Nearly hyperharmonic functions and Jensen measures

Let (X,H) be a P-harmonic space and assume for simplicity that constants are harmonic. Given a numerical function φ on X which is locally lower bounded, let Jφ(x) := sup{ ∫ ∗ φdμ(x) : μ ∈ Jx(X)}, x ∈ X, where Jx(X) denotes the set of all Jensen measures μ for x, that is, μ is a compactly supported measure on X satisfying ∫ u dμ ≤ u(x) for every hyperharmonic function on X. The main purpose of the paper is to show that, assuming quasi-universal measurability of φ, the function Jφ is the smallest nearly hyperharmonic function majorizing φ and that Jφ = φ∨ Ĵφ, where Ĵφ is the lower semicontinuous regularization of Jφ. So, in particular, Jφ turns out to be at least “as measurable as” φ. This improves recent results, where the axiom of polarity was assumed. The preparations about nearly hyperharmonic functions on balayage spaces are closely related to the study of strongly supermedian functions triggered by J.-F. Mertens more than forty years ago.