Measurability of Semimartingale Characteristics with Respect to the Probability Law

Given a c`adl`ag 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 ${\mathfrak{P}}_{sem}$ be the set of all probability measures $P$ under which $X$ is a semimartingale. We construct processes $(B^P,C,{\nu }^P)$ 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 $P\in {\mathfrak{P}}_{sem}$. 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.