Singularities of bounded harmonic functions. (Q1852317)

In the context of Brelot harmonic spaces the authors show that: If \(u\) is harmonic in \(D\setminus K\) where \(K\) is a compact subset of an open set \(D\) implies its extension to \(D\) then the same result holds independently of \(D\). They also show that (assuming points are polar) the above is true without assuming existence of positive superharmonic functions.