Choquet integration on Riesz spaces and dual comonotonicity

We give a general integral representation theorem for nonadditive functionals defined on an Archimedean Riesz space X X with unit. Additivity is replaced by a weak form of modularity, or, equivalently, dual comonotonic additivity, and integrals are Choquet integrals. Those integrals are defined through the Kakutani isometric identification of X X with a C ( K ) C\left (K\right ) space. We further show that our notion of dual comonotonicity naturally generalizes and characterizes the notions of comonotonicity found in the literature when X X is assumed to be a space of functions.