ext{ATR}_0$, and the full theorem with $ ext{Pi}_1^1$-$ ext{CA}_0$, using novel forcing methods.

Findings

Analytic sets are Lebesgue-regular iff $ ext{Sigma}_1^1$-$ ext{IND}$ over $ ext{ATR}_0$.

Full Lusin's theorem equivalent to $ ext{Pi}_1^1$-$ ext{CA}_0$ over $ ext{ATR}_0$.

Employs class forcing over models of weak set theories.

Abstract

A classical theorem of Lusin states that all analytic sets are Lebesgue-measurable. In this article we established the reverse mathematical strength of Lusin's theorem, which depends on how precisely it is formalized. By doing so, we answer to a question of Simpson. Our main proof is motivated towards proving a specific version of that result, namely that analytic sets are Lesbesgue-regular, which requires the equality of the outer and inner measures of the set in question. We prove this statement to be equivalent to $\Sigma^{1}_{1}$-$\mathrm{IND}$ over $\mathrm{ATR}_{0}$. The full statement of the theorem, that is the one implying the existence of the measure as a real number, is equivalent to $\Pi^{1}_{1}$-$\mathrm{CA}_{0}$, again provably over $\mathrm{ATR}_{0}$. In our main proof, we draw inspiration from Solovay's construction of a model of Zermelo-Fraenkel set theory where every set is Lebesgue measurable. In our case the argument requires the use of class forcing over a family of standard and non-standard models of a very weak set theory obtained through the method of pseudohierarchies.