On Continuous Capacities

Abstract

Let $(X, W)$ be a balayage space, $\gamma$ a Choquet capacity on $X$, $\beta(E)$ the essential base of $E \subset X$ and, for a compact set $K \subset X, \alpha (K) = \gamma (\beta(K))$. Then some properties of the set function $\alpha$ are investigated. In particular, it is shown when $\alpha$ is the Choquet capacity. Further, some relation a to the so-called continuous capacity deduced from a kernel on $X$ is given. At last, some open problems from the book [1] by G. Anger are solved.