PUB - Publikationen an der Universität Bielefeld

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(x) for every x is an element of X and every relatively compact open neighborhood V of x, where tau(V) denotes the exit time of V. For every such function u, its lower semicontinuous regularization (u) over cap is excessive. The main purpose of the paper is to give a short, complete and understandable proof for the statement that u = inf{w is an element of epsilon(X) : w >= 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) < infinity the expected number of times the process X visits the set of points y. X, where <(u)over cap>(y) := lim inf(z) -> y u(z) < u(y), is finite. 2. The consequent use of (only) the strong Markov property. 3. The proof of the equality integral u d mu = inf{integral w d mu: w is an element of epsilon(X), w >= 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.