Theorem 4.5.32C4B36B5F3CD

Proof. The minimum $q=\min (w,G\mu )$ is a nonnegative superharmonic minorant of a potential, hence is a potential $G\nu $ with $\nu \in \mathcal E^+$. The hypothesis gives $\int (G\mu -G\nu )d\mu =0$, whereas $\int (G\mu -G\nu )d\nu \ge 0$. Their difference is $I(\mu -\nu )\le 0$. Strict definiteness makes $\mu =\nu $, proving the assertion. The nonnegativity of $w$ is necessary, including when $\mu =0$. $\square$