Representing measures in potential theory and an ideal boundary

Abstract

A primary motivating application of the author’s work on measure theory was a nonstandard construction of standard representing measures for positive harmonic functions. That work yielded new standard weak convergence methods for constructing such measures on spaces of extreme harmonic functions in very general settings. The search for a Martin-type ideal boundary for the placement of those measures resulted in a new almost everywhere regular boundary that supported the representing measures for a large proper subclass of all nonnegative harmonic functions. In this note, we outline the construction of the rich measure spaces that are now called Loeb measure spaces in the literature. We then review the application of these measure spaces to the construction of representing measures. We finish with the problem of constructing an appropriate boundary associated with the nonstandard construction of general representing measures that supports all of those measures.