PomeD0AF90DDC7A5Theorem 7.3.11.D0AF90DDC7A5 Let $D\subset \mathbb {C}^ n$ be a bounded open set, and let $u$ and $v$ be locally bounded plurisubharmonic functions on $D$ such that $\varliminf _{D\ni z\to x}(u(z)-v(z))\ge 0$ for every $x\in \partial D$. Then
where both sides may be infinite. If $(\mathrm{dd}^{\mathrm{c}}u)^ n=0$, then $u\ge v$ on $D$.
Proof. Let $\varepsilon >0$ and $w=\max \{ u,v-\varepsilon \} $. We show that $(\mathrm{dd}^{\mathrm{c}}w)^ n$ and $(\mathrm{dd}^{\mathrm{c}}u)^ n$ have the same mass near a compact set outside which $w=u$, and then identify $(\mathrm{dd}^{\mathrm{c}}w)^ n$ by proposition 7.3.10. Every $x\in \partial D$ has an open neighborhood $U_ x$ with $u>v-\varepsilon $ on $U_ x\cap D$, and finitely many of them cover the compact set $\partial D$; let $U$ be their union. Then $F=D\setminus U=\overline D\setminus U$ is a compact subset of $D$, it contains $\{ u\le v-\varepsilon \} $, and $w=u$ on $D\setminus F$. For $0\le l\le n-1$ put $R_ l=(\mathrm{dd}^{\mathrm{c}}w)^ l\wedge (\mathrm{dd}^{\mathrm{c}}u)^{n-1-l}$. By additivity and symmetry of the products (proposition 7.2.2, theorem 7.3.7),
and summing over $l$, the right-hand sides telescope:
The current in parentheses vanishes outside $F$, since $w-u$ does. Hence the signed measure $(\mathrm{dd}^{\mathrm{c}}w)^ n-(\mathrm{dd}^{\mathrm{c}}u)^ n$ is supported in $F$, and if $\chi \in C^\infty _ c(D)$ equals $1$ on a neighborhood of $F$, then its total mass is
because $\mathrm{dd}^{\mathrm{c}}\chi =0$ on a neighborhood of $F$. Choose a relatively compact open set $D'\subset D$ containing $F$. Both measures are finite on $D'$, and $\int _{D'}(\mathrm{dd}^{\mathrm{c}}w)^ n=\int _{D'}(\mathrm{dd}^{\mathrm{c}}u)^ n$. By proposition 7.3.10, $(\mathrm{dd}^{\mathrm{c}}w)^ n$ coincides with $(\mathrm{dd}^{\mathrm{c}}v)^ n$ on $\{ u<v-\varepsilon \} $ and with $(\mathrm{dd}^{\mathrm{c}}u)^ n$ on $\{ u>v-\varepsilon \} $; here we use that adding the constant $-\varepsilon $ does not change $(\mathrm{dd}^{\mathrm{c}}v)^ n$. Since $\{ u<v-\varepsilon \} $ and $D'\cap \{ u>v-\varepsilon \} $ are disjoint Borel subsets of $D'$, and $\{ u\le v-\varepsilon \} \subset F\subset D'$,
As $\varepsilon \downarrow 0$, the sets $\{ u<v-\varepsilon \} $ increase to $\{ u<v\} $, and monotone convergence proves the comparison inequality.
The last assertion is the case of corollary 7.3.12 below in which the roles of $u$ and $v$ are interchanged: if $(\mathrm{dd}^{\mathrm{c}}u)^ n=0$, then $v\le u$ holds almost everywhere for $(\mathrm{dd}^{\mathrm{c}}u)^ n$. The proof of that corollary uses only the comparison inequality. $\square$
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