Infima of superharmonic functions

Abstract

Let Ω be a Greenian domain in ℝ d , d≥2, or—more generally—let Ω be a connected $\mathcal{P}$ -Brelot space satisfying axiom D, and let u be a numerical function on Ω, $u\not\equiv\infty$ , which is locally bounded from below. A short proof yields the following result: The function u is the infimum of its superharmonic majorants if and only if each set {x:u(x)>t}, t∈ℝ, differs from an analytic set only by a polar set and $\int u\,d\mu_{x}^{V}\le u(x)$ , whenever V is a relatively compact open set in Ω and x∈V.