Spatial models for boolean actions in the infinite measure-preserving setup

Abstract

We show that up to a null set, every infinite measure-preserving action of a locally compact Polish group can be turned into a continuous measure-preserving action on a locally compact Polish space where the underlying measure is Radon. We also investigate the distinction between spatial and boolean actions in the infinite measure-preserving setup. In particular, we extend Kwiatkowska and Solecki's Point Realization Theorem to the infinite measure setup. We finally obtain a streamlined proof of a recent result of Avraham-Re'em and Roy: Lévy groups cannot admit nontrivial continuous measure-preserving actions on Polish spaces when the measure is locally finite.

Numerous fixes following referee comments; Spatial Realization result extended to isometry groups of locally compact separable metric spaces. Comments welcome!