Metrization of the space of weakly additive order-preserving homogeneous functionals

The work is devoted to the study of the space of weakly additive, order-preserving, normalized, and homogeneous functionals on a compact metric space. For a metric compact space $X$, we propose a formula for calculating the Kantorovich–Rubinstein metric on the space of weakly additive, order-preserving, homogeneous functionals $S(X)$. Also, we prove that the superextension $\lambda(X)$ of the compact set $X$ is isometrically embedded in the space $S(X)$.