Predicates and connections to effectus theory

Develop an explicit notion of predicates within semiadditive CD categories (or more generally CMon-enriched categorical probability frameworks) and formalize its connection to effectus theory to make precise the proposed approach to existence of Lebesgue decompositions.

Background

In discussing avenues toward general existence results for Lebesgue decompositions, the authors suggest that a suitable Radon–Nikodym theorem in the enriched categorical setting will likely require an explicit notion of predicates.

They point out a potential connection to effectus theory and state that making this precise is deferred, identifying this as specific future work needed to advance the program.