On the characterization of certain similarly ordered super-additive functionals

Functionals which behave (sub-, super-) additively on similarly ordered functions occur quite naturally in many contexts. In the present paper we characterize (super-) additive functionals which are defined on a family of functions with the Stone-property in terms of their naturally adjoined dyadic martingales. As corollaries we obtain essential generalizations of integral representations as derived by Schmeidler (1986) and discussed in a recent monograph of Denneberg (1994).