On concavity and supermodularity

Concavity and supermodularity are in general independent properties. A class of functionals defined on a lattice cone of a Riesz space has the Choquet property when it is the case that its members are concave whenever they are supermodular. We show that for some important Riesz spaces both the class of positively homogeneous functionals and the class of translation invariant functionals have the Choquet property. We extend in this way the results of Choquet [G. Choquet, Theory of capacities, Ann. Inst. Fourier (Grenoble) 5 (1953–1954) 131–295] and König [H. König, The (sub/super) additivity assertion of Choquet, Studia Math. 157 (2003) 171–197].