A random set characterization of possibility measures

Formal Representations of Uncertainty

“…However, in the infinite case, the relation between consonant random sets and possibility measures is more complex in the general case (see Miranda et al [91,92]). …”

Consonant Random Sets: Structure and Properties

“…This order provides a common background to the images of the different elements of the initial space, so there is not contradiction between them (hence the term consonant). Although there are other conditions (see for instance [17,19]), the ones we recall here are the strongest and the most interesting ones for the purposes of this paper. …”

“…Goodman proved in [10] that for any possibility measure Π on a measurable space (X, P(X)) there exists an antitone random set whose upper probability is Π. In [19], we considered the problem of the representability when we fix also the initial space. We proved that for any random set Γ inducing a possibility measure there is a C1 random set Γ 1 defined between the same spaces and with the same upper probability.…”

“…Let us define C x := {y|P * ({y}) ≥ P * ({x})} for any x ∈ X, and Γ 1 : Ω → P(X) by Γ 1 (ω) = ∪ x∈Γ (ω) C x . We check in [19,Theorem 4.7] that Γ 1 is strongly measurable, C1 and that P From the definition of Γ 1 we deduce that there are only two alternatives: either ∩ ω∈A Γ 1 (ω) = {y : P * ({y}) ≥ z A }, and then given a sequence {ω n } n of elements of A such that z n = inf x∈Γ (ωn) P * ({x}) converges to z A , it is…”

“…They have been studied for instance within stochastic geometry ( [16]), economy ( [13]), or from the measure-theoretic point of view ( [12]). Within random sets, those which are consonant constitute a subclass of particular interest, as the works in [4,10,17,19] testify. In spite of all this work, there is not a unique definition of consonant random set; on the contrary, the term 'consonancy' has been used whenever there is some relationship of nestedness between the images of the multi-valued mapping.…”