A Lemma on Positive Harmonic Functions
Maurice H. Heins · Annals of Mathematics · 1950
1. Let F denote a Riemann surface tolerating non-trivial positive harmonic functions. We shall be concerned with the class P of non-negative harmonic functions with domain F and the additive real-valued functionals on P which are continuous with respect to pointwise convergence on F of the elements of P. That is, we admit real-valued functionals X having domain P and satisfying: i) X(u + v) = X(u) + X(v) for u, v e P, ii) If {u,} is a sequence of elements of P converging pointwise on F with limit function u, then limnOO X(un) = X(u) (we recall that limb_ u. e P). It is immediate from i) and ii) that if k is a positive constant, then X(ku) = kX(u) for u e P. A member u of P is said to be minimal (R. S. Martin [6]) provided that u # 0 and that whenever v e P satisfies