Theorem 3.5.32791D284428C

Proof. Lowering makes the sequence decrease and leaves it fixed outside $W$. On each fixed ball, the subsequence of stages just lowered there is harmonic and decreases to the same limit; the checked harmonic monotone convergence theorem makes that limit harmonic or $-\infty $. Connectedness propagates the alternative as in the preceding proof. If $v$ is any harmonic minorant on $W$, positivity of the Poisson kernel proves inductively $v\le u_ j$ there. Thus $v\le u_\infty \le u$, which excludes $-\infty $ and proves maximality and uniqueness. For two functions use the same sequence of balls. Linearity of each lowering gives $(u+v)_ j=u_ j+v_ j$; the limits are finite on $W$ when harmonic minorants exist, so their sum is the corresponding greatest harmonic minorant. No claim is made that the unregularized $u_\infty $ is superharmonic across $\partial W$. $\square$