An uncountable Mackey–Zimmer theorem
Volume 266 / 2022
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.
