Theorem 7.5.842828E3BDDFA

Proof. Weak convergence is local (lemma 7.3.6), so it suffices to prove the convergence on every ball $B\Subset D$. By the remark following definition 7.5.5, the convergence in capacity holds with $\operatorname {Cap}_ B$ in place of $\operatorname {Cap}_ D$ on every ball $B\Subset D$. The limit $f$ is quasi-continuous for $\operatorname {Cap}_ B$ as well: if $K\subset B$ is compact, $K'$ is a compact neighborhood of $K$ in $B$, and $G\subset D$ is open with $f$ continuous on $K'\setminus G$, then $G\cap \operatorname{Int}K'$ is open in $B$, $f$ is continuous on $K\setminus (G\cap \operatorname{Int}K')=K\setminus G$, and $\operatorname {Cap}_ B(G\cap \operatorname{Int}K')\le C\operatorname {Cap}_ D(G)$ by proposition 7.4.3(4) applied to $K'$. Finally, the functions $u_{l,j}$ and $u_ l$, restricted to $B$, form a family which is uniformly bounded on compact subsets of $B$. We may therefore work on such a ball and use lemma 7.5.7.

We argue by induction on $q$. For $q=0$ the product is $1$, and the weighted assertion is the second part of the lemma with $S_ j=1$. Let $q\ge 1$, and assume that $S_ j=\mathrm{dd}^{\mathrm{c}}u_{2,j}\wedge \cdots \wedge \mathrm{dd}^{\mathrm{c}}u_{q,j}$ converges weakly to $S=\mathrm{dd}^{\mathrm{c}}u_2\wedge \cdots \wedge \mathrm{dd}^{\mathrm{c}}u_ q$. The functions $u_{1,j}$ converge in capacity to $u_1$, which is quasi-continuous by theorem 7.5.2; hence the second part of the lemma gives $u_{1,j}S_ j\to u_1S$, and applying $\mathrm{dd}^{\mathrm{c}}$ proves the unweighted assertion for $q$ factors. The weighted assertion then follows from the second part of the lemma once more. $\square$