Extensions of subcopulas

In view of Sklar's Theorem the probability distribution function of every (not necessarily continuous) random vector can be uniquely decomposed in terms of the marginal distributions of its components and a suitable subcopula. The study of such latter functions is therefore of interest for understanding the dependence information of non-continuous variables. Here, we investigate some analytical properties of the class of subcopulas, including compactness (with respect to a novel metric), approximations and Baire category results. Moreover, under a suitable assumption, we describe all possible extensions from a subcopula to a copula in any dimension.