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By the Choquet theorem, distributions of random closed sets can be characterized by a certain class of set functions called capacity functionals. In this paper a generalization to the multivariate case is presented, that is, it is proved that the joint distribution of finitely many random sets can be characterized by a multivariate set function being completely alternating in each component, or alternatively, by a capacity functional defined on complements of cylindrical sets. For the specialdoi:10.1016/j.ijar.2012.06.017 fatcat:lr72oty2u5fitmsh5blxtvtkti