A powerdomain of possibility measures

Abstract We provide a domain-theoretic framework for possibility theory by studying possibility measures on the lattice of opens 𝒪(X) of a topological space X. The powerspaces P[0,∞] (X) and P[0,1] (X) of all such maps extend to functors in the natural way. We may think of possibility measures as continuous valuations by replacing ‘+’ with ‘V’ in their modular law. The functors above send continuous maps to sup-maps and continuous domains to completely distributive lattices; in the latter case they are locally continuous. Finite suprema of scalar multiples of point valuations form a basis of the powerdomains above if 𝒪(X) is the Scott-topology of a continuous domain. The notions of [0,1]- and [0,∞]-modules corresponds to that of continuous cones if addition on the reals and on the module is replaced by suprema. The powerdomain P[0,∞] (D) is the free [0, ∞]-module over a continuous domain D.