The (sub//super)additivity assertion of Choquet

Volume 157 / 2003

Heinz König Studia Mathematica 157 (2003), 171-197 MSC: 26A51, 26D15, 28A12, 28A25, 28C05, 28C15, 46G12, 52A40. DOI: 10.4064/sm157-2-4


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.


  • Heinz KönigFakultät für Mathematik und Informatik
    Universität des Saarlandes
    D-66041 Saarbrücken, Germany

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