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

Abstract

By reformulating the classical proof as a Baire Category argument, we show that Besicovitch's Theorem on the existence of subsets of finite Hausdorff measure is provable in $\mathsf{ACA}_0$, and additionally that the witnessing subset is computable from one jump of the original set. We show that the corresponding formulation of Baire Category, which we call Baire Category Theorem for Closed Sets ($\mathsf{BCTC}$), is equivalent to $\mathsf{ACA}_0$, contrasting with previous results on the reverse math strength of Baire Category variants. We also examine the implications of $\mathsf{BCTC}$ for a class of 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.