Lemma 13.7.4.88D1663D1111 Let $D\subset \mathbb C^ n$ be a bounded open set and $1\leq p\leq n$. For $1\leq j\leq p$ let $(v_{j,k})_{k\geq 1}$ be a sequence of plurisubharmonic functions on $D$ that is locally uniformly bounded above and converges in extended capacity to a plurisubharmonic function $v_ j$, and suppose that the non-pluripolar products $R_ k=\langle \mathrm{dd}^{\mathrm{c}}v_{1,k}\wedge \cdots \wedge \mathrm{dd}^{\mathrm{c}}v_{p,k}\rangle $ and $R=\langle \mathrm{dd}^{\mathrm{c}}v_1\wedge \cdots \wedge \mathrm{dd}^{\mathrm{c}}v_ p\rangle $ are locally finite. For $C,\varepsilon >0$ put
\[ v_{j,k}^ C=\max \{ v_{j,k},-C\} ,\qquad a_{j,k}=\frac{v_{j,k}^ C+C}{v_{j,k}^ C+C+\varepsilon },\qquad a_ k=\prod _ j a_{j,k}, \]
and define $v_ j^ C$, $a_ j$ and $a$ from the $v_ j$ in the same way. Then:
the functions $a_ k$ and $a$ are Borel and take values in $[0,1]$, the function $a$ is quasi-continuous, and $a_ k\to a$ locally in capacity as $k\to \infty $;
$a_ kR_ k=a_ k\, \mathrm{dd}^{\mathrm{c}}v_{1,k}^ C\wedge \cdots \wedge \mathrm{dd}^{\mathrm{c}}v_{p,k}^ C$ and $aR=a\, \mathrm{dd}^{\mathrm{c}}v_1^ C\wedge \cdots \wedge \mathrm{dd}^{\mathrm{c}}v_ p^ C$, where the products on the right are Bedford–Taylor products;
for fixed $C$, the weight $a$ increases to the indicator function of $\bigcap _ j\{ v_ j>-C\} $ as $\varepsilon \downarrow 0$, and these indicator functions increase to the indicator function of the complement of the union of the pole sets of the $v_ j$ as $C\to \infty $.
Proof.
(1) The weights are Borel functions of the Borel functions $v_{j,k}$ and $v_ j$, and they take values in $[0,1]$ because $v_{j,k}^ C+C\geq 0$ and $v_ j^ C+C\geq 0$. Let $K\subset D$ be compact and $\delta >0$. By theorem 7.5.2 there are open sets $G_ j\subset D$ with $\operatorname {Cap}_ D(G_ j)<\delta /p$ such that $v_ j$ is continuous on $K\setminus G_ j$. Then $a$, a continuous function of $v_1^ C,\ldots ,v_ p^ C$, is continuous on $K\setminus \bigcup _ jG_ j$, and $\operatorname {Cap}_ D(\bigcup _ jG_ j)<\delta $ by subadditivity; thus $a$ is quasi-continuous. By lemma 13.7.3, $v_{j,k}^ C\to v_ j^ C$ locally in capacity. Since $s\mapsto (s+C)/(s+C+\varepsilon )$ is Lipschitz with constant $1/\varepsilon $ on $[-C,\infty )$, and since $|\prod _ jb_ j-\prod _ jb_ j'|\leq \sum _ j|b_ j-b_ j'|$ for numbers in $[0,1]$, the set where $|a_ k-a|>\delta $ is contained in the union over $j$ of the sets where $|v_{j,k}^ C-v_ j^ C|>\varepsilon \delta /p$. By subadditivity of outer capacity, $a_ k\to a$ locally in capacity.
(2) Fix $k$, and let $O=\bigcap _ j\{ v_{j,k}>-C\} $, a plurifine open set. For every integer $m\geq C$, the set $O$ is contained in the level-$m$ set $O_ m=\bigcap _ j\{ v_{j,k}>-m\} $ of the tuple $(v_{j,k})_ j$, and on $O$ the truncations $\max \{ v_{j,k},-m\} $ coincide with $v_{j,k}$, hence with $v_{j,k}^ C$. By lemma 13.2.2, $\mathbf1_{O_ m}R_ k$ is the Bedford–Taylor product of the truncations at level $m$ restricted to $O_ m$, and by theorem 5.47.22 this product coincides on $O$ with $\bigwedge _ j\mathrm{dd}^{\mathrm{c}}v_{j,k}^ C$. Hence $\mathbf1_ OR_ k=\mathbf1_ O\bigwedge _ j\mathrm{dd}^{\mathrm{c}}v_{j,k}^ C$. The weight $a_ k$ vanishes outside $O$, because $a_{j,k}=0$ wherever $v_{j,k}\leq -C$, and multiplying by $a_ k$ gives the first identity. The same argument, applied to the tuple $(v_ j)_ j$, gives the second.
(3) Where $v_ j>-C$ we have $v_ j^ C+C>0$, so $a_ j$ increases to $1$ as $\varepsilon \downarrow 0$; where $v_ j\leq -C$ we have $a_ j=0$. Hence $a$ increases to the indicator function of $\bigcap _ j\{ v_ j>-C\} $, and these sets increase to $\bigcap _ j\{ v_ j>-\infty \} $ as $C\to \infty $.
$\square$
Pome
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