Theorem 13.21.1453D4DA50352

Proof. Step 1: the envelopes. Translate so that $u\leq 0$, and set $p_ j=P_\theta [u_ j]$, $p=P_\theta [u]$ and $s=(\lim _ jp_ j)^*$. The $p_ j$ increase, and $u\leq s\leq p\leq V_\theta $. By theorem 13.19.5, the lower volume of $p$ is positive and $u\in \mathcal E_{\mathrm{env}}(X,\theta ,p)$. Upward stability (proposition 16.28.62) puts $s$ in the same class.

The ordinary envelope can use $V_\theta $ in place of zero as obstacle (see the remark after definition 5.40.4); that is, $p_ j=P_\theta [u_ j](V_\theta )$. Hence lemma 13.19.4 gives

The fixed bounded quasi-continuous weight $\chi =\min (1,(p-s)_+)$, defined arbitrarily on the pluripolar set where $p-s$ is undefined, vanishes on $\{ p_ j=V_\theta \} $, because $p_ j\leq s\leq p\leq V_\theta $. Since $p_ j\to s$ in capacity by lemma 5.29.19, lemma 13.14.8 gives $\int _ X\chi \, \mu _ s=0$. Thus $\mu _ s$ vanishes on $\{ s<p\} $, and theorem 13.19.1, with reference $p$ and coefficient $c=0$, yields $s=p$. This proves the assertions about the envelopes.

Step 2: the upper volumes of the envelopes. Choose any $w\in \mathrm{PSH}(X,\theta )$ with $w\sim p$, and $K\geq 0$ with $p-K\leq w\leq p+K$. Define $r_ j=P_\theta [u_ j](w)$; this is meaningful because $u_ j-K\leq p-K\leq w$, so that $P_\theta (u_ j,w)\not\equiv -\infty $. For $C\geq 0$ the competitors compare as follows:

The leftmost potential lies below $u_ j+C$ and below $p_ j-K\leq p-K\leq w$. For the upper bound, note that the middle term minus $K$ lies below $w-K\leq p\leq 0$ and is more singular than $u_ j$, so it is at most $p_ j$. Taking the increasing limit in $C$ gives

The $r_ j$ increase with $j$ and converge in capacity to $r=(\lim _ jr_ j)^*$, and $p-K\leq r\leq w\leq p+K$. Thus $r\sim p$, so $r\in \mathcal E_{\mathrm{env}}(X,\theta ,p)$. Lemma 13.19.4 gives $\mu _{r_ j}\leq \mathbf1_{\{ r_ j=w\} }\mu _ w$, and weighted lower semicontinuity with the weight $\min (1,(w-r)_+)$ shows that $\mu _ r$ vanishes on $\{ r<w\} $. Domination with reference $p$ gives $r=w$. Therefore

where $r_ j\sim p_ j$ justifies the last inequality. Taking the supremum over $w\sim p$ and using volume monotonicity proves

Step 3: the volumes of $u_ j$. Put $\delta =\underline{\operatorname {vol}}_\theta (p)>0$ and $D_ j=\overline{\operatorname {vol}}_\theta (p)-\overline{\operatorname {vol}}_\theta (p_ j)$. Since $D_ j\to 0$, we can choose $b_ j>1$ with $b_ j\to \infty $ and $b_ j^ nD_ j<\delta $ for all large $j$. With $a_ j=1-b_ j^{-1}$, lemma 13.18.1 gives $h_ j\in \mathrm{PSH}(X,\theta )$ such that

for all sufficiently large $j$. Addition, homogeneity and monotonicity of the lower volume, together with theorem 13.19.5, give

Since $a_ j\to 1$, this proves the convergence of the lower volumes, and shows that $\underline{\operatorname {vol}}_\theta (u_ j)>0$ for all large $j$. For such $j$, theorem 13.19.5 identifies the upper volumes of $u_ j$ and $p_ j$, and also those of $u$ and $p$; Step 2 therefore gives the asserted upper-volume limit. Finally, proposition 13.18.3, applied to $u_ j\preceq u$, gives the uniform convergence of Lelong numbers on each fixed modification. $\square$