Typicality for Stratified Measures

We define stratified measures on Euclidean space as convex combinations of rectifiable measures. They are possibly singular with respect to the Lebesgue measure and generalize discrete-continuous mixtures. A stratified measure
$\rho $
can thus be represented as
$\sum _{i=1}^{k} q_{i} \rho _{i}$
, where
$(q_{1},..,q_{k})$
is a probability vector and each
$\rho _{i}$
is absolutely continuous with respect to the
$m_{i}$
-Hausdorff measure
$\mu _{i}$
on a
$m_{i}$
-rectifiable set
$E_{i}$
(e.g., a smooth
$m_{i}$
-manifold if
$m_{i}>0$
or a countable set if
$m_{i}=0$
). We introduce a set of strongly typical realizations of
$\rho ^{\otimes n}$
that occur with high probability; they are supported on a finite union of strata
$E_{i_{1}}\times \cdots \times E_{i_{n}}$
whose dimensions “per factor” concentrate around the mean dimension
$\sum _{i=1}^{k} q_{i} m_{i}$
. For each
$n$
, an appropriate sum of Hausdorff measures on the different strata gives a notion of reference “volume”; the exponential growth rate of the typical set’s volume is quantified by Csiszar’s generalized entropy of
$\rho $
with respect to
$\mu =\sum _{i=1}^{k} \mu _{i}$
. This entropy satisfies a chain rule; the conditional term is related to the volume growth of the typical realizations in each stratum. Under suitable hypotheses, our notion of mean dimension coincides with Rényi’s information dimension when applied to stratified measures, but the generalized entropy used here differs from Rényi’s dimensional entropy.