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.

翻译：我们总结了几何测度论中关于可整流集与可整流测度的一些结果。结合分解的熵链式法则(Vigneaux, 2021)，这些结果解释了(Koliander等人，2016)首次研究的关于Hausdorff测度的可整流测度熵的某些性质。随后我们介绍了分层测度（即可整流测度的凸组合）的最新工作。此类测度推广了离散-连续混合分布，可能包含奇异连续部分。其熵服从链式法则，其中条件项为所涉及可整流测度熵的平均值。我们陈述了分层测度的渐近等分性质(AEP)，该性质表明测度集中在少数"典型维数"的层上，并将链式法则的条件项与每层典型序列的体积增长联系起来。