Axiom of proportionality and the integral representation of potentials (Q1841548)

In the axiomatic potential theory, let \(X\) be a \(P\)-harmonic space in the sense of Constantinescu-Cornea. A function \(G(x,y)\) l.s.c. on \(X\times X\) and continuous outside the diagonal is called a Green function if \(G_y: x\to G(x,y)\) is a potential with harmonic support \(\{y\}\). If any two potentials with the same harmonic point support are proportional, the axiom of proportionality is said to be satisfied. There are many results, when \(G(x,y)\) exists, studying the connection between this axiom of proportionality and the property of representing any potential as an integral of \(G\) with respect to a suitable Radon measure. In this note, assuming that (1) \(X\) is a \(P\)-harmonic space with a countable base, (2) the constant 1 is superharmonic on \(X\), (3) \(X\) satisfies the axiom of proportionality, and (4) there is a Green function, the author shows that given any potential \(p\), there exists a unique Radon measure \(\mu\) such that \(p(x)= \int G(x,y) d\mu(y)\); and as a consequence, \(X\) has the Doob convergence property.