Equicontinuity on Harmonic Spaces

Abstract

G. Mokobodzki proved [5] that on any harmonic space with countable basis satisfying the axioms 1, 2, T+, K0 [2] [1] any equally bounded set of harmonic functions is equicontinuous. P. Loeb and B. Walsh showed [4] that the same property holds on a harmonic space without countable basis, if Brelot’s axiom 3 is fulfilled. The aim of this paper is to generalize these results to a harmonic space X satisfying only the axioms 1, 20, K1, [2] [1] where 20 is a weakened form of axiom 2. As a corollary we get: if any point of X possesses two open neighbourhoods U, V such that the set of harmonic functions on U separates the points of U∩V, then X has locally a countable basis.