Concentration of measure, classification of submeasures, and dynamics of $L_0$

Abstract

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 diffuse 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.

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Cited by in corpus (2)

• Concentration of invariant means and dynamics of chain stabilizers in continuous geometries
• Groups without unitary representations, submeasures, and the escape property