Localization in fine potential theory and uniform approximation by subharmonic functions

Abstract

It is shown how the cone l (U) of superharmonic functions ⩾0 on an open set U in R n , n ⩾ 3, can be recovered from the cone l of superharmonic functions ⩾0 on the whole of R n 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 R n . 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 R n .