Locally uniform approximation by solutions of the classical Dirichlet problem (Q1803184)

Let \(U\) be an open subset of a \({\mathcal P}\)-harmonic space. It is shown that, in the topology of local uniform convergence, the space of classical solutions of the Dirichlet problem on \(U\) is dense in the space of Perron-Wiener-Brelot solutions of the Dirichlet problem if and only if the irregular boundary points of \(U\) form a set of harmonic measure 0.