Concentration of measure, classification of submeasures, and dynamics of $L_{0}$

Friedrich Martin Schneider, Sławomir Solecki

Published 2019-07-29Version 1

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 entropic version of the weighted Loomis--Whitney inequality. 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.