The Balayage onto Closed Sets with Respect to Continuous Function-Kernels

概要

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Let X be a locally compact Hausdorff space with a countable base and G be a continuous function-kernel on X such that each non-empty open set is non-negligible with respect to G. Under the assumption that G and the adjoint kernel G satisfies the continuity principle, R. Durier proved that, if G or G satisfies the domination principle, G or G does the balayaged principle and conversely ([2]). Further, I. Higuchi and M. Ito obtained the same conclusion without the assumption of the continuity principle ([3]). In this paper we shall consider the balayage onto any closed non-negligible set with respect to a continuous function-kernel G satisfying the domination principle. We shall show that, if each non-empty open set is non-negligible and the convex cone of continuous potentials is adapted, then it is possible to balayage onto any closed non-negligible set. Further, we shall show that there exists a "minimum" balayaged potential uniquely up to a negligible set.
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