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

TL;DR

This paper demonstrates how infinite measure-preserving actions of locally compact Polish groups can be realized as continuous actions on Radon measure spaces, extending existing theorems and clarifying distinctions between spatial and boolean actions.

Contribution

It extends the Point Realization Theorem to infinite measures and provides a streamlined proof that Lévý groups cannot have nontrivial continuous actions on Polish spaces with locally finite measures.

Findings

Every infinite measure-preserving action can be realized on a Radon measure space.

Extension of the Point Realization Theorem to infinite measures.

Lévý groups do not admit nontrivial continuous measure-preserving actions on Polish spaces.

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: L\'evy groups cannot admit nontrivial continuous measure-preserving actions on Polish spaces when the measure is locally finite.