PomeTheorem 13.22.7.62ECCE0A41E6 Every $d_\theta $-Cauchy sequence in $\mathcal M_\delta (\theta )$ converges for $d_\theta $ to a unique element of $\mathcal M_\delta (\theta )$. It also converges to that element in $L^1(X,\mathrm{d}V)$.
Theorem 13.22.762ECCE0A41E6
Proof. The plan is to dominate a fast subsequence by the ordinary envelopes of its tail suprema; these decrease to the limit, and the stretching lemma controls the volume they add. Choose a subsequence $a_ j$ of the given sequence such that
\[ d_\theta (a_ j,a_{j+1})<\delta \, 2^{-(j+1)n} \qquad (j\geq 1). \]
For $k\geq 0$, set
\[ v_ j^ k=\max (a_ j,\ldots ,a_{j+k}),\qquad v_ j=\left(\sup _{l\geq j}a_ l\right)^*. \]
For every $l\geq j$, lemma 13.18.1 with $b=2^ l$, applied to $a_ l\preceq \max (a_ l,a_{l+1})$, gives $h_ l\leq a_ l$ satisfying
\[ (1-2^{-l})\max (a_ l,a_{l+1})+2^{-l}h_ l\leq a_ l. \]
The argument by cases in the proof of proposition 13.22.5 gives
\[ (1-2^{-l})v_ j^{l+1-j}+2^{-l}h_ l\leq v_ j^{l-j}. \]
Consequently
\[ U(v_ j^{l+1-j})-U(v_ j^{l-j})\leq nM2^{-l}. \]
Summing from $l=j$ to $j+k-1$ and then letting $k\to \infty $, using theorem 13.21.1, yields
\[ 0\leq U(v_ j)-U(a_ j)\leq nM2^{1-j}. \]
Here $v_ j\in \mathrm{PSH}(X,\theta )$ has lower volume at least $\delta $, because $a_ j\leq v_ j\leq 0$; this also justifies the use of that theorem.
Put $q_ j=P_\theta [v_ j]$. By theorem 13.19.5, $q_ j$ is an ordinary model with the same two volumes as $v_ j$. The sequence $q_ j$ decreases, and each $q_ j$ has supremum zero. By normalized compactness, its decreasing limit $q$ is not identically $-\infty $, so $q\in \mathrm{PSH}(X,\theta )$. By theorem 13.21.2,
\[ q=P_\theta [q],\qquad \underline{\operatorname {vol}}_\theta (q) =\lim _ j\underline{\operatorname {vol}}_\theta (q_ j)\geq \delta , \qquad U(q_ j)\longrightarrow U(q). \]
Both $a_ j$ and $q$ lie below $q_ j$. Hence
\[ \begin{split} d_\theta (a_ j,q) & \leq \bigl (U(q_ j)-U(a_ j)\bigr )+\bigl (U(q_ j)-U(q)\bigr )\\ & \leq nM2^{1-j}+U(q_ j)-U(q)\longrightarrow 0. \end{split} \]
To treat the whole sequence, apply (13.22.6) with a late member $a_ j$ of the subsequence as the middle point. The Cauchy property makes the first term small for all sufficiently late members of the original sequence, while $d_\theta (a_ j,q)$ makes the other two terms small. Thus the original sequence converges to $q$. Uniqueness follows from proposition 13.22.5, and the $L^1$ estimate in proposition 13.22.4 proves the last assertion. $\square$
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