Corollary 7.11.3811945FC061EC

Proof. We compare the products with those of the truncations at a level $-T$: the latter converge by theorem 7.5.8, and by theorem 7.11.33 the two differ by a term of order $1/T$. For $T>0$ put $a_{l,j}=\max \{ u_{l,j},-T\} $, and write

By lemma 7.11.19, $a_{l,j}\in \mathcal E_0(B)$ and $e_1(a_{l,j})\leq E$. For each $l$, theorem 7.11.33 with $u_0=u_{l,j}$, respectively $u_0=a_{l,j}$, gives

Let $D_{j,T}=\bigcup _{l=1}^ n\{ u_{l,j}\leq -T\} $. On $\{ u_{l,j}\leq -T\} $ both $-u_{l,j}$ and $-a_{l,j}$ are at least $T$, so Chebyshev's inequality gives

On the complement of $D_{j,T}$, every $u_{l,j}$ is greater than $-T$. For $0\leq k\leq n$ put

so that $P_0=R_ j$ and $P_ n=R_{j,T}$. By the symmetry of the products, $P_ k=\mathrm{dd}^{\mathrm{c}}a_{k,j}\wedge \Theta _ k$ and $P_{k-1}=\mathrm{dd}^{\mathrm{c}}u_{k,j}\wedge \Theta _ k$ with the same closed positive current $\Theta _ k$ formed by the other $n-1$ factors, and maximum locality (proposition 7.3.10 with $p=1$, $u=u_{k,j}$, $v=-T$ and base current $\Theta _ k$) gives $P_ k=P_{k-1}$ on $\{ u_{k,j}>-T\} $. Hence $R_ j=R_{j,T}$ on the complement of $D_{j,T}$, and for every $\chi \in C_ c(B)$,

Now fix $\chi \in C_ c(B)$, choose an open set $B'\Subset B$ containing $\operatorname{Supp}\chi $, and let $T>\max _ l\sup _{B'}|u_ l|$. By (7.11.17), the uniformly bounded functions $a_{l,j}$ converge to the locally bounded plurisubharmonic function $\max \{ u_ l,-T\} $ locally in capacity, and theorem 7.5.8 shows that $R_{j,T}\to \bigwedge _ l\mathrm{dd}^{\mathrm{c}}\max \{ u_ l,-T\} $ weakly on $B$. On $B'$ we have $\max \{ u_ l,-T\} =u_ l$, so by locality (proposition 7.2.2) this limit equals $\mathrm{dd}^{\mathrm{c}}u_1\wedge \cdots \wedge \mathrm{dd}^{\mathrm{c}}u_ n$ there. Consequently

for all large $T$, and letting $T\to \infty $ completes the proof. $\square$