The (sub//super)additivity assertion of Choquet

Heinz König¹

¹ Fakultät für Mathematik und Informatik Universität des Saarlandes D-66041 Saarbrücken, Germany

Studia Mathematica, Tome 157 (2003) no. 2, pp. 171-197

Résumé

The assertion in question comes from the short final section in Theory of capacities of Choquet (1953/54), in connection with his prototype of the subsequent Choquet integral. The problem was whether and when this operation is additive. Choquet had the much more abstract idea that all functionals in a certain wide class must be subadditive, and similarly for superadditivity. His treatment of this point was more like an outline, and his proof limited to a rather narrow special case. Thus the proper context and scope of the assertion has remained open. In this paper we present a counterexample which shows that the initial context has to be modified, and then in a new context we prove a comprehensive theorem which fulfils all the needs that have turned up so far.

Détail

DOI : 10.4064/sm157-2-4

Keywords: assertion question comes short final section theory capacities choquet connection his prototype subsequent choquet integral problem whether operation additive choquet had much abstract idea functionals certain wide class subadditive similarly superadditivity his treatment point outline his proof limited rather narrow special proper context scope assertion has remained paper present counterexample which shows initial context has modified context prove comprehensive theorem which fulfils needs have turned far