Measures of distortion in possibilistic approximations of consistent random sets and intervals

The available of probability or possibility measures for random set (Dempster-Shafer evidence theoretical) structures are highly desirable. Probabilistic conditions involve disjointness or specificity, while possibilistic conditions involve consonance of the underlying focal elements. Consistency results in possibilistic distributions, but not measures, and a unique approximation is available. Especially in random interval measurement situations, this condition is common. In this paper we develop some of the mathematical ideas necessary to develop a measure of the distortion introduced by this consonant approximation of a consistent random set.