一种关于贝西科维奇定理和测度正则性的贝尔纲方法
A Baire Category Approach to Besicovitch's Theorem and Measure Regularity
AI总结：
通过将经典证明重构为贝尔纲论证，证明贝西科维奇定理在\(\mathsf{ACA}_0\)中可证且见证子集可从原集的一次跳跃计算得出，还表明\(\mathsf{BCTC}\)与\(\mathsf{ACA}_0\)等价，并研究其对闭集上一类单调函数的影响。
摘要：
通过将经典证明重新表述为贝尔纲论证，我们表明关于有限豪斯多夫测度子集存在性的贝西科维奇定理在\(\mathsf{ACA}_0\)中是可证的，此外，见证子集可从原始集合的一次跳跃计算得出。我们表明相应的贝尔纲表述，我们称之为闭集的贝尔纲定理（\(\mathsf{BCTC}\)），与\(\mathsf{ACA}_0\)等价，这与之前关于贝尔纲变体的反向数学强度的结果形成对比。我们还研究了\(\mathsf{BCTC}\)对闭集上一类单调函数的影响，并探讨改变闭集的表示如何影响其测度正则性性质的反向数学强度。
英文摘要：
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.
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