Exhibiting a new type of measure concentration, we prove uniform concentration bounds for measurable Lipschitz functions on product spaces, where Lipschitz is taken with respect to the metric induced by a weighted covering of the index set of the product. Our proof combines the Herbst argument with an analogue of Shearer's lemma for differential entropy. We give a quantitative ``geometric'' classification of diffused submeasures into elliptic, parabolic, and hyperbolic. We prove that any non-elliptic submeasure (for example, any measure, or any pathological submeasure) has a property that we call covering concentration. Our results have strong consequences for the dynamics of the corresponding topological $L_{0}$-groups.

中文翻译：

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度量的浓度、子度量的分类和 L0 的动态

展示了一种新型的度量浓度，我们证明了乘积空间上可测 Lipschitz 函数的统一浓度边界，其中 Lipschitz 是根据乘积的索引集的加权覆盖引起的度量取的。我们的证明将 Herbst 论证与希勒微分熵的引理的类似物相结合。我们将扩散的子度量定量地“几何”分类为椭圆形、抛物线形和双曲线形。我们证明任何非椭圆子度量（例如，任何度量或任何病理子度量）都具有我们称之为覆盖集中的特性。我们的结果对相应的拓扑 $L_{0}$-groups 的动力学有很强的影响。