  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 group 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. To achieve the desired generalization we had to change our
perspective from systems defined on concrete probability spaces to systems
defined on abstract probability spaces, and we had to introduce novel tools to
analyze systems relative to a factor. However, the change of perspective and
the new tools lead to some simplifications in the arguments and the
presentation which we arrange in a self-contained manner in this work. As an
independent byproduct, we establish a new connection between the relative
analysis of systems in ergodic theory and the internal logic in certain Boolean
topoi.
