Applications of nonstandard analysis to ideal boundaries in potential theory

A solution is given of the generalized Dirichlet problem for an arbitrary compactification of a Brelot harmonic space. A method of obtaining the Martin-Choquet integral representation of positive harmonic functions is given, and the existence is established of an ideal boundary Δ supporting the maximal representing measures for positive bounded and quasibounded harmonic functions with almost all points of Δ being regular for the Dirichlet problem. This work was supported by a grant from the U. S. National Science Foundation. The results in Sections 1–5 were presented at the 1974 Oberwolfach Conferences on Potential Theory and Nonstandard Analysis; Sections 1–6 were discussed at the Abraham Robinson Memorial Conference, Yale, University, May 1975.