Kernels and Choquet capacities (Q2366039)

Let \(X\) be a locally compact space with countable base. A lower semicontinuous function \(K\): \(X\times X\to[0,\infty]\) is called a kernel on \(X\) and the \(K\)-potential of a nonnegative measure \(\mu\) is denoted by \(K\mu\). The author considers the following two principles for \(K\): [CDP] (continuous domination principle) If \(K\mu\), \(K\nu\) are continuous and \(K\nu\leq K\mu\) on Supp \(\nu\), then \(K\nu\leq K\mu\) on \(X\); [CPE] (continuous principle of equilibrium) For any relatively compact open set \(G\) and a compact subset \(L\) of \(G\), there exists a nonnegative measure \(\mu\) such that Supp \(\mu\subset G\), \(K\mu\leq 1\) on \(X\), \(K\mu\) is continuous and \(K\mu=1\) on a neighborhood of \(L\). The main theorem of this paper states that if \(K\) and its adjoint kernel \(\tilde K\) satisfy CPE and \(K\) satisfies CDP, then the \(K\)-capacity coincides with the \(\tilde K\)-capacity and is a Choquet capacity. Then this theorem is applied to a kernel \(K\) on a balayage space with adjoint structure with respect to \(K\).