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 phi on X which is locally lower bounded, let J(phi) (x) := sup {integral* phi d mu : mu is an element of J(x) (X)}, x is an element of X, where J(x) (X) denotes the set of all Jensen measures mu for x, that is, mu is a compactly supported measure on X satisfying integral u d mu <= u(x) for every hyperharmonic function u on X. The main purpose of the paper is to show that, assuming quasi-universal measurability of phi, the function J(phi) is the smallest nearly hyperharmonic function majorizing phi and that J(phi) = phi boolean OR (J) over cap (phi), where (J) over cap (phi) J(phi) is the lower semicontinuous regularization of J(phi). So, in particular, J(phi) turns out to be at least "as measurable as" phi. This improves recent results, where the axiom of polarity was assumed. The preliminaries on nearly hyperharmonic functions in the framework of balayage spaces are closely related to the study of strongly supermedian functions triggered by Mertens more than forty years ago.
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