Theorem 4.3.6C651523E22C3

Proof. The family of majorants is nonempty (it contains $u$), closed under minimum and locally bounded below. The checked Foundations subharmonic envelope and polar limit discrepancy, applied after sign reversal, give superharmonic regularization and a polar discrepancy set. On $E$ the unregularized infimum equals $u$, and locality gives equality of the regularization on the interior. Outside $\overline E$, the family is saturated under ball lowering, so its infimum is harmonic by Lemma 3.5.2; it is nonnegative, excluding $-\infty $. Order and scaling follow from the defining families. Sums of majorants majorize the sum; the unregularized inequality passes to regularizations almost everywhere by the polar discrepancy assertion, and then everywhere by ball-average representatives.

For bounded $u$ on $K$, in each component meeting $K$ choose a pole $a$. A multiple of $G(a,\cdot )$ majorizes $u$ on that component's compact intersection with $K$, since its positive lsc kernel has a positive minimum there. On all other components use $u$. This competitor shows $\widehat R_ Ku\le C G(a,\cdot )$ on the chosen component, so it has zero greatest harmonic minorant there. Components disjoint from $K$ have zero reduction. The global Riesz theorem makes the reduction a potential. For general $u=p+h$ with $p$ a potential and $h\ge 0$ harmonic, $h$ is bounded on $K$. The reduction of $p$ is dominated by $p$, hence has zero harmonic minorant; that of $h$ is a potential by the bounded case. Subadditivity bounds the reduction of $u$ by their sum, so it too has zero minorant and is a potential. $\square$