Regular capacities on metrizable spaces - Cherkovskyi, T.M. (Carpathian Mathematical Publications, 2014)
It is proved that for a (not necessarily compact) metric space: the metrics on the space of capacities in the sense of Zarichnyi and Prokhorov are equal; completeness of the space of capacities is equivalent to completeness of the original space. It is shown that for the capacities on metrizable spaces the properties of $ω$-smoothness and of $τ$-smoothness are equivalent precisely on the separable spaces, and the properties of $ω$-smoothness and of regularity w.r.t. some (then w.r.t. any) admissible metric are equivalent precisely on the compact spaces.