Typicality for stratified measures

Abstract

Stratified measures on Euclidean space are defined here as convex combinations of rectifiable measures. They are possibly singular with respect to the Lebesgue measure and generalize continuous-discrete 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 $m_i$-rectifiable for some integer $m_i$ i.e. 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). We introduce a set of strongly typical realizations of $\rho^{\otimes n}$ (memoryless source) that occur with high probability. The typical realizations are supported on a finite union of strata $\{E_{i_1}\times \cdots \times E_{i_n}\}$ whose dimension concentrates 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 natural 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 $\mu=\sum_{i=1}^k \mu_i$. Moreover, we prove that this generalized entropy satisfies a chain rule and that the conditional term is related to the volume growth of the typical realizations in each stratum. The chain rule and its asymptotic interpretation hold in the more general framework of piecewise continuous measures: convex combinations of measures restricted to pairwise disjoint sets equipped with reference $\sigma$-finite measures. Finally, we establish that our notion of mean dimension coincides with R\'enyi's information dimension when applied to stratified measures, but the generalized entropy used here differs from R\'enyi's dimensional entropy.