Let X be a separable compact
Hausdorff space, and Iet B be a linear space of continuous real functions on X, where
le B and B separates the points of X. Let Γ der.ote the Silov boundary of B in
X, and assume that Δ = X ∼ Γ≠0. Further assumptions on B are made
which are in the nature of axioms for an abstract potential theory. These
assumptions are more globaI than is usual, and in particular a sheaf axiom is not
assumed, nor is the existence of a base of regular neighborhoods. Instead the
assumptions are concerned with equicontinuity properties of B on Δ, and the
consequences of Δ being a single Gleason part of X. With suitable hypotheses on
B and Δ there is an integral kernel representation of the following sort:
u(x) =∫Γu(𝜃)Q(x,𝜃)d1J(𝜃), where Q is a jointly measurable function on Δ × Γ
which is “in B” (i.e., abstractly harmonic) as a function of x for each fixed
𝜃 ∈ Γ.
