Universität Des Saarlandes Fachrichtung 6.1 – Mathematik the (sub/super)additivity Assertion of Choquet the (sub/super)additivity Assertion of Choquet the (sub/super)additivity Assertion of Choquet

The assertion in question comes from the short final section in the Theory of Capacities of Choquet 1953/54, in connection with his prototype of the subsequent Choquet integral. The problem was whether and when this formation is additive. Choquet had the much more abstract idea that all functionals in a certain wide class must be subadditive, and the counterpart with superadditive. His treatment of this point was kind of an outline, and his proof limited to a rather narrow special case. Thus the adequate context and scope of the assertion remained open even up to now. In this paper we present a counterexample which shows that the initial context has to be modified, and then in new context a comprehensive theorem which fulfils all needs turned up so far. In section 48 of his famous Theory of Capacities [2] Gustave Choquet introduced a certain class of functionals with the flavour of an integral, but invented for an important issue in capacities and not at all for the sake of measure and integration. Yet the concept showed basic qualities in that other respect too: It was in the initial spirit of Lebesgue [10] to construct the integral via decomposition into horizontal strips rather than into vertical ones, which had fallen into oblivion in the course of the 20th century, and was simpler and much more comprehensive than the usual constructions. Thus in subsequent decades the concept developed into a universal one in measure and integration, called the Choquet integral. One could even wonder why the Choquet integral did not become the foundation for all of integration theory. But the fact that this did not happen had an immediate reason: The basic hardship with the Choquet integral is that it is a priori obscure whether and when it is additive, which one best even subdivides into subadditive and superadditive. To this issue Choquet contributed in his final section 54 a spectacular, because much more abstract idea: On certain lattice cones all submodular and positive-homogeneous real-valued functionals must be subadditive, and the same for super in place of sub. It is this assertion which forms the theme of the present paper (in the sequel the two cases will be united via an obvious sub/super shorthand notation). The treatment of Choquet was kind of an outline, and his proof limited to a rather narrow special case. While the Choquet integral has been explored in subsequent decades, the abstract assertion remained unsettled up to now. In recent years the present author became motivated because he needed an assertion of this kind for the further development of his extended Daniell-Stone and Riesz representation theorems [7]. In 1998 he obtained an intermediate result which 1991 Mathematics Subject Classification. 26A51, 26D15, 28A12, 28A25, 28C05, 28C15, 46G12, 52A40.