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 ${\mathbb{R}}^n$ ($n\ge 2$) (i.e., a set U which is open and connected in the H. Cartan fine topology on ${\mathbb{R}}^n$, 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 $\mathrm{\Delta }(U)$ 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 $\stackrel{￣}{U}=U\cup \mathrm{\Delta }(U)$ instead of the natural topology. The two corresponding concepts of resolutivity are compatible.