  We study a generalized notion of a homogeneous skew-product extension of a
probability-preserving system in which the homogeneous space fibres are allowed
to vary over the ergodic decomposition of the base. The construction of such
extensions rests on a simple notion of `direct integral' for a `measurable
family' of homogeneous spaces, which has a number of precedents in older
literature. The main contribution of the present paper is the systematic
development of a formalism for handling such extensions, including non-ergodic
versions of the results of Mackey describing ergodic components of such
extensions, of the Furstenberg-Zimmer Structure Theory and of results of
Mentzen describing the structure of automorphisms of such extensions when they
are relatively ergodic. We then offer applications to two structural results
for actions of several commuting transformations: firstly to describing the
possible joint distributions of three isotropy factors corresponding to three
commuting transformations; and secondly to describing the characteristic
factors for a system of double nonconventional ergodic averages. Although both
applications are modest in themselves, we hope that they point towards a
broader usefulness of this formalism in ergodic theory.
