Symmetric function kernels and sweeping of measures

This is a potential theoretic study of balayage (sweeping) of a positive Radon measure ω on a locally compact (Hausdorff) space X onto a closed, or, more generally, a quasiclosed set A ? X (that is, a set which can be approximated in outer capacity by closed sets). The setting is that of potentials with respect to a suitable symmetric function kernel G: X × X → 0,+∞]. We consider energy capacity, not as a set function, but as a functional, acting on positive numerical functions on X. The finiteness of the upper capacity of the function 1 _A Gω is sufficient for the possibility of the sweeping in question (1 _A denoting the indicator function of A and Gω the G-potential of ω).