Lemma 3.5.2AC0C4669DCBF

Proof. On a ball $B\Subset \Omega $ its infimum is also $\inf _{u\in \mathcal F}L_ Bu$, since $L_ Bu\le u$ and $L_ Bu\in \mathcal F$. The lowered harmonic functions form a downward directed family: $L_ B(\min (u,v))\le \min (L_ Bu,L_ Bv)$ by positivity of the Poisson kernel. A downward directed family of harmonic functions has harmonic infimum or infimum identically $-\infty $ on a ball. To see this from the checked Harnack convergence theorem, negate and restrict to the cofinal subfamily above one fixed member; its differences from that member are nonnegative harmonic. If bounded at one point, Harnack chains bound them on compact subsets, and harmonic compactness shows their directed supremum harmonic. If unbounded at a point, Harnack chains make the supremum infinity everywhere. Thus the infimum is locally harmonic or locally identically $-\infty $. The regions of these two alternatives are disjoint open sets; connectedness of each component leaves only one alternative. $\square$