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

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.

💡 Research Summary

This paper brings reverse mathematics into the realm of geometric measure theory by analysing Besicovitch’s theorem and related measure‑regularity statements within subsystems of second‑order arithmetic. The authors begin by recalling that while reverse mathematics has been extensively applied to real analysis and point‑set topology, its use in measure theory remains relatively unexplored. Besicovitch’s theorem (originally proved in Euclidean space) asserts that any closed set of infinite Hausdorff measure contains a closed subset of positive finite measure. When the setting is shifted to Cantor space (2^\omega), the theorem can be formalised using trees that code closed sets.

The central technical contribution is a reformulation of the classical proof as an application of a Baire Category argument. The authors introduce a variant called the Baire Category Theorem for Closed Sets (BCTC): given a non‑empty closed set (F) in a complete metric space and a sequence of open sets ({U_n}) each dense in (F), the intersection (\bigcap_n U_n) is also dense in (F). Crucially, the open sets are coded as unions of cylinder sets and the closed set as a tree. Under this coding, the usual proof of Baire Category in RCA(_0) does not go through, because one cannot decide computably whether a basic open set meets the tree. The authors prove that over RCA(_0), BCTC is equivalent to ACA(_0) (Theorem 1.7). The equivalence is established by showing that BCTC can be used to construct the range of any one‑to‑one function (the hallmark of ACA(_0)), and conversely that ACA(_0) suffices to prove BCTC.

Using BCTC, the authors give a new proof of Besicovitch’s theorem in Cantor space that works inside ACA(_0) (Theorem 1.1). Moreover, they show that the witnessing closed subset (E\subseteq F) can be computed from the first Turing jump of the code for (F) (Theorem 1.2). This dramatically reduces the computational strength required compared with the original argument, which seemed to need arbitrarily many jumps.

The paper then investigates how the representation of closed sets influences the reverse‑mathematical strength of various measure‑regularity statements. Three notions of closed sets are considered: (i) “standard closed” sets coded by arbitrary trees, (ii) “pruned closed” sets coded by trees that have at least one extension at each node, and (iii) “separably closed” sets. For standard closed sets, RCA(_0) already proves that for any target value (c) below the Hausdorff measure of (F) and any (\varepsilon>0) there is a closed subset (E\subseteq F) with measure in (