The Vitali Covering Theorem in the Weihrauch Lattice

TL;DR

This paper analyzes the computational complexity of various formulations of the Vitali Covering Theorem within the Weihrauch lattice, revealing diverse degrees of non-computability and connections to Weak Weak König's Lemma.

Contribution

It provides a detailed Weihrauch degree classification of the Vitali Covering Theorem's formulations, extending previous reverse mathematics results with a uniform computational perspective.

Findings

Different versions have varying computational content.

Some formulations are computable.

Others relate to uniform Weak Weak König's Lemma.

Abstract

We study the uniform computational content of the Vitali Covering Theorem for intervals using the tool of Weihrauch reducibility. We show that a more detailed picture emerges than what a related study by Giusto, Brown, and Simpson has revealed in the setting of reverse mathematics. In particular, different formulations of the Vitali Covering Theorem turn out to have different uniform computational content. These versions are either computable or closely related to uniform variants of Weak Weak K\H{o}nig's Lemma.