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.

Whereas Shannon entropy is related to the growth rate of multinomial coefficients, we show that the quadratic entropy (Tsallis 2-entropy) is connected to their q-deformation; when q is a prime power, these q-multinomial coefficients count flags of finite vector spaces with prescribed length and dimensions. In particular, the q-binomial coefficients count vector subspaces of a given dimension. We obtain in this way a combinatorial explanation for the nonadditivity of the quadratic entropy, which arises from a recursive counting of flags. We show that the statistical systems whose configurations are described by flags provide a frequentist justification for the maximum entropy principle with Tsallis statistics. We then introduce a discretetime stochastic process associated to the q-binomial probability distribution, that generates at time n a vector subspace of Fqn(here Fqis the finite field of order q). The concentration of measure on certain “typical subspaces” allows us to extend the asymptotic equipartition property to this setting. The size of the typical set is quantified by the quadratic entropy. We discuss the applications to Shannon theory, particularly to source coding, when messages correspond to vector spaces.