The (sub/super)additivity assertion of Choquet
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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.
Keywords(sub/super)modular and (sub/super)additive functionals convex functions Choquet integral Stonean function classes Stonean and truncable functionals Daniell–Stone and Riesz representation theorems
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