GENERALIZED COMONOTONICALLY ADDITIVE OPERATORS: REPRESENTATIONS BY CHOQUET INTEGRALS

Abstract

Operators which behave (sub-, super-) additive on comonotonic functions occur quite naturally in many contexts, e.g. in decision theory, artificial intelligence, and fuzzy set theory. In the present paper we define comonotonicity for Riesz spaces with the principal projection property and obtain integral representations (in terms of Bochner integrals) for comonotonically additive operators acting on Riesz with the principal projection property and taking values in certain Riesz- or Banach spaces. As easy corollaries we obtain essential generalizations of representation theorems à la schmeidler, Proc. Am. Math. Soc. 97 (1986), 255 – 261. The existence of the necessary convergence theorems makes it possible to extend our results to set-valued operators. This is the topic of a further paper.