Title: On the entropy of rectifiable and stratified measures Show Chinese title

Title: 关于可求长和分层测度的熵

Abstract: We summarize some results of geometric measure theory concerning rectifiable sets and measures. Combined with the entropic chain rule for disintegrations (Vigneaux, 2021), they account for some properties of the entropy of rectifiable measures with respect to the Hausdorff measure first studied by (Koliander et al., 2016). Then we present some recent work on stratified measures, which are convex combinations of rectifiable measures. These generalize discrete-continuous mixtures and may have a singular continuous part. Their entropy obeys a chain rule, whose conditional term is an average of the entropies of the rectifiable measures involved. We state an asymptotic equipartition property (AEP) for stratified measures that shows concentration on strata of a few "typical dimensions" and that links the conditional term of the chain rule to the volume growth of typical sequences in each stratum. Show Chinese abstract

Abstract: 我们总结了几何测度论中关于可测集和测度的一些结果。 结合分解的熵链法则（Vigneaux，2021），它们解释了可测测度相对于Hausdorff测度的熵的一些性质，这些性质最初由（Koliander等，2016）研究。 然后我们介绍了关于分层测度的一些最新工作，这些测度是可测测度的凸组合。 这些测度推广了离散-连续混合测度，并可能具有奇异连续部分。 它们的熵遵循一个链法则，其条件项是所涉及的可测测度熵的平均值。 我们陈述了一个分层测度的渐近等价分割性质（AEP），该性质表明在少数“典型维度”的层上出现集中现象，并将链法则的条件项与每个层中典型序列的体积增长联系起来。