Nearly Hyperharmonic Functions are Infima of Excessive Functions

Let $$\mathfrak {X}$$ X be a Hunt process on a locally compact space X such that the set $$\mathcal {E}_{\mathfrak {X}}$$ E X of its Borel measurable excessive functions separates points, every function in $$\mathcal {E}_{\mathfrak {X}}$$ E X is the supremum of its continuous minorants in $${\mathcal {E}}_{{\mathfrak {X}}}$$ E X , and there are strictly positive continuous functions $$v,w\in {\mathcal {E}}_{{\mathfrak {X}}}$$ v , w ∈ E X such that v / w vanishes at infinity. A numerical function $$u\ge 0$$ u ≥ 0 on X is said to be nearly hyperharmonic, if $$\int ^*u\circ X_{\tau _V}\,\text {d}P^x\le u(x)$$ ∫ ∗ u ∘ X τ V d P x ≤ u ( x ) for every $$x\in X$$ x ∈ X and every relatively compact open neighborhood V of x, where $$\tau _V$$ τ V denotes the exit time of V. For every such function u, its lower semicontinuous regularization $$\hat{u}$$ u ^ is excessive. The main purpose of the paper is to give a short, complete and understandable proof for the statement that $$ u=\inf \{w\in {\mathcal {E}}_{{\mathfrak {X}}}:w\ge u\}$$ u = inf { w ∈ E 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 $$x\in X$$ x ∈ X with $$u(x)