Spatial models for boolean actions in the infinite measure-preserving setup
Fabien Hoareau, François Le Maître
TL;DR
The paper develops a comprehensive framework relating boolean and spatial actions for infinite-measure-preserving group actions on standard σ-finite spaces. It proves that every infinite measure-preserving action of a locally compact Polish group can be realized as a continuous action on a locally compact Polish space with a Radon measure, via a constructive G-continuity approach and a Gelfand-space realization. It also characterizes when boolean actions admit spatial (Radon) models, generalizes Kwiatkowska–Solecki-type results to the infinite-measure setting, and shows Lévy groups act trivially on Polish spaces with locally finite measures, using Poisson point processes as a key tool. The work highlights subtle obstructions in non-locally compact settings, including examples with S_∞ and Aut(X,λ) that deny spatial realizations, thereby clarifying the landscape of spatial versus boolean realizations. Overall, the findings provide concrete criteria, explicit constructions, and robust techniques for embedding infinite-measure dynamics into well-behaved topological-measure spaces with broad implications for ergodic theory and descriptive set theory.
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.