Lemma 13.7.1AB4F8DB9BF3E

Proof. For every $x\in K$ choose $r_ x>0$ such that the closed ball $\overline{B(z(x),2r_ x)}$ lies in $z(W\cap W')$. Finitely many of the sets $z^{-1}(B(z(x_ i),r_ i))$, $r_ i=r_{x_ i}$, cover $K$; put $B_ i=B(z(x_ i),2r_ i)$ and $K_ i=K\cap z^{-1}(\overline{B(z(x_ i),r_ i)})$. The map $F_ i=z'\circ z^{-1}$ is a biholomorphism from $B_ i$ onto the bounded open set $B_ i'=F_ i(B_ i)\subset D'$. Let $E\subset K$ and $E_ i=E\cap K_ i$. Every competitor for $\operatorname {Cap}_{D'}$ restricts to a competitor for $\operatorname {Cap}_{B_ i'}$, so $\operatorname {Cap}_{D'}(G)\leq \operatorname {Cap}_{B_ i'}(G)$ for Borel sets $G\subset B_ i'$, and taking infima over open sets, $\operatorname {Cap}^*_{D'}\leq \operatorname {Cap}^*_{B_ i'}$ on subsets of $B_ i'$. By proposition 7.4.3(5) and (4), applied to the ball $B_ i\Subset D$ and the compact set $\overline{B(z(x_ i),r_ i)}\subset B_ i$, there is a constant $C_ i$ with

By subadditivity of $\operatorname {Cap}^*_{D'}$ (proposition 7.4.3), the assertion holds with $C=\sum _ iC_ i$. If $E$ is a Borel set, then so are $E_ i$, $z(E_ i)$ and $z'(E_ i)$, because $z$ and $z'$ are homeomorphisms onto their images. The restriction argument above and proposition 7.4.3(4),(5) also apply to the capacities of Borel sets, so the same chain of inequalities holds with $\operatorname {Cap}$ in place of $\operatorname {Cap}^*$, and the subadditivity of $\operatorname {Cap}_{D'}$ gives the second assertion with the same constant.

For the consequences, let $(W',z')$ be a chart with bounded image and $K'\subset W'$ compact. Cover $K'$ by finitely many compact sets $K_ i'\subset W'\cap W_ i$, where the $(W_ i,z_ i)$ are charts of the given atlas. Suppose that $(f_ a)$ converges to $f$ locally in capacity in the charts $W_ i$. By subadditivity, the outer capacity in $D'=z'(W')$ of the image of $K'\cap \{ f_ a>f+\delta \} $ is at most the sum over $i$ of the outer capacities of the images of $K_ i'\cap \{ f_ a>f+\delta \} $. By the first part, each of these is at most a constant times the corresponding outer capacity in $z_ i(W_ i)$, which tends to zero; similarly for $\{ f_ a<f-\delta \} $.

If $f$ is quasi-continuous in the charts $W_ i$ and $\varepsilon >0$, choose open sets $G_ i\subset z_ i(W_ i)$ of small capacity such that $f$ is continuous on $K_ i'\setminus z_ i^{-1}(G_ i)$. By the first part, $z'(K_ i'\cap z_ i^{-1}(G_ i))$ has outer capacity at most $C_ i\operatorname {Cap}_{z_ i(W_ i)}(G_ i)$ in $D'$, so it is contained in an open set $G_ i'\subset D'$ with $\operatorname {Cap}_{D'}(G_ i')<\varepsilon /N$, where $N$ is the number of sets $K_ i'$, provided the $G_ i$ are chosen small enough. Then $G'=\bigcup _ iG_ i'$ satisfies $\operatorname {Cap}_{D'}(G')<\varepsilon $, and the set $F=K'\setminus z'^{-1}(G')$ is the union of the finitely many relatively closed sets $F\cap K_ i'$, on each of which $f$ is continuous; hence $f$ is continuous on $F$. $\square$