7.6 Increasing limits and exceptional envelopesE5550B44EC7B

Proof. We first note that $u_{l,j}\le u_ l$ everywhere. Indeed, $u_{l,j}\le u_ l$ almost everywhere, so the plurisubharmonic functions $\max \{ u_{l,j},u_ l\} $ and $u_ l$ agree almost everywhere, and hence everywhere, by the uniqueness of plurisubharmonic representatives in theorem 5.9.1, applied on each connected component of $D$. Consequently $u_{l,1}\le u_{l,j}\le u_ l$, and the sequences are uniformly bounded on each compact subset of $D$.

As in the proof of theorem 7.3.7, we prove the weighted assertion by induction on $q$, for all bounded open sets $D$ and all sequences as in the statement; convergence in capacity is proved at the end. For $q=0$ it follows from dominated convergence with respect to Lebesgue measure. Let $q\ge 1$ and assume the weighted assertion for $q-1$ factors; applying $\mathrm{dd}^{\mathrm{c}}$ to it, with the weight $u_{1,j}$, gives the unweighted assertion for $q$ factors. Put $S_ j=\mathrm{dd}^{\mathrm{c}}u_{1,j}\wedge \cdots \wedge \mathrm{dd}^{\mathrm{c}}u_{q,j}$ and $S=\mathrm{dd}^{\mathrm{c}}u_1\wedge \cdots \wedge \mathrm{dd}^{\mathrm{c}}u_ q$, so that $S_ j\to S$ weakly.

Weak convergence is a local property (lemma 7.3.6). Fix $x\in D$ and $r>0$ with $\overline{B(x,4r)}\subset D$, and put $B=B(x,3r)$ and $B_1=B(x,4r)$; we prove the convergence on $B(x,2r)$. Subtracting constants does not affect the assertions, since the unweighted assertion for $q$ factors is known, so we may assume that $-N\le u_{l,j},u_ l\le -1$ on $B_1$ for all $l$ and $j$, where $N\ge 1$. Put $\psi =|z-x|^2-9r^2$, choose $\delta _0>0$ such that $\overline{B(x,2r)}\subset \{ \psi <-\delta _0\} $, and replace each function $v$ among the $u_{l,j}$ and $u_ l$ by the function $\max \{ v,A\psi \} $ of (7.3.2), with $U=B_1$, $\delta _0$ in place of $\delta $ and $A=N/\delta _0$. The new functions are plurisubharmonic on $B_1$ and take values in $[-N,0)$ on $B$; the new sequences still increase, satisfy $u_{l,j}\le u_ l$ and converge almost everywhere. The functions do not change on $B(x,2r)$, and by locality neither do the products. On the collar $V=\{ z\in B_1:\psi (z)>-1/A\} $ of $\partial B$ the new functions coincide with $A\psi $.

By locality, on $V$ all the currents considered below coincide with fixed currents, independent of $j$ and $k$. Hence, if $\chi \in C^\infty _ c(B)$ equals $1$ on $B\setminus V$, the integral over $B$ of each of these currents splits into its integral against $\chi $, to which weak convergence applies, and its integral against $(1-\chi )\mathbf1_ B$, which only involves values on $V$ and therefore does not change; we use this without further comment. The currents $-u_{0,j}S_ j$ are positive on $B$, and their masses on compact subsets of $B$ are bounded uniformly in $j$ by theorem 7.2.5; by the compactness part of theorem 5.3.7, every subsequence of $(u_{0,j}S_ j)$ has a further subsequence converging weakly on $B$. Let $\Theta $ be the limit of such a subsequence.

Since $u_{0,j}\le u_0$, the currents $(u_0-u_{0,j})S_ j$ are positive. Let $\varphi $ be a strongly positive test form on $B$ of complementary bidegree. The positive measures $-u_0S_ j\wedge \varphi $ have bounded masses on compact sets, so by theorem 2.6.5 a further subsequence of $u_0S_ j\wedge \varphi $ converges weakly to some $\nu $, and lemma 7.3.4, applied to the constant sequence $f_ j=u_0$ and the measures $S_ j\wedge \varphi \to S\wedge \varphi $, gives $\nu \le u_0S\wedge \varphi $. The measures $(u_0-u_{0,j})S_ j\wedge \varphi $ are positive and converge along this subsequence to $\nu -\Theta \wedge \varphi $, which is therefore positive. Hence $\Theta \wedge \varphi \le u_0S\wedge \varphi $. Evaluating this inequality of measures on a nonnegative test function equal to $1$ on $\operatorname{Supp}\varphi $, we get $(u_0S-\Theta )(\varphi )\ge 0$. Thus $u_0S-\Theta $ is a positive current on $B$, and it vanishes on $V$, since all the currents $u_{0,j}S_ j$ and $u_0S$ coincide there with $A\psi (A\mathrm{dd}^{\mathrm{c}}\psi )^ q$. Write $\gamma =\beta ^{n-q}$.

For fixed $k$ and all $j\ge k$, positivity of $S_ j\wedge \gamma $ and $u_{0,j}\ge u_{0,k}$ give

Let $j\to \infty $ along the subsequence defining $\Theta $. The left-hand side tends to $\int _ B\Theta \wedge \gamma $. The fixed weight $u_{0,k}$ is quasi-continuous for $\operatorname {Cap}_ B$ by theorem 7.5.2, and the functions $u_{l,j}$, $u_ l$ form a family which is uniformly bounded on $B$; so by lemma 7.5.7(2) the right-hand side tends to $\int _ Bu_{0,k}S\wedge \gamma $. By the second assertion of corollary 7.3.8, applied with $a=u_{0,k}$, $b=u_1$ and $R=\mathrm{dd}^{\mathrm{c}}u_2\wedge \cdots \wedge \mathrm{dd}^{\mathrm{c}}u_ q\wedge \gamma $, and by symmetry,

As $k\to \infty $, the unweighted assertion for $q$ factors, already established, gives $\mathrm{dd}^{\mathrm{c}}u_{0,k}\wedge \mathrm{dd}^{\mathrm{c}}u_2\wedge \cdots \wedge \mathrm{dd}^{\mathrm{c}}u_ q\to \mathrm{dd}^{\mathrm{c}}u_0\wedge \mathrm{dd}^{\mathrm{c}}u_2\wedge \cdots \wedge \mathrm{dd}^{\mathrm{c}}u_ q$ weakly, and lemma 7.5.7, with the fixed quasi-continuous weight $u_1$, passes the integral to the limit. A second application of corollary 7.3.8 identifies the limit with $\int _ Bu_0\, \mathrm{dd}^{\mathrm{c}}u_1\wedge \cdots \wedge \mathrm{dd}^{\mathrm{c}}u_ q\wedge \gamma =\int _ Bu_0S\wedge \gamma $. We have shown that $\int _ B\Theta \wedge \gamma \ge \int _ Bu_0S\wedge \gamma $.

Since $u_0S-\Theta $ is positive with compact support in $B$, its trace measure has total mass $\int _ B(u_0S-\Theta )\wedge \gamma \le 0$ and therefore vanishes, and $\Theta =u_0S$ by theorem 5.3.7. Thus every subsequence of $(u_{0,j}S_ j)$ has a further subsequence converging weakly to $u_0S$ on $B$. Hence the whole sequence converges to $u_0S$ on $B$: otherwise there would be a test form $\varphi $ on $B$ and a subsequence along which $|(u_{0,j}S_ j-u_0S)(\varphi )|$ stays bounded away from zero, and no further subsequence of it could converge to $u_0S$. Since the modifications change neither the functions nor the weighted products on $B(x,2r)$, the original weighted products converge on $B(x,2r)$; this completes the induction.

Now that both assertions are known for every $q\le n$, we prove convergence in capacity. Fix $l$, a compact set $K\subset D$ and $\delta >0$. Since $u_{l,j}\le u_ l$, the sets $\{ u_{l,j}>u_ l+\delta \} $ are empty. Covering $K$ by finitely many balls and using subadditivity of outer capacity, we may assume that $K\subset \overline{B(x,r)}$ with $\overline{B(x,4r)}\subset D$; put $L=\overline{B(x,3r/2)}$. Let $\hat u_{l,j}$ and $\hat u_ l$ be the functions obtained by the modification above, on $B=B(x,3r)$; on $\{ \psi <-\delta _0\} \supset L$ we have $\hat u_ l-\hat u_{l,j}=u_ l-u_{l,j}$. These functions are bounded and plurisubharmonic on $B_1$, they satisfy $\hat u_{l,j}\le \hat u_ l$, and both coincide with $A\psi $ on the collar $V$ of $\partial B$. By (7.4.6), applied with $u=\hat u_{l,j}$ and $h=\hat u_ l$,

The right-hand side tends to zero: $\int _ B\hat u_ l(\mathrm{dd}^{\mathrm{c}}\hat u_{l,j})^ n\to \int _ B\hat u_ l(\mathrm{dd}^{\mathrm{c}}\hat u_ l)^ n$ by the unweighted assertion and lemma 7.5.7(2) with the fixed quasi-continuous weight $\hat u_ l$, and $\int _ B\hat u_{l,j}(\mathrm{dd}^{\mathrm{c}}\hat u_{l,j})^ n\to \int _ B\hat u_ l(\mathrm{dd}^{\mathrm{c}}\hat u_ l)^ n$ by the weighted assertion with all factors and the weight equal to $\hat u_{l,j}$; as before, the integrals over $B$ are split by a cutoff. The real function $u_ l-u_{l,j}$ is quasi-continuous on $D$ (theorem 7.5.2 and the remark following definition 7.5.1), so lemma 7.5.4 gives

Proof. By theorem 5.10.2, $u=F^*$ is plurisubharmonic, and there is a countable subfamily $\{ w_1,w_2,\ldots \} \subset \mathcal F$ with $(\sup _ kw_ k)^*=u$. The functions $v_ j=\max \{ w_1,\ldots ,w_ j\} $ are plurisubharmonic (proposition 5.9.5) and increase to $G=\sup _ kw_ k$, and $G=G^*=u$ almost everywhere by theorem 5.10.2, applied to the countable family. Fix an integer $m\ge 1$. The functions $\max \{ v_ j,-m\} $ are locally bounded and plurisubharmonic on $D$, and they increase almost everywhere to the locally bounded plurisubharmonic function $\max \{ u,-m\} $, so by theorem 7.6.1 they converge to it locally in capacity. For rational $\delta >0$, the set where $\sup _ j\max \{ v_ j,-m\} <\max \{ u,-m\} -\delta $ is contained in $\{ \max \{ v_ k,-m\} <\max \{ u,-m\} -\delta \} $ for every $k$, and the outer capacities of the intersections of the latter sets with a compact set tend to zero as $k\to \infty $. Taking the union over rational $\delta $, countable subadditivity shows that the set

meets every compact set in a set of outer capacity zero.

If $\sup _ jv_ j(x)<u(x)$, then $u(x)>-\infty $, so $u(x)>-m$ for some integer $m$, and then $x\in E_ m$. Hence $\{ \sup _ jv_ j<u\} \subset \bigcup _ mE_ m$ also meets every compact set in a set of outer capacity zero. Finally, $\sup _ jv_ j\le F\le u$, so $\{ F<F^*\} \subset \{ \sup _ jv_ j<u\} $. (Since outer capacity is defined for all sets, we need not know whether $\{ F<F^*\} $ is Borel.) $\square$