On the upper semicontinuity of Choquet capacities
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(infinity) of E with the "north pole" added to the projection. This leads to a random compact set (X) over bar 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) over bar of (X) over bar is u.s.c. on the space of all closed subsets of E(infinity) Further, consequences of the upper semicontinuity of (T) over bar are explored, and a proof of the equivalence between the upper semicontinuity of T and continuity from above on F(E) is provided. (C) 2009 Elsevier Inc. All rights reserved.
upper semicontinuity choquet capacity upper probability hit-or-miss topology stereographic projection continuity from above induced random closed set hausdorff metric