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

## TL;DR

This paper introduces a new measure concentration phenomenon for Lipschitz functions on product spaces, classifies diffuse submeasures into geometric types, and explores implications for topological $L_{0}$-groups.

## Contribution

It provides a novel uniform concentration bound using Herbst's argument and Shearer's lemma, along with a geometric classification of submeasures and their dynamical consequences.

## Key findings

- Established a new measure concentration bound for Lipschitz functions.
- Classified diffuse submeasures into elliptic, parabolic, and hyperbolic types.
- Proved non-elliptic submeasures exhibit covering concentration.

## 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.