A Baire Category Approach to Besicovitch's Theorem and Measure Regularity

TL;DR

This paper reformulates Besicovitch's Theorem using Baire Category in Cantor space, proving its provability in $ACA_0$ and analyzing the reverse math strength of measure regularity.

Contribution

It introduces the Baire Category Theorem for Closed Sets (BCTC), showing its equivalence to $ACA_0$ and exploring its implications for measure regularity and representations.

Findings

Besicovitch's Theorem is provable in $ACA_0$ using Baire Category arguments.

The witnessing subset is computable from one jump of the original set.

BCTC is equivalent to $ACA_0$, contrasting with other Baire Category variants.

Abstract

By reformulating the classical proof as a Baire Category argument, we show that Besicovitch's Theorem in Cantor space is provable in $ACA_0$, and additionally that the witnessing subset is computable from one jump of the original set. We show that the necessary formulation of Baire Category, which we call Baire Category Theorem for Closed Sets (BCTC), is equivalent to $ACA_0$, contrasting with previous results on the reverse math strength of Baire Category variants. We also examine the implications of BCTC for more general monotone functions on closed sets, and explore how changing the representation of a closed set affects the reverse math strength of its measure regularity properties.