Simultaneous Boolean sections of measure quotients Finite compatibility, obstructions, and separable products

Given a complete probability space and a family of measurable quotient structures, we study Boolean sections of the measure quotient which respect all these structures simultaneously. Relative to a prescribed normalized lower density, we characterize exis- tence by finite assignments of atoms of Boolean algebras. The entire space of compatible sections is an inverse limit of finite solution spaces. An obstruction example shows that commutation of quotient formation with intersections of the distinguished subalgebras is not sufficient. As an application, we prove that a completed countable product of probability spaces with separable measure algebras admits a Boolean lifting preserving dependence on every coordinate subset. The proof uses finite-partition approximation and compatible finite-point couplings; neither point separation nor countable generation of the original sigma-algebras is assumed.