On the Choquet charge of \(\delta\)-superharmonic functions (Q1977860)

Let \((E,{\mathcal E})\) be a measurable space and \(\{V_\lambda\}_\lambda\) be a resolvent with proper potential operator \(V_0\). Assume that \(\{V_\lambda\}_\lambda\) is absolutely continuous with respect to a \(\sigma\)-finite measure \(\gamma\) on \(E\) and that the cone \({\mathcal S}\) of superharmonic functions is inf-stable, separating and contains some \(s>0\). It is known that \(u\in{\mathcal S}\cap L^1(\gamma)\) admits a Choquet-type representation \(u(x)= \int_{E_\gamma^*} s(x) \mu_u(ds)\) with a unique \(\mu_u\) on the set \(E_\gamma^*\) of minimal functions in \({\mathcal S}\) with \(\int s d\gamma= 1\). The author shows that any two superharmonic \(u,v\) with \(u\geq v\) have representing measures satisfying \(\mu_u\leq \mu_v\) on the contact set \(\{s\in{\mathcal S}: \limsup_{\tau(s)} v/u= 1\}\) where \(\tau(s)\) is the co-fine neighbourhood filter for \(s\). This result is an abstract generalization of \textit{B. Fuglede}'s paper [Potential Anal. 1, 355-371 (1992; Zbl 0766.31010)] where a similar question is considered for the Laplacian and its resolvent.