Localization in fine potential theory and uniform approximation by subharmonic functions

AbstractIt is shown how the cone l(U) of superharmonic functions ⩾0 on an open set U in Rn, n ⩾ 3, can be recovered from the cone l of superharmonic functions ⩾0 on the whole of Rn by a process involving the operator of localization associated with U. Actually we treat the more general case where U is open in the Cartan-Brelot fine topology on Rn. As an application we obtain a new proof of a theorem of J. Bliedtner and W. Hansen on uniform approximation by continuous subharmonic functions in open sets containing a given compact set K in Rn