An uncountable Furstenberg--Zimmer structure theory

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.