Abstract
Let X be a Hunt process on a locally compact space X such that the set epsilon(X) of its Borel measurable excessive functions separates points, every function in epsilon(X) is the supremum of its continuous minorants in epsilon(X), and there are strictly positive continuous functions v, w is an element of epsilon(X) such that v/w vanishes at infinity. A numerical function u = 0 on X is said to be nearly hyperharmonic, if integral* u omicron X-tau V dP(x) = u} for every Borel measurable nearly hyperharmonic function on X.
Principal novelties of our approach are the following: 1. A quick reduction to the special case, where starting at x is an element of X with u(x) (y) := lim inf(z) -> y u(z) = u} not only for measures mu satisfying integral w d mu < infinity for some excessive majorant w of u, but also for all finite measures.
At the end, the measurability assumption on u is weakened considerably.