Regular capacities on metrizable spaces

Keywords:
regular capacity, -smoothness, -smoothness, Hausdorff metric, complete metric space, separable spaceAbstract
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 -smoothness and of -smoothness are equivalent precisely on the separable spaces, and the properties of -smoothness and of regularity w.r.t. some (then w.r.t. any) admissible metric are equivalent precisely on the compact spaces.