On a capacitability problem raised in connection with the Gauss variational problem

Abstract

In a locally compact Hausdorff space, there are many ways to consider a set function for compact sets which is similar to the capacity in the classical sense. Starting from such a set function, we can define an inner quantity and an outer quantity. The problem of capacitability is to discuss when they coincide. A very useful tool is the general theory of capacitability which was estabilished by G. Choquet [2Γ\. In this paper we shall examine the capacitability problem in relation to the Gauss variational problem. More precisely, let Ω be a locally compact Hausdorff space and Φ(χ, γ) be a lower semicontinuous function on ΩxΩ. Throughout this paper, we shall assume that Φ takes values in Q0, + oo], A measure μ will be always a non-negative Radon measure and Sμ the support of μ. The potential of μ is defined by