Proposition 7.4.397EFDB6FDEE3

Proof. For each competitor $v$, the measure $(\mathrm{dd}^{\mathrm{c}}v)^ n$ is increasing and countably subadditive on Borel sets: $\int _{\bigcup _ jE_ j}(\mathrm{dd}^{\mathrm{c}}v)^ n\le \sum _ j\int _{E_ j}(\mathrm{dd}^{\mathrm{c}}v)^ n\le \sum _ j\operatorname {Cap}_ D(E_ j)$. Taking the supremum over $v$ gives the same properties for $\operatorname {Cap}_ D$. Outer capacity is increasing by definition. For countable subadditivity, given sets $A_ j$ with $\sum _ j\operatorname {Cap}^*_ D(A_ j)<\infty $ and $\varepsilon >0$, choose open sets $G_ j\supset A_ j$ with $\operatorname {Cap}_ D(G_ j)<\operatorname {Cap}^*_ D(A_ j)+\varepsilon 2^{-j}$; then $\operatorname {Cap}^*_ D(\bigcup _ jA_ j)\le \operatorname {Cap}_ D(\bigcup _ jG_ j)\le \sum _ j\operatorname {Cap}^*_ D(A_ j)+\varepsilon $.

Finiteness on a compact set $K$ follows from theorem 7.2.5 with $T=1$, applied to $K$ and a compact neighborhood $L$ of $K$ in $D$, since $\| v\| _{L^\infty (L)}\le 1$ for every competitor. Inner regularity of each Radon measure $(\mathrm{dd}^{\mathrm{c}}v)^ n$ (definition 2.2.1) gives $\int _ E(\mathrm{dd}^{\mathrm{c}}v)^ n=\sup _ K\int _ K(\mathrm{dd}^{\mathrm{c}}v)^ n\le \sup _ K\operatorname {Cap}_ D(K)$ over compact $K\subset E$, and taking the supremum over $v$ gives (1). For (2), $\operatorname {Cap}_ D(E_ j)\le \operatorname {Cap}_ D(E)$ by monotonicity, and for every competitor $v$ monotone convergence gives $\int _ E(\mathrm{dd}^{\mathrm{c}}v)^ n=\lim _ j\int _{E_ j}(\mathrm{dd}^{\mathrm{c}}v)^ n\le \lim _ j\operatorname {Cap}_ D(E_ j)$; taking the supremum over $v$ gives (2). For (3), choose $a\in D$ and $d>0$ with $D\subset B(a,d)$; the function $|z-a|^2/d^2$ is a competitor, and by (7.2.4) and proposition 7.2.2(3) its Monge–Ampère measure is $n!\pi ^{-n}d^{-2n}\, \mathrm dV_{2n}$. Hence (3) holds with $C_ D=\pi ^ nd^{2n}/n!$.

For (4), the first inequality holds because the restriction to $B$ of a competitor for $D$ is a competitor for $B$, with the same Monge–Ampère measure on $B$. For the second, write $B=B(a,r)$ and choose $\lambda >0$ such that $\rho =\lambda (|z-a|^2-r^2)<-1$ on a neighborhood of $K$. Given a competitor $v$ for $B$, let $w$ be the function equal to $\max \{ v-1,\rho \} $ on $B$ and to $\rho $ on $D\setminus B$. Since $v-1\le 0$ on $B$ and $\rho =0$ on $\partial B$, proposition 5.9.5 shows that $w$ is plurisubharmonic on $D$. It equals $v-1$ near $K$, satisfies $w\ge -1$ on $D$, and is bounded above by $\Lambda =\max \{ 0,\sup _ D\rho \} $, which does not depend on $v$. Thus $(w+1)/(\Lambda +1)$ is a competitor for $D$, and by locality its Monge–Ampère measure near $K$ is $(\Lambda +1)^{-n}(\mathrm{dd}^{\mathrm{c}}v)^ n$. This gives the second inequality with $C=(\Lambda +1)^ n$; the first inequality holds for all Borel subsets of $B$.

For the outer capacities in (4), let $K'$ be a compact neighborhood of $K$ in $B$; the constant obtained for $K'$ also serves for $K$, and we use it. If $A\subset K$ and $G$ is an open subset of $D$ or of $B$ containing $A$, then $G\cap \operatorname{Int}K'$ is open in both $B$ and $D$, contains $A$, lies in $K'$, and has no larger capacity than $G$. Applying the inequalities to these sets and taking infima over $G$ gives them for the outer capacities of $A$.

For (5), let $F\colon D\to D'$ be biholomorphic. By proposition 5.9.3, applied to $F$ and to $F^{-1}$, the map $v\mapsto v\circ F$ is a bijection from the competitors for $D'$ onto the competitors for $D$, and by proposition 7.2.2(5), $(\mathrm{dd}^{\mathrm{c}}(v\circ F))^ n=F^*\bigl ((\mathrm{dd}^{\mathrm{c}}v)^ n\bigr )$. We claim that for every positive Radon measure $\mu $ on $D'$, regarded as a current of bidegree $(n,n)$, the pullback $F^*\mu $ is the image of $\mu $ under $F^{-1}$, that is, $\int \psi \, \mathrm d(F^*\mu )=\int \psi \circ F^{-1}\, \mathrm d\mu $ for every test function $\psi $ on $D$.

If $\mu =g\, \mathrm dV_{2n}$ with $g$ smooth, then $F^*\mu =(g\circ F)\, |\det F'|^2\, \mathrm dV_{2n}$, because $F^*(\mathrm dz_1\wedge \cdots \wedge \mathrm dz_ n)=\det F'\, \mathrm dz_1\wedge \cdots \wedge \mathrm dz_ n$ and similarly for the conjugate forms; since $|\det F'|^2$ is the Jacobian determinant of $F$ as a map of $\mathbb R^{2n}$, the change of variables formula gives the claim. Both sides of the claimed identity are weakly continuous in $\mu $ (proposition 5.6.2), and on relatively compact open subsets of $D'$ every $\mu $ is the weak limit of its smooth regularizations, so the claim holds in general. Consequently $(F^*\mu )(F^{-1}(E'))=\mu (E')$ for Borel sets $E'\subset D'$, since two Radon measures that agree on test functions agree on Borel sets (proposition 2.5.3(4)). Applying this to $\mu =(\mathrm{dd}^{\mathrm{c}}v)^ n$ and $E'=F(E)$ and taking the supremum over $v$ gives $\operatorname {Cap}_ D(E)=\operatorname {Cap}_{D'}(F(E))$. Since $F$ is a homeomorphism, it maps the open sets containing a set $A$ bijectively onto the open sets containing $F(A)$, so the outer capacities agree as well. $\square$