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Let \(\mathfrak {X}\) be a Hunt process on a locally compact space X such that the set \(\mathcal {E}_{\mathfrak {X}}\) of its Borel measurable excessive functions separates points, every function in \(\mathcal {E}_{\mathfrak {X}}\) is the supremum of its continuous minorants in \({\mathcal {E}}_{{\mathfrak {X}}}\), and there are strictly positive continuous functions \(v,w\in {\mathcal {E}}_{{\mathfrak {X}}}\) such that v / w vanishes at infinity. A numerical function \(u\ge 0\) on X is said to be nearly hyperharmonic, if \(\int ^*u\circ X_{\tau _V}\,\text {d}P^x\le u(x)\) for every \(x\in X\) 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 \(\hat{u}\) is excessive. The main purpose of the paper is to give a short, complete and understandable proof for the statement that \( u=\inf \{w\in {\mathcal {E}}_{{\mathfrak {X}}}:w\ge u\}\) 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 \(x\in X\) with \(u(x)<\infty \) the expected number of times the process \({\mathfrak {X}}\) visits the set of points \(y\in X\), where \(\hat{u}(y):=\liminf _{z\rightarrow y} u(z)<u(y)\), is finite.2.The consequent use of (only) the strong Markov property.3.The proof of the equality \(\int u\,\text {d}\mu =\inf \{\int w\,\text {d}\mu :w\in \mathcal {E}_{\mathfrak {X}},\,w\ge u\}\) not only for measures \(\mu \) satisfying \(\int w\,\text {d}\mu <\infty \) for some excessive majorant w of u, but also for all finite measures.At the end, the measurability assumption on u is weakened considerably.

Nearly Hyperharmonic Functions are Infima of Excessive Functions