Fuzzy Measures on Finite Scales as Families of Possibility Measures

Abstract

We show that any capacity or fuzzy measure ranging on a qualitative scale can be viewed both as the lower bound of a set of possibility measures, and the upper bound of a set of necessity measures. An algorithm is provided to compute the minimal set of possibility measures dominating a given capacity. This algorithm relies on the representation of the capacity by means of its qualitative Moebius transform, and the use of selection functions of the corresponding focal sets. We also introduce the counterpart of a contour function, that turns out to be the union of all most specific possibility distributions dominating the capacity. Finally we show the connection between Sugeno integrals and lower possibility measures.