Measurability of semimartingale characteristics with respect to the probability law

Given a càdlàg process X on a filtered measurable space, we construct a version of its semimartingale characteristics which is measurable with respect to the underlying probability law. More precisely, let P s e m be the set of all probability measures P under which X is a semimartingale. We construct processes ( B P , C , ν P ) which are jointly measurable in time, space, and the probability law P , and are versions of the semimartingale characteristics of X under P for each P ∈ P s e m . This result gives a general and unifying answer to measurability questions that arise in the context of quasi-sure analysis and stochastic control under the weak formulation.