Concentration of measure, classification of submeasures, and dynamics of L0

Exhibiting a new type of measure concentration, we prove uniform concentration bounds for measurable Lipschitz functions on product spaces, where is taken with respect to the metric induced by weighted covering index set product. Our proof combines Herbst argument an analogue Shearer's lemma differential entropy. We give quantitative “geometric” classification diffuse submeasures into elliptic, parabolic, and hyperbolic. that any non-elliptic submeasure (for example, measure, or pathological submeasure) has property call concentration. results have strong consequences dynamics corresponding topological L0-groups.