\[ d_\theta (u,v)=2\, \overline{\operatorname {vol}}_\theta (\max (u,v)) -\overline{\operatorname {vol}}_\theta (u)-\overline{\operatorname {vol}}_\theta (v) \]
and, for $\delta >0$, the set $\mathcal M_\delta (\theta )$ of ordinary models $u=P_\theta [u]$ with $\underline{\operatorname {vol}}_\theta (u)\geq \delta $. For models $\varphi _ j\to \varphi $ in this comparison, theorem 13.23.3 provides increasing lower models whose upper volumes approach that of $\varphi $. Projecting finite families of potentials onto these lower models makes their rooftops comparable. To keep the limit in the relative class of $\varphi $, domination of the measures is needed only where the potentials lie below one fixed member of that class; domination on all of $X$ gives, in addition, convergence in capacity and convergence of the measures.
Proof.
We first construct the lower models $\psi _ j$, then a common lower bound for rooftops of the $u_ l$ in the classes $\mathcal E_{\psi _ j}$, and finally treat domination on all of $X$.
By Hartogs' lemma, $u_ j$, $u$ and $\chi $ have a common upper bound; translate them to be nonpositive. The envelope identity then gives $u_ j\leq \varphi _ j$. Since $\varphi _ j\to \varphi $ in $L^1$ by proposition 13.22.4, it follows that $u\leq \varphi $. Put $\Lambda =\nu +\mu _\chi $. Using lemma 5.44.2, extract a subsequence along which $u_ j\to u$ almost everywhere for smooth volume and
\[ I_ j:=\int _ X|\mathrm{e}^{u_ j}-\mathrm{e}^ u|\, \mathrm{d}\Lambda \leq 2^{-j}. \]
Apply theorem 13.23.3 to extract further, keeping these estimates, and obtain increasing models $w_ j\leq \varphi _ j,\varphi $ with lower volume at least $\delta /2$ and $d_\theta (w_ j,\varphi )\to 0$. For $l\geq j$ we have $w_ j\leq w_ l\leq \varphi _ l$. Hence $\psi _ j$ is not identically $-\infty $, and
\[ w_ j\leq \psi _ j\leq \varphi . \]
The upper bound follows by passing to the $L^1$ limit in $\psi _ j\leq \varphi _ l$ as $l\to \infty $. Monotonicity of the ordinary envelope gives $P_\theta [\psi _ j]\leq \varphi _ l$ for all $l\geq j$, so $\psi _ j$ is an ordinary model. It increases with $j$, has lower volume at least $\delta /2$, and
2A2692F560B3
10.31.31
\begin{equation} \overline{\operatorname {vol}}_\theta (\psi _ j) \longrightarrow \overline{\operatorname {vol}}_\theta (\varphi ). \end{equation}
Set $g_ j=P_\theta [\psi _ j](\chi )$. Proposition 10.31.26, with $(\varphi ,\eta )$ replaced by $(\psi _ j,\varphi )$, gives $g_ j\in \mathcal E_{\psi _ j}$, $g_ j\leq \chi $ and $\mu _{g_ j}\leq \mu _\chi $. Moreover $g_ j$ increases with $j$. For $N\geq j$, consider
\[ q_{j,N}=P_\theta (g_ j,u_ j,\ldots ,u_ N). \]
For each $l\geq j$, the projection $P_\theta [\psi _ j](u_ l)$ belongs to $\mathcal E_{\psi _ j}$ by the same proposition, now with upper reference $\varphi _ l$. The finite rooftop of these projections and $g_ j$ lies below $q_{j,N}$. Since $q_{j,N}\leq g_ j$, the finite-rooftop proposition and upward stability give $q_{j,N}\in \mathcal E_{\psi _ j}$, and hence $\mu _{q_{j,N}}(X)\geq \delta /2$.
Partition the contact set of $q_{j,N}$ into disjoint pieces, one for each obstacle. On a piece where $q_{j,N}=u_ l$, we have $u_ l\leq g_ j\leq \chi $, so the hypothesis bounds the measure there by $\nu $. On the piece where $q_{j,N}=g_ j$, the measure is bounded by $\mu _{g_ j}\leq \mu _\chi $. Consequently $\mu _{q_{j,N}}\leq \Lambda $. Put $F_ j=\min (\mathrm{e}^ u,\mathrm{e}^{g_ j})$. On the $u_ l$ piece,
\[ |\mathrm{e}^{q_{j,N}}-F_ j|\leq |\mathrm{e}^{u_ l}-\mathrm{e}^ u|. \]
On the $g_ j$ piece, $g_ j=q_{j,N}\leq u_ N$, so the error is $(\mathrm{e}^{g_ j}-\mathrm{e}^ u)_+\leq |\mathrm{e}^{u_ N}-\mathrm{e}^ u|$. The $u_ N$ piece and the $g_ j$ piece are controlled by the two summands $\nu $ and $\mu _\chi $ of $\Lambda $, so together they contribute at most $I_ N$. Thus
4D376181A695
10.31.32
\begin{equation} \int _ X|\mathrm{e}^{q_{j,N}}-F_ j|\, \mu _{q_{j,N}} \leq \sum _{l=j}^ N I_ l\leq 2^{1-j}. \end{equation}
Since $F_ j\geq F_1>0$ outside a pluripolar set, we may choose $\epsilon >0$ with $\Lambda (\{ F_1\leq \epsilon \} )<\delta /4$. The lower mass bound and the domination by $\Lambda $ give
\[ \epsilon \delta /4 \leq \int _ X F_ j\, \mu _{q_{j,N}} \leq 2^{1-j}+\Lambda (X)\mathrm{e}^{\sup _ Xq_{j,N}}. \]
For every sufficiently large $j$, this bounds $\sup _ Xq_{j,N}$ from below, uniformly in $N$. Hence the decreasing limit $q_ j$ as $N\to \infty $ is not identically $-\infty $. Applying theorem 10.31.5 in the fixed class $\mathcal E_{\psi _ j}$, with dominating measure $\Lambda $, gives $q_ j\in \mathcal E_{\psi _ j}$. Now
\[ q_ j\leq h_ j\leq \psi _ j, \]
so $h_ j\in \mathcal E_{\psi _ j}$ by upward stability. We discard the finitely many earlier indices. Since $h_ j\leq u\leq \varphi $, monotonicity and (10.31.31) show that $u$ and $\varphi $ have the same upper volume, and theorem 13.18.5 gives $u\in \mathcal E_\varphi $.
Suppose now that $\mu _{u_ j}\leq \nu $ on $X$. The finite rooftops $p_{j,N}=P_\theta (u_ j,\ldots ,u_ N)$ are not identically $-\infty $, since they lie above $h_ j$. Partitioning their contact sets gives
\[ \mu _{p_{j,N}}\leq \nu ,\qquad \int _ X|\mathrm{e}^{p_{j,N}}-\mathrm{e}^ u|\, \mu _{p_{j,N}}\leq 2^{1-j}. \]
For fixed $j$, they decrease to $h_ j$ as $N\to \infty $. These lower envelopes increase to a regularized limit $h\leq u\leq \varphi $, and their upper volumes tend to that of $\varphi $; hence $h\in \mathcal E_\varphi $. Both monotone limits hold in capacity by lemma 5.29.19. Applying lemma 13.14.8 twice gives
\[ \int _ X|\mathrm{e}^ h-\mathrm{e}^ u|\, \mu _ h=0. \]
Hence $\mu _ h(\{ h<u\} )=0$, and zero domination gives $h=u$. The upper envelopes $(\sup _{k\geq j}u_ k)^*$ decrease to $u$ by Hartogs' lemma and the almost-everywhere convergence, and squeezing gives convergence in capacity. Finally,
\[ \mu _{u_ j}(\{ u_ j\leq -T\} )\leq \nu (\{ h_1\leq -T\} )\longrightarrow 0 \]
uniformly in $j$, and the tail criterion theorem 13.15.23 gives weak convergence of the measures. Since the extraction can start from any subsequence, both conclusions hold for the original sequence.
$\square$
Pome
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