Representing Functionals as Integrals

Abstract

We have seen already how useful a representation theorem (Theorem 11.2) can be applied within the theory (for proving Corollary 11.3, Exercise 11.5 d) and Fubinis Theorem). As in Theorem 11.2 the crucial properties of a functional to be representable as an integral are mono-tonicity and comonotonic additivity (or, as in Greco 1982, a somewhat weaker condition). In Theorem 11.2 the domain of the functional is rather large. In decision situations one often has only restricted information, i.e. the domain of the functional is small. Representation theorems with minimal requirements on the domain are treated here. They are closely related to the extension theorems for set functions of Chapter 2. A further important question (e.g. in decision theory) is under what conditions the representing set function is sub- or supermodular and continuous from below. A corollary of the respective Representation Theorem is the classical Daniell-Stone Representation Theorem, where the representing set function is a measure.