PomeDefinition 7.5.1.A2CACA24130B A Borel function $f\colon D\to [-\infty ,\infty ]$ is quasi-continuous (for $\operatorname {Cap}_ D$) if for every compact set $K\subset D$ and every $\varepsilon >0$ there is an open set $G\subset D$ with $\operatorname {Cap}_ D(G)<\varepsilon $ such that the restriction of $f$ to $K\setminus G$ is continuous. Here $[-\infty ,\infty ]$ carries its order topology.
7.5 Quasi-continuity and convergence in capacity887374BF3465
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Throughout this section and the next, $D\subset \mathbb {C}^ n$, $n\ge 1$, is a bounded open set. By theorem 4.1.30, locally integrable subharmonic functions are quasi-continuous in the sense of 1: they are continuous outside open sets of arbitrarily small relative capacity. We now define quasi-continuity with respect to the Monge–Ampère capacity instead, which is the meaning of the term in the following constructions, prove that plurisubharmonic functions are quasi-continuous in this sense, and then show that the products are continuous under convergence in capacity.
Theorem 7.5.2.C39FEC1AB985 Let $u\in \mathrm{PSH}(D)$. For every $\varepsilon >0$ there is an open set $G\subset D$ with $\operatorname {Cap}_ D(G)<\varepsilon $ which contains the pole set $\{ u=-\infty \} $ and such that the restriction of $u$ to $D\setminus G$ is finite and continuous. In particular $u$ is quasi-continuous.
Proof. Step 1: locally bounded $u$. Let $u$ be locally bounded, $K\subset D$ compact and $\varepsilon >0$; we find an open set $G\subset D$ with $\operatorname {Cap}_ D(G)<\varepsilon $ such that $u$ is continuous on $K\setminus G$. For each $x\in K$ choose $r_ x>0$ with $\overline{B(x,3r_ x)}\subset D$; finitely many balls $B(x_ i,r_{x_ i})$, $1\le i\le m$, cover $K$. Suppose that for each $i$ we find an open set $G_ i$ with $\operatorname {Cap}_ D(G_ i)<\varepsilon /m$ such that $u$ is continuous on $\overline{B(x_ i,r_{x_ i})}\setminus G_ i$. Then $G=\bigcup _ iG_ i$ satisfies $\operatorname {Cap}_ D(G)<\varepsilon $, and $u$ is continuous on $K\setminus G$, because the finitely many sets $(K\setminus G)\cap \overline{B(x_ i,r_{x_ i})}$ are closed in $K\setminus G$ and cover it, and $u$ is continuous on each of them. We may therefore assume that $K=\overline{B'}$, where $B'=B(x,r)$, $B=B(x,2r)$ and $B_1=B(x,3r)$ satisfy $\overline{B_1}\subset D$.
Subtracting a constant, we may assume that $-N\le u\le -1$ on $B_1$. Let $j>2/r$. Since $U=B(x,5r/2)$, a neighborhood of $\overline B$, lies at distance $r/2>1/j$ from $\mathbb C^ n\setminus B_1$, the regularization $u*\rho _{1/j}$ is smooth and plurisubharmonic on $U$ and takes values in $[-N,-1]$ there; moreover, $u*\rho _{1/j}$ decreases to $u$ as $j\to \infty $ (proposition 5.9.2). Put $\psi =|z-x|^2-4r^2$, choose $\delta >0$ with $\overline{B'}\subset \{ \psi <-\delta \} $, and let $g=\max \{ u,A\psi \} $ and $g_ j=\max \{ u*\rho _{1/j},A\psi \} $, with $A=N/\delta $, be the functions (7.3.2) on $U$. The $g_ j$ are continuous and plurisubharmonic on $U$ and decrease to $g$; moreover, $g=u$ on $\{ \psi <-\delta \} \supset \overline{B'}$, and $g_ j=g=A\psi $ on a collar of $\partial B$.
For $\delta '>0$ put $G_{j,\delta '}=\{ z\in B:g_ j(z)-g(z)>\delta '\} $; it is open, because $g_ j$ is continuous and $g$ is upper semicontinuous. By (7.4.6), applied with $h=g_ j$, $u=g$ and $\delta '$ in place of $\delta $,
\[ \operatorname {Cap}_ D(G_{j,\delta '})\le \frac{(n+1)!}{\delta '^{n+1}}\int _ B(g_ j-g)(\mathrm{dd}^{\mathrm{c}}g)^ n. \]
The right-hand side tends to zero as $j\to \infty $ by dominated convergence: $(\mathrm{dd}^{\mathrm{c}}g)^ n$ has finite mass on $B$, since $g$ is bounded and plurisubharmonic near $\overline B$, and the functions $g_ j-g\ge 0$ are bounded on $B$ uniformly in $j$ and decrease to zero. Choose $j_1<j_2<\cdots $ such that $G_ l=G_{j_ l,2^{-l}}$ satisfies $\operatorname {Cap}_ D(G_ l)<\varepsilon 2^{-l}$, and let $G=\bigcup _ lG_ l$. Then $\operatorname {Cap}_ D(G)<\varepsilon $, and on $B\setminus G$ we have $0\le g_{j_ l}-g\le 2^{-l}$ for every $l$. Thus $g$ is a uniform limit of continuous functions on $B\setminus G$, hence continuous there; in particular $u=g$ is continuous on $\overline{B'}\setminus G$.
Step 2: arbitrary $u$. Let $K\subset D$ be compact and $\varepsilon >0$. By proposition 7.4.4, applied to $K$ and a compact neighborhood $L$ of $K$ in $D$, there is $s>0$ with $\operatorname {Cap}^*_ D(K\cap \{ u<-s\} )<\varepsilon /2$, and hence an open set $G_1\supset K\cap \{ u<-s\} $ with $\operatorname {Cap}_ D(G_1)<\varepsilon /2$. By Step 1, applied to the locally bounded plurisubharmonic function $\max \{ u,-s\} $, there is an open set $G_2$ with $\operatorname {Cap}_ D(G_2)<\varepsilon /2$ such that $\max \{ u,-s\} $ is continuous on $K\setminus G_2$. On $K\setminus (G_1\cup G_2)$ we have $u\ge -s$, hence $u=\max \{ u,-s\} $. Thus $G=G_1\cup G_2$ contains $K\cap \{ u=-\infty \} $, $\operatorname {Cap}_ D(G)<\varepsilon $, and $u$ is finite and continuous on $K\setminus G$.
Step 3: a single exceptional set. Choose compact sets $K_ l$ with $K_ l\subset \operatorname{Int}K_{l+1}$ and $\bigcup _ lK_ l=D$. By Step 2 there are open sets $G^{(l)}\supset K_ l\cap \{ u=-\infty \} $ with $\operatorname {Cap}_ D(G^{(l)})<\varepsilon 2^{-l}$ such that $u$ is finite and continuous on $K_ l\setminus G^{(l)}$. Put $G=\bigcup _ lG^{(l)}$. Then $\operatorname {Cap}_ D(G)<\varepsilon $, and $G$ contains the pole set of $u$. Every $x\in D\setminus G$ lies in some $\operatorname{Int}K_ l$, and on the neighborhood $\operatorname{Int}K_ l\setminus G$ of $x$ in $D\setminus G$ the function $u$ is a restriction of the continuous function $u|_{K_ l\setminus G^{(l)}}$. Hence $u|_{D\setminus G}$ is finite and continuous. $\square$
In the proofs below we extend continuous functions from compact sets, using the following elementary case of Tietze's extension theorem.
Lemma 7.5.3.D127F8356D96 Let $(X,d)$ be a metric space, let $F\subset X$ be compact, and let $f\colon F\to \mathbb R$ be continuous with $|f|\le a$. Then $f$ has a continuous extension $g\colon X\to \mathbb R$ with $|g|\le a$.
Proof. If $F=\varnothing $ or $a=0$, take $g=0$. Otherwise, for $L>0$ put
\[ f_ L(x)=\max \Bigl \{ -a,\min \bigl \{ a,\inf _{y\in F}(f(y)+L\, d(x,y))\bigr \} \Bigr \} ,\qquad x\in X. \]
The infimum is finite, since $f\ge -a$, and as an infimum of functions that are Lipschitz with constant $L$ it is Lipschitz with constant $L$; hence $f_ L$ is continuous and $|f_ L|\le a$. We claim that $f_ L\to f$ uniformly on $F$ as $L\to \infty $. Taking $y=x$ shows $f_ L\le f$ on $F$. Given $\eta >0$, uniform continuity of $f$ on the compact set $F$ gives $r>0$ with $|f(y)-f(x)|<\eta $ for $x,y\in F$ with $d(x,y)<r$. Let $Lr>2a$ and $x\in F$. Points $y\in F$ with $d(x,y)\ge r$ give $f(y)+L\, d(x,y)\ge -a+Lr>a\ge f(x)$, and the other points give $f(y)+L\, d(x,y)>f(x)-\eta $. Hence the infimum is at least $f(x)-\eta $, and so is $f_ L(x)$, because $a\ge f(x)>f(x)-\eta $ and $-a\le f(x)$. This proves the claim. Consequently there is a continuous $g_1$ on $X$ with $|g_1|\le a$ and $|f-g_1|\le a/2$ on $F$. Applying the same construction to $f-g_1$ on $F$, we find a continuous $g_2$ with $|g_2|\le a/2$ and $|f-g_1-g_2|\le a/4$ on $F$, and inductively continuous $g_ k$ with $|g_ k|\le 2^{1-k}a$ and $|f-g_1-\cdots -g_ k|\le 2^{-k}a$ on $F$. The series $\sum _ kg_ k$ converges uniformly on $X$; its sum is continuous and equals $f$ on $F$, and clamping it to $[-a,a]$, that is, replacing it by $\max \{ -a,\min \{ a,\sum _ kg_ k\} \} $, does not change its values on $F$. $\square$
Lemma 7.5.4.56B45ADBA326 Let $K,L\subset D$ be compact sets with $K\subset \operatorname{Int}L$, and let $f$ be a real Borel function which is defined and quasi-continuous on a neighborhood of $L$. Then, for every $\delta >0$,
\[ \operatorname {Cap}_ D^*(K\cap \{ |f|>\delta \} ) \le \operatorname {Cap}_ D(L\cap \{ |f|>\delta /2\} ). \]
Proof. Let $\eta >0$. By quasi-continuity, applied to the compact set $L$, there is an open set $G\subset D$ with $\operatorname {Cap}_ D(G)<\eta $ such that $f$ is continuous on $L\setminus G$; let $g$ be a continuous extension of $f|_{L\setminus G}$ to $\mathbb {C}^ n$ (lemma 7.5.3); here $f|_{L\setminus G}$ is bounded, being continuous on a compact set. The open set $G\cup (\operatorname{Int}L\cap \{ |g|>\delta /2\} )$ contains $K\cap \{ |f|>\delta \} $, since $K\subset \operatorname{Int}L$ and $g=f$ on $K\setminus G$. Its part outside $G$ is a Borel set contained in $L\cap \{ |f|>\delta /2\} $, because $g=f$ on $L\setminus G$. By monotonicity and subadditivity,
\[ \operatorname {Cap}_ D^*(K\cap \{ |f|>\delta \} )\le \eta +\operatorname {Cap}_ D(L\cap \{ |f|>\delta /2\} ). \]
Let $\eta \downarrow 0$. $\square$
Definition 7.5.5.DAD5D0809372 A net $(f_ a)$ of Borel functions $f_ a\colon D\to [-\infty ,+\infty ]$ converges locally in capacity to a Borel function $f\colon D\to [-\infty ,+\infty ]$ if, for every compact set $K\subset D$ and every $\delta >0$,
\[ \operatorname {Cap}_ D^*(K\cap \{ f_ a>f+\delta \} )\longrightarrow 0, \qquad \operatorname {Cap}_ D^*(K\cap \{ f_ a<f-\delta \} )\longrightarrow 0. \]
Here $\pm \infty +\delta =\pm \infty $ and $\pm \infty -\delta =\pm \infty $.
Proposition 7.5.6.9542ED69E1C1 Let $u_ j$ be locally bounded plurisubharmonic functions on $D$ decreasing to a locally bounded function $u$. Then $u_ j\to u$ locally in capacity. Moreover, a net of real Borel functions on $D$ which is uniformly bounded on each compact subset of $D$ and converges locally in capacity to a locally bounded real Borel function also converges in $L^1_{\mathrm{loc}}(D)$.
Proof. Let $K\subset D$ be compact and $\delta >0$. Since $u_ j\ge u$, the sets $\{ u_ j<u-\delta \} $ are empty, and we must show that $\operatorname {Cap}^*_ D(K\cap \{ u_ j-u>\delta \} )\to 0$. By theorem 5.10.1, $u$ is plurisubharmonic. Covering $K$ by finitely many balls $B(x,r)$ with $\overline{B(x,4r)}\subset D$ and using subadditivity of outer capacity, we may assume that $K\subset \overline{B(x,r)}$ for such a ball; put $L=\overline{B(x,3r/2)}$, $B=B(x,3r)$ and $B_1=B(x,4r)$.
We first make the functions coincide near $\partial B$. Since $u\le u_ j\le u_1$, all these functions are bounded on $B_1$ uniformly in $j$, so there are constants $c$ and $N\ge 1$ with $-N\le u_ j-c,\, u-c\le -1$ on $B_1$. Put $\psi =|z-x|^2-9r^2$, choose $\delta _0>0$ with $L\subset \{ \psi <-\delta _0\} $, and let $\hat u_ j=\max \{ u_ j-c,A\psi \} $ and $\hat u=\max \{ u-c,A\psi \} $, with $A=N/\delta _0$, be the functions (7.3.2) on $B_1$, with $\delta _0$ in place of $\delta $. They are bounded and plurisubharmonic on $B_1$, $\hat u_ j\downarrow \hat u$, $\hat u_ j-\hat u=u_ j-u$ on $L$, and $\hat u_ j=\hat u$ on a collar of $\partial B$. By (7.4.6), applied with $u=\hat u$ and $h=\hat u_ j$,
\[ \operatorname {Cap}_ D\bigl (B\cap \{ \hat u_ j-\hat u>\delta /2\} \bigr ) \le (n+1)!\, (2/\delta )^{n+1}\int _ B(\hat u_ j-\hat u)(\mathrm{dd}^{\mathrm{c}}\hat u)^ n, \]
and the right-hand side tends to zero by dominated convergence, since $(\mathrm{dd}^{\mathrm{c}}\hat u)^ n$ has finite mass on $B$ and $0\le \hat u_ j-\hat u\le \hat u_1-\hat u$ decreases to zero. The real function $w_ j=u_ j-u$ is quasi-continuous on $D$, as a difference of two real quasi-continuous functions (theorem 7.5.2 and the remark following definition 7.5.1), and $w_ j=\hat u_ j-\hat u$ on $L$. Hence lemma 7.5.4 gives
\[ \operatorname {Cap}^*_ D(K\cap \{ w_ j>\delta \} )\le \operatorname {Cap}_ D(L\cap \{ w_ j>\delta /2\} )\le \operatorname {Cap}_ D\bigl (B\cap \{ \hat u_ j-\hat u>\delta /2\} \bigr )\longrightarrow 0. \]
For the second assertion, let $f_ a\to f$ be as in the statement, let $K\subset D$ be compact, and let $M$ bound $|f_ a|$ and $|f|$ on $K$. Splitting $K$ into the Borel set where $|f_ a-f|>\delta $ and its complement, and using proposition 7.4.3(3) together with $\operatorname {Cap}_ D\le \operatorname {Cap}^*_ D$ on Borel sets, we get
\[ \int _ K|f_ a-f|\, \mathrm{d}V_{2n} \le 2MC_ D\operatorname {Cap}_ D^*(K\cap \{ |f_ a-f|>\delta \} ) +\delta |K|. \]
Taking the limit along the net and then letting $\delta \downarrow 0$ proves $L^1$ convergence on $K$. $\square$
Lemma 7.5.7.880C30054932 Let $B\subset \mathbb C^ n$ be a ball, and let $\mathcal U$ be a family of plurisubharmonic functions on $B$ which is uniformly bounded on each compact subset of $B$.
For every compact set $K\subset B$ there is a constant $C$ such that, for all integers $0\le q\le n$, all $u_1,\ldots ,u_ q\in \mathcal U$ and all Borel sets $E\subset K$, the trace measure $\mathrm{dd}^{\mathrm{c}}u_1\wedge \cdots \wedge \mathrm{dd}^{\mathrm{c}}u_ q\wedge \beta ^{n-q}$, and the total variation of every coefficient measure of $\mathrm{dd}^{\mathrm{c}}u_1\wedge \cdots \wedge \mathrm{dd}^{\mathrm{c}}u_ q$, give $E$ mass at most $C\operatorname {Cap}_ B(E)$.
Let $S_ j=\mathrm{dd}^{\mathrm{c}}u_{1,j}\wedge \cdots \wedge \mathrm{dd}^{\mathrm{c}}u_{q,j}$ with $u_{l,j}\in \mathcal U$, and suppose that $S_ j\to S$ weakly. Let $f_ j$ be real Borel functions on $B$, uniformly bounded on each compact subset of $B$, which converge locally in capacity (for $\operatorname {Cap}_ B$) to a real Borel function $f$ that is locally bounded and quasi-continuous. Then $f_ jS_ j\to fS$ weakly, and the same holds for the trace measures of $S_ j$ and $S$. This applies in particular to a fixed weight $f_ j=f$.
Proof. (1) Let $B=B(a,R)$, fix $K$, and choose $R'<R$ with $K\subset B'=B(a,R')$. Let $M\ge 1$ be a common bound for $|u|$ on $B'$, $u\in \mathcal U$. For $u_1,\ldots ,u_ q\in \mathcal U$ put $v_ l=(u_ l+M)/(2M)$ and $\phi =|z-a|^2/R'^2$; these functions take values in $[0,1]$ on $B'$, and so does
\[ F=\frac1n\bigl (v_1+\cdots +v_ q+(n-q)\phi \bigr ), \]
which is therefore a competitor for $\operatorname {Cap}_{B'}$. Expand $(\mathrm{dd}^{\mathrm{c}}F)^ n$ by additivity, positive homogeneity and symmetry (proposition 7.2.2, theorem 7.3.7). All terms are positive measures, and those in which each $\mathrm{dd}^{\mathrm{c}}v_ l$ occurs once and $\mathrm{dd}^{\mathrm{c}}\phi $ occurs $n-q$ times add up to $c_{n,q}\, \mathrm{dd}^{\mathrm{c}}v_1\wedge \cdots \wedge \mathrm{dd}^{\mathrm{c}}v_ q\wedge (\mathrm{dd}^{\mathrm{c}}\phi )^{n-q}$, where $c_{n,q}=n^{-n}\, n!\, (n-q)^{n-q}/(n-q)!>0$ (with $0^0=1$) is the corresponding multinomial coefficient. Hence $\mathrm{dd}^{\mathrm{c}}v_1\wedge \cdots \wedge \mathrm{dd}^{\mathrm{c}}v_ q\wedge (\mathrm{dd}^{\mathrm{c}}\phi )^{n-q}\le c_{n,q}^{-1}(\mathrm{dd}^{\mathrm{c}}F)^ n$. Since $\mathrm{dd}^{\mathrm{c}}\phi =\beta /(\pi R'^2)$ and $\mathrm{dd}^{\mathrm{c}}u_ l=2M\mathrm{dd}^{\mathrm{c}}v_ l$, it follows that $\int _ E\mathrm{dd}^{\mathrm{c}}u_1\wedge \cdots \wedge \mathrm{dd}^{\mathrm{c}}u_ q\wedge \beta ^{n-q}\le C\operatorname {Cap}_{B'}(E)$ for Borel $E\subset B'$. The bound for the coefficient measures follows from theorem 5.3.7, and proposition 7.4.3(4), applied to $B'\Subset B$ and $K$, replaces $\operatorname {Cap}_{B'}$ by $\operatorname {Cap}_ B$ on subsets of $K$.
(2) The weak limit $S$ is a positive current (theorem 5.3.7). Let $\sigma _ j$ be either the trace measure or a coefficient measure of $S_ j$, and $\sigma $ the corresponding measure of $S$; then $\int \psi \, \mathrm d\sigma _ j\to \int \psi \, \mathrm d\sigma $ for every test function $\psi $, since this integral is the value of $S_ j$ on a test form. Fix a continuous function $\chi $ with compact support in $B$ and a compact set $K\subset B$ with $\operatorname{Supp}\chi \subset \operatorname{Int}K$, and let $C$ be the constant of (1) for $K$; in particular $|\sigma _ j|(K)\le C\operatorname {Cap}_ B(K)$, which is finite by proposition 7.4.3(1). A similar bound holds for $\sigma $ on open subsets $U\subset \operatorname{Int}K$: for the trace measure, which is the vague limit of the trace measures of $S_ j$ (section 5.3), proposition 2.6.2 gives $\sigma (U)\le \varliminf _ j\sigma _ j(U)\le C\operatorname {Cap}_ B(U)$, and the total variations of the coefficient measures of the positive current $S$ are bounded by a constant multiple of its trace measure.
Since the total variations $|\sigma _ j|(K)$ are bounded, the convergence $\int \psi \, \mathrm d\sigma _ j\to \int \psi \, \mathrm d\sigma $ extends to continuous functions $\psi $ with support in $\operatorname{Int}K$: approximating $\psi $ uniformly by test functions supported in a fixed compact subset of $\operatorname{Int}K$ (lemma 2.5.1(5)) changes the integrals by at most $\sup |\psi -\psi _\varepsilon |$ times a bound for $|\sigma _ j|(K)$ and $|\sigma |(K)$.
Consider first the fixed weight $f$, and let $\eta >0$. Since $f$ is quasi-continuous, there is an open set $G\subset B$ with $\operatorname {Cap}_ B(G)<\eta $ such that $f$ is continuous on $K\setminus G$; let $g$ be a continuous extension of $f|_{K\setminus G}$ to $\mathbb C^ n$ with $\sup |g|\le \sup _ K|f|$ (lemma 7.5.3). Then
\[ \Bigl |\int \chi f\, \mathrm{d}\sigma _ j-\int \chi g\, \mathrm{d}\sigma _ j\Bigr |\le 2\sup _ K|f|\sup |\chi |\, |\sigma _ j|(G\cap \operatorname{Supp}\chi )\le C'\eta , \]
and similarly for $\sigma $, using the open set $G\cap \operatorname{Int}K$. Since $\chi g$ is continuous with support in $\operatorname{Int}K$, we have $\int \chi g\, \mathrm{d}\sigma _ j\to \int \chi g\, \mathrm{d}\sigma $. Hence $\varlimsup _ j|\int \chi f\, \mathrm{d}\sigma _ j-\int \chi f\, \mathrm{d}\sigma |\le 2C'\eta $ for every $\eta >0$, which proves $f\sigma _ j\to f\sigma $.
For varying weights, let $M_ K$ bound $|f_ j|$ and $|f|$ on $K$, and let $\delta >0$. By (1),
\[ \Bigl |\int \chi (f_ j-f)\, \mathrm{d}\sigma _ j\Bigr | \le \sup |\chi |\Bigl (\delta \, C\operatorname {Cap}_ B(K)+2M_ KC\operatorname {Cap}^*_ B\bigl (K\cap \{ |f_ j-f|>\delta \} \bigr )\Bigr ). \]
As $j\to \infty $, the second term tends to zero by convergence in capacity; then we let $\delta \downarrow 0$. Together with the fixed-weight case, this proves $f_ j\sigma _ j\to f\sigma $, and hence $f_ jS_ j\to fS$. $\square$
Theorem 7.5.8.42828E3BDDFA Let $0\le q\le n$ be an integer. For $1\le l\le q$, let $(u_{l,j})_{j\ge 1}$ be a sequence of plurisubharmonic functions on $D$, uniformly bounded on each compact subset of $D$, which converges locally in capacity to a locally bounded plurisubharmonic function $u_ l$. Then
\[ \mathrm{dd}^{\mathrm{c}}u_{1,j}\wedge \cdots \wedge \mathrm{dd}^{\mathrm{c}}u_{q,j} \longrightarrow \mathrm{dd}^{\mathrm{c}}u_1\wedge \cdots \wedge \mathrm{dd}^{\mathrm{c}}u_ q \]
weakly. More generally, if $f_ j$ are real Borel functions on $D$, uniformly bounded on each compact subset of $D$, which converge locally in capacity to a locally bounded, quasi-continuous real Borel function $f$, then $f_ j\mathrm{dd}^{\mathrm{c}}u_{1,j}\wedge \cdots \wedge \mathrm{dd}^{\mathrm{c}}u_{q,j}\to f\mathrm{dd}^{\mathrm{c}}u_1\wedge \cdots \wedge \mathrm{dd}^{\mathrm{c}}u_ q$ weakly.
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. $\square$
Unlike theorem 7.3.7, theorem 7.5.8 concerns only products with the base current $T=1$.
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