Theorem 4.3.7ABDF7610F7B2

Proof. The reduction theorem proves the representation and, since $V_ K$ equals one on the interior and is harmonic outside $K$, its Riesz measure is supported on $\partial K$. If $p=G\mu \le 1$ is supported on $K$ and $v$ is admissible for $W_ K$, then $v-p$ is superharmonic and bounded below outside $K$. Its boundary lower limits are nonnegative: at $K$ use $v\ge 1$, at the ambient boundary use admissibility, outside a polar exception. At infinity use decay when required. The polar boundary minimum principle gives $v\ge p$, including on $K$. Infimize and regularize to get $V_ K\ge p$.

Choose a compact neighborhood $L$ of $K$, with $K\subset \operatorname {int}L$. Its potential $G\mu _ L=1$ on $K$. For measures $\mu ,\nu $ supported on $K$ with $G\mu \le G\nu $, reciprocity gives $\mu (K)=\int G\mu \, d\mu _ L\le \int G\nu \, d\mu _ L=\nu (K)$. Taking $\nu =\mu _ K$ proves the supremum formula and monotonicity.

If $K_ j\downarrow K$, put $g=\lim W_{K_ j}$. Its regularization is a potential dominated by $V_{K_1}$, harmonic off $K$ by harmonic convergence, and at most one. The greatest-potential characterization bounds it by $V_ K$, while monotonicity bounds it below by $V_ K$. Hence $g=V_ K=W_ K$ off $K$, and all $W_{K_ j}=W_ K=1$ on $K$, proving $W_{K_ j}\downarrow W_ K$. For a compact neighborhood $L$ of $K_1$ with its boundary outside $K_1$, reciprocity gives $C(K_ j)=\int W_{K_ j}\, d\mu _ L$. Bounded convergence then gives decreasing compact continuity.

For strong subadditivity first prove $W_{K\cup L}+W_{K\cap L}\le W_ K+W_ L$. On $K\cup L$ it follows from $W_{K\cup L}=1$ and order of the other reductions. On its complement the left side is harmonic. If $v,w$ are majorants for $K,L$, their sum minus that left side has nonnegative boundary lower limits: at the compact boundary, use that $W_{K\cup L}\le 1$ and that $W_{K\cap L}$ is continuous off $K\cap L$, while at points of the intersection it is bounded by one. At the ambient boundary use admissibility and at infinity use decay. Apply the polar minimum principle and infimize over $v,w$. Integrate the inequality against $\mu _ N$ for a larger compact neighborhood $N$; reciprocity changes each integral to its capacity, giving strong subadditivity.

Here are the extension details, to specify the precise Choquet interface. Compact continuity and compact neighborhoods imply $C^*(K)=C(K)$. For open sets strong subadditivity follows by decomposing each compact subset of their union into finitely many compact pieces in the respective opens, adding a compact subset of the intersection, and applying compact strong subadditivity. Taking almost optimal open majorants gives strong subadditivity of $C^*$. Repeated use gives the finite excess inequality

when the right side is finite; if a term is infinite use the equivalent inequality with sums on opposite sides. Continuity on increasing open sets is immediate from compactness. For increasing arbitrary $A_ i$ with bounded capacities choose open $O_ i\supset A_ i$ with $C(O_ i)<C^*(A_ i)+\varepsilon 2^{-i}$. The excess inequality bounds $C(\bigcup _{i\le m}O_ i)$ by $C^*(A_ m)+\varepsilon $. Continuity for open unions and then $\varepsilon \downarrow 0$ proves $C^*(\bigcup A_ i)=\lim C^*(A_ i)$. If the limit is infinite, monotonicity already proves equality. This also gives countable subadditivity.

The capacitability argument is purely axiomatic. In a compact Hausdorff space, a set $D=\bigcap _ i\bigcup _ kK_{ik}$ with compact $K_{ik}$ is capacitable for any such outer capacity: replace each row by increasing unions and successively choose row bounds so that the capacity of the remaining part stays above any fixed number below $C^*(D)$. The resulting decreasing compact row intersections have capacity at least that number, and their intersection lies in $D$. For a continuous map on $D$ into a compact subset of $\Omega $, take its graph in the product of the ambient compact space and that target compact. The graph is relatively closed in $D$ times the target, hence is again of the same compact-intersection form. Pull back capacity by the projection and apply the preceding argument. Continuous images commute with decreasing compact intersections because the closed fibers form nested nonempty compact sets. This proves capacitability of K-analytic sets in compact target neighborhoods; an increasing compact exhaustion and capacity continuity remove that restriction. Countable unions/intersections of these images are again of that form, by the product/equal-image construction for intersections and a disjoint union with one-point compactification for unions. Closed and open subsets of Euclidean space are countable unions of compacts, so the class of sets whose complements are also K-analytic is a sigma algebra containing the open sets. Thus all Borel sets are capacitable.

If $A\Subset \Omega $ is polar, a finite compactly supported fundamental measure producing its poles can be localized to a compact neighborhood in $\Omega $; its Green potential $p$ is infinite on $A$. For $O_ t=\{ p>t\} $ and compact $L\subset O_ t$, reciprocity gives $C(L)\le t^{-1}\int p\, d\mu _ L\le \mu (\Omega )/t$. Thus $C^*(A)=0$, and exhaustions handle arbitrary polar $A$. Conversely, locally separate a set $A$ of outer capacity zero from one interior point $a$ in a connected relatively compact neighborhood. Choose open $O_ j\supset A$ with $C(O_ j)<2^{-j}$, still separated from $a$. Increasing compact exhaustions of $O_ j$ have potentials increasing to $w_ j\le 1$, equal one on $O_ j$. The Green kernel at $a$ is bounded on the separated region, so $w_ j(a)\le M2^{-j}$. Hence $\sum _ jw_ j$ is superharmonic on that component, finite at $a$ and infinite on $A$. To perform the separation for a general local set, split it into its intersections with finitely many smaller closed neighborhoods, each avoiding a different interior ball; then use a countable local covering. Countable unions of polar sets are polar, proving the converse. $\square$