Metrization of the space of weakly additive positively-homogeneous functionals

The present paper is devoted to study of the space of all weakly additive, order-preserving, normalized and positively-homogeneous functionals on a metric compactum. We construct an analogue of a modified Kantorovich-Rubinstein metric on the space OH(X) of all weakly additive, order-preserving, normalized and positively-homogeneous functionals on a metric compactum X. We prove that the functor OH is metrizable. We also show that for any metric compactum X the hyperspace exp(X) equipped with the Hausdorff metric can be isometrically embedded into OH(X).