  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.
