A functional representation of the capacity multiplication monad

Анотація

Functional representations of the capacity monad based on the max and min operations were considered in [10] and [7]. Nykyforchyn considered in [8] some alternative monad structure for the possibility capacity functor based on the max and usual multiplication operations. We show that such a capacity monad (which we call the capacity multiplication monad) has a functional representation, i.e. the space of capacities on a compactum $X$ can be naturally embedded (with preservation of the monad structure) in some space of functionals on $C\left(X,I\right)$. We also describe this space of functionals  in terms of properties of functionals.