On Choquet theorem for random upper semicontinuous functions

Motivated by the problem of modeling of coarse data in statistics, we investigate in this paper topologies of the space of upper semicontinuous functions with values in the unit interval [0, 1] to define rigorously the concept of random fuzzy closed sets. Unlike the case of the extended real line by Salinetti and Wets, we obtain a topological embedding of random fuzzy closed sets into the space of closed sets of a product space, from which Choquet theorem can be investigated rigorously. Precise topological and metric considerations as well as results of probability measures and special properties of capacity functionals for random u.s.c. functions are also given.