标题： 分层测度的典型性 显示英文标题

标题： Typicality for stratified measures

摘要： 欧几里得空间上的分层测度在这里定义为可测测度的凸组合。 它们可能相对于勒贝格测度是奇异的，并且推广了连续-离散混合体。 因此，一个分层测度$\rho$可以表示为$\sum_{i=1}^k q_i \rho_i$，其中$(q_1,..,q_k)$是一个概率向量，每个$\rho_i$对于某个整数$m_i$来说都是$m_i$-可测的，即 与$m_i$-豪斯多夫测度在$\mu_i$上的绝对连续性，在$m_i$-可求长集$E_i$上（例如，光滑的$m_i$-流形）。我们引入了一组强典型实现的$\rho^{\otimes n}$（无记忆信源），这些实现出现的概率很高。 典型的实现支持在有限个层的并集上 $\{E_{i_1}\times \cdots \times E_{i_n}\}$，其维度集中在平均维度$\sum_{i=1}^k q_i m_i$附近。 对于每个$n$，不同层上的豪斯多夫测度的适当和给出了参考“体积”的自然概念；典型集的体积的指数增长速率由 Csiszar 的$\rho$相对于$\mu=\sum_{i=1}^k \mu_i$的广义熵来量化。 此外，我们证明了这种广义熵满足链式法则，并且条件项与每个层中典型实现的体积增长有关。 链式法则及其渐近解释在更一般的分段连续测度框架中成立：在成对不相交集合上限制的测度的凸组合，这些集合配备了参考$\sigma$-有限测度。 最后，我们证明了当我们将该平均维度概念应用于分层测度时，它与 Rényi 的信息维数一致，但此处使用的广义熵与 Rényi 的维度熵不同。 显示英文摘要

摘要： 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.