Vol. 23, No. 2, 1967

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Integral kernel for one-part function spaces

Herbert Stanley Bear, Jr. and Bertram John Walsh

Vol. 23 (1967), No. 2, 209–215

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 Γ0. 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.

Mathematical Subject Classification
Primary: 31.50
Secondary: 46.00
Received: 19 September 1966
Revised: 30 November 1966
Published: 1 November 1967
Herbert Stanley Bear, Jr.
Bertram John Walsh