Bĥer's theorem in a space of dimension one (Q1578879)

In the axiomatic potential theory, let \(X\) be a harmonic space in the sense of Brelot; \(X\) has a countable base; the constants are assumed to be harmonic; and \(X\) satisfies the local axiom of proportionality (that is, for any \(x\in X\), if \(p_1\) and \(p_2\) are two potentials in a domain \(\delta\), \(x\in \delta\), with harmonic point support \(x\), then \(p_1\) and \(p_2\) are proportional). Then one can define a kernel \(q_y(x)\) in \(X\), which has most of the properties of the Newtonian kernel \(\frac{1}{|x-y|}\) in \(\mathbb{R}^3\) if \(X\) has potentials \(>0\) and those of the logarithmic kernel \(\log|x-y|\) in \(\mathbb{R}^2\) if \(X\) has no positive potentials. The authors prove in this note, that if \(h\) is a harmonic function defined outside a compact set in \(X\), then there exists a signed (Radon) measure \(\mu\) with compact support and a unique harmonic function \(u\) on \(X\) such that \(h(x)= \int q_y(x) d\mu(y)+ u(x)\) near infinity; moreover, if the harmonic dimension at infinity of \(X\) is one, then \(u\) is a constant if and only if \(h\) is bounded on one side near infinity. They interpret the last remark when \(X= \mathbb{R}^n\), \(n\geq 2\), after an inversion, as the classical Bôcher theorem in \(\mathbb{R}^n\) concerning the point singularity of a positive harmonic function.