A functional representation of capacity monad

We show that the capacity 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 $C\left(X\right)$. We also describe this space of functionals in terms of properties of functionals. Using such a representation we obtain some results about geometric properties of the capacity monad.