On the upper semicontinuity of Choquet capacities

Abstract

The Choquet capacity T of a random closed set X on a metric space E is regarded as or related to a non-additive measure, an upper probability, a belief function, and in particular a counterpart of the distribution functions of ordinary random vectors. While the upper semicontinuity of T on the space of all closed subsets of E (hit-or-miss topology) is highly desired, T is not necessarily u.s.c. if E is not compact, e.g. E = R n . For any locally compact separable metric space E, this controversial situation can be resolved in the probabilistic context by stereographically projecting X into the Alexandroff compactification E ∞ of E with the “north pole” added to the projection. This leads to a random compact set X ¯ that is defined on the same probability space, takes values in a space homeomorphic to the space of X, and possesses an equivalent probability law. Particularly, the Choquet capacity T ¯ of X ¯ is u.s.c. on the space of all closed subsets of E ∞ . Further, consequences of the upper semicontinuity of T ¯ are explored, and a proof of the equivalence between the upper semicontinuity of T and continuity from above on F ( E ) is provided.