  Furstenberg-Zimmer structure theory refers to the extension of the dichotomy
between the compact and weakly mixing parts of a measure preserving dynamical
system and the algebraic and geometric descriptions of such parts to a
conditional setting, where such dichotomy is established relative to a factor
and conditional analogues of those algebraic and geometric descriptions are
sought. Although the unconditional dichotomy and the characterizations are
known for arbitrary systems, the relative situation is understood under certain
countability and separability hypotheses on the underlying groups and spaces.
The aim of this article is to remove these restrictions in the relative
situation and establish a Furstenberg-Zimmer structure theory in full
generality. As an independent byproduct, we establish a connection between the
relative analysis of systems in ergodic theory and the internal logic in
certain Boolean topoi.
