Proposition 7.2.62F39A3F97E8E

Proof. We first show that $u_ j$ is plurisubharmonic for $1\le j\le q$. Since plurisubharmonicity is a local property, it suffices to prove this on every ball $W\Subset \Omega $. Choose $k_1<k_2<\cdots $ such that $|u_{j,k_ m}-u_ j|\le 2^{-m}$ on $W$, and put $w_ m=u_{j,k_ m}+3\cdot 2^{-m}$. These functions are plurisubharmonic on $W$, and they decrease, because

Their limit $u_ j$ is finite, so it is plurisubharmonic on $W$ by theorem 5.10.1.

Since the $u_{j,k}$ converge locally uniformly to continuous functions, they are bounded on each compact set uniformly in $k$. Since $T_ k\to T$ weakly, the masses $\| T_ k\| _ L$ are bounded for each compact set $L$: if $\chi \in C^\infty _ c(\Omega )$ satisfies $0\le \chi \le 1$ and $\chi =1$ on $L$, then $\| T_ k\| _ L\le T_ k(\chi \beta ^ r)/r!$, and the right-hand side converges. By theorem 7.2.5, the masses of all the products considered below are therefore bounded on compact sets uniformly in $k$, and so are the total variations of their coefficient measures (theorem 5.3.7).

We claim: if $S_ k\to S$ weakly, where $S_ k$ and $S$ are closed positive currents whose masses are bounded on compact sets uniformly in $k$, and if locally bounded Borel functions $v_ k$ converge locally uniformly to a continuous function $v$, then $v_ kS_ k\to vS$ weakly. Applying this to $S_ k=\mathrm{dd}^{\mathrm{c}}u_{2,k}\wedge \cdots \wedge \mathrm{dd}^{\mathrm{c}}u_{q,k}\wedge T_ k$ and $v_ k=u_{1,k}$, and then $\mathrm{dd}^{\mathrm{c}}$, which is weakly continuous, proves the first assertion by induction on $q$; the case $q=0$ is the hypothesis. The second assertion is the claim with $v_ k=u_{0,k}$.

To prove the claim, fix a test form $\varphi $ with compact support $K$ and $\varepsilon >0$. By lemma 2.5.1(1), applied to $v$ on a relatively compact open neighborhood of $K$, there is a smooth function $v_\varepsilon $ on a neighborhood of $K$ with $|v-v_\varepsilon |\le \varepsilon $ on $K$. Write

Evaluated on $\varphi $, a term $fS_ k$ with a bounded Borel function $f$ is at most $C_\varphi \sup _ K|f|$ times the total variation on $K$ of the coefficient measures of $S_ k$, where $C_\varphi $ bounds the coefficients of $\varphi $; the same holds for $S$. Hence the first term tends to zero, since $v_ k\to v$ uniformly on $K$; the second and fourth terms are bounded by a constant times $\varepsilon $, uniformly in $k$; and the third tends to zero as $k\to \infty $, since $v_\varepsilon \varphi $ is a test form. Letting first $k\to \infty $ and then $\varepsilon \downarrow 0$ proves the claim. $\square$