The Dirichlet problem at the Martin boundary of a fine domain

We adapt the Perron-Wiener-Brelot method of solving the Dirichlet problem at the Martin boundary of a Euclidean domain so as to cover also the Dirichlet problem at the Martin boundary of a fine domain U in Rn (n≥2) (i.e., a set U which is open and connected in the H. Cartan fine topology on Rn, the coarsest topology in which all superharmonic functions are continuous). It is a complication that there is no Harnack convergence theorem for so-called finely harmonic functions. We define resolutivity of a numerical function on the Martin boundary Δ(U) of U. Our main result Theorem 4.14 implies the corresponding known result for the classical case. We also obtain analogous results for the case where the upper and lower PWB-classes are defined in terms of the minimal-fine topology on the Riesz-Martin space U‾=U∪Δ(U) instead of the natural topology. The two corresponding concepts of resolutivity are compatible.