Relations between capacities and maximum principle

There are various definitions of capacities in potential theory. These capacities are usually used to determine potential theoretic exceptional sets. The aim of this paper is to study some properties of these capacities as set functions. More precisely, let Ω be a locally compact Hausdorff space and Φ be a kernel, i. e., a lower semicontinuous function on Ω x Ω which takes values in (0, + oo]. The adjoint kernel Φ is defined by Φ(x, y) = Φ(y, x). Φ is called symmetric if Φ = Φ. A measure μ will always be a non-negative Radon measure with compact support Sμ. The ^-potential of μ is defined by