On amenability and groups of measurable maps

TL;DR

This paper investigates the relationship between amenability of topological groups and the properties of the group of measurable maps into them, establishing equivalences and new properties for these groups.

Contribution

It proves that the group of measurable maps inherits amenability properties and characterizes amenability of the original group via the measurable map group.

Findings

If G is amenable, then L^0(G) is whirly and extremely amenable.

L^0(G) being amenable implies G is amenable.

L^0(G) inherits amenability properties from G.

Abstract

We show that if $G$ is an amenable topological group, then the topological group $L^{0}(G)$ of strongly measurable maps from $([0,1],\lambda)$ into $G$ endowed with the topology of convergence in measure is whirly amenable, hence extremely amenable. Conversely, we prove that a topological group $G$ is amenable if $L^{0}(G)$ is.