An uncountable Mackey-Zimmer theorem

TL;DR

This paper extends the Mackey-Zimmer theorem to uncountable settings, removing previous countability and separability assumptions, and develops a more abstract framework for measure-preserving systems with potential applications in ergodic theory.

Contribution

It generalizes the Mackey-Zimmer theorem to uncountable groups and systems by removing countability assumptions, using a new abstract framework and a canonical model for measure-preserving systems.

Findings

The theorem now applies to uncountable groups and systems.

Introduces a more abstract notion of measure-preserving systems.

Lays groundwork for uncountable Host-Kra structural theory.

Abstract

The Mackey-Zimmer theorem classifies ergodic group extensions $X$ of a measure-preserving system $Y$ by a compact group $K$, by showing that such extensions are isomorphic to a group skew-product $X \equiv Y \rtimes_\rho H$ for some closed subgroup $H$ of $K$. An analogous theorem is also available for ergodic homogeneous extensions $X$ of $Y$, namely that they are isomorphic to a homogeneous skew-product $Y \rtimes_\rho H/M$. These theorems have many uses in ergodic theory, for instance playing a key role in the Host-Kra structural theory of characteristic factors of measure-preserving systems. The existing proofs of the Mackey-Zimmer theorem require various "countability", "separability", or "metrizability" hypotheses on the group $\Gamma$ that acts on the system, the base space $Y$, and the group $K$ used to perform the extension. In this paper we generalize the Mackey-Zimmer theorem to "uncountable" settings in which these hypotheses are omitted, at the cost of making the notion of a measure-preserving system and a group extension more abstract. However, this abstraction is partially counteracted by the use of a "canonical model" for abstract measure-preserving systems developed in a companion paper. In subsequent work we will apply this theorem to also obtain uncountable versions of the Host-Kra structural theory.