The Riesz representation theorem and weak${}^{\ast }$ compactness of semimartingales

We show that the sequential closure of a family of probability measures on the canonical space of c{`a}dl{`a}g paths satisfying Stricker’s uniform tightness condition is a weak${}^{\ast }$ compact set of semimartingale measures in the pairing of the Riesz representation theorem under topological assumptions on the path space. Similar results are obtained for quasi- and supermartingales under analogous conditions. In particular, we give a full characterization of the strongest topology on the Skorokhod space for which these results are true.