  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.
