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

Emma Gruner, Jan Reimann

TL;DR

This work analyzes the reverse mathematical strength of Besicovitch's theorem in Cantor space by recasting its proof as a Baire category argument on the space of closed sets. It introduces the Baire Category Theorem for Closed Sets (BCTC) and shows its strength is exactly $ACA_0$ for standard closed sets, while related representations (separably closed, pruned closed) admit weaker provability (RCA$_0$). The authors further connect measure-regularity properties of $s$-dimensional Hausdorff measures to monotone-minimum principles, proving DMMin and SMMin are equivalent to $ACA_0$ and using these to derive regularity results. The Besicovitch-Davies theorem is established in $ACA_0$ via BCTC, with a computability bound: the witnessing subset is computable from one jump of the input’s tree code, and the results extend to density arguments and computability considerations across representations. Collectively, the paper clarifies the logical strength of measure-regularity in the Cantor setting and highlights precise buy-in points for ACA$_0$ and related subsystems in geometric measure-theory reverse mathematics.

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.

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

TL;DR

This work analyzes the reverse mathematical strength of Besicovitch's theorem in Cantor space by recasting its proof as a Baire category argument on the space of closed sets. It introduces the Baire Category Theorem for Closed Sets (BCTC) and shows its strength is exactly for standard closed sets, while related representations (separably closed, pruned closed) admit weaker provability (RCA). The authors further connect measure-regularity properties of -dimensional Hausdorff measures to monotone-minimum principles, proving DMMin and SMMin are equivalent to and using these to derive regularity results. The Besicovitch-Davies theorem is established in via BCTC, with a computability bound: the witnessing subset is computable from one jump of the input’s tree code, and the results extend to density arguments and computability considerations across representations. Collectively, the paper clarifies the logical strength of measure-regularity in the Cantor setting and highlights precise buy-in points for ACA and related subsystems in geometric measure-theory reverse mathematics.

Abstract

By reformulating the classical proof as a Baire Category argument, we show that Besicovitch's Theorem in Cantor space is provable in , 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 , 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.
Paper Structure (11 sections, 49 theorems, 94 equations)

This paper contains 11 sections, 49 theorems, 94 equations.

Table of Contents

  1. Introduction
  2. Background and Preliminaries
  3. Subsystems of Second Order Arithmetic
  4. Closed Sets and their Representations
  5. Hausdorff Measure and its Representations
  6. The Baire Category Theorem
  7. Baire Category Theorem Formulations
  8. Monotone Functions on Closed Sets
  9. Measure Regularity Results
  10. Regularity of $H^s_n$ measure
  11. The Besicovitch Theorem

Key Result

Theorem 1.1

In Cantor space $2^{\omega}$, Besicovitch's Theorem is provable in $\text{ACA}_0$.

Theorems & Definitions (99)

  • Theorem 1.1
  • Theorem 1.2
  • Proposition 1.3
  • Proposition 1.4
  • Proposition 1.5
  • Proposition 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Proposition 2.1
  • Proposition 2.2
  • ...and 89 more