Regular capacities on metrizable spaces
Keywords: regular capacity, $\omega$-smoothness, $\tau$-smoothness, Hausdorff metric, complete metric space, separable space
Published online: 2014-07-15
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 $\omega$-smoothness and of $\tau$-smoothness are equivalent precisely on the separable spaces, and the properties of $\omega$-smoothness and of regularity w.r.t. some (then w.r.t. any) admissible metric are equivalent precisely on the compact spaces.