We develop a theory of inner balayage positive Radon measure ? finite energy on locally compact space X to arbitrary A ? X, thereby generalizing Cartan’s Newtonian ?n, n ? 3, suitable function kernel X. As an application the developed, we show that if is ?-compact and perfectly normal, then recent result by B. Fuglede (Anal. Math., 2016) outer quasiclosed remains valid for Borel A. give in part...