Corollary 16.9.14.F84D53FDA599 Suppose that $(\varphi _ i)_{i\in I}$ is a net in $\mathrm{PSH}(X,\theta )$ such that
\[ \varphi _ j\preceq _ P\varphi _ i \]
whenever $j\geq i$. Let $\varphi \in \mathrm{PSH}(X,\theta )$ be such that $\varphi \preceq _ P\varphi _ i$ for all $i\in I$. Then the following are equivalent:
3D1A7D703ADF
We have
\[ \varphi _ i\xrightarrow {d_ S} \varphi ; \]
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for each $k=0,\ldots ,n$, we have
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16.9.15
\begin{equation} \lim _{j\in I}\int _ X \theta _{\varphi _ j}^ k\wedge \theta _{V_{\theta }}^{n-k} = \int _ X \theta _{\varphi }^ k\wedge \theta _{V_{\theta }}^{n-k}. \end{equation}
If we assume furthermore that $\int _ X \theta _{\varphi }^ n>0$, then the above conditions are equivalent to the following:
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We have
\[ \lim _{j\in I}\int _ X \theta _{\varphi _ j}^ n=\int _ X \theta _{\varphi }^ n. \]
When $\int _ X \theta _{\varphi }^ n>0$ and these equivalent conditions are met, we also have
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16.9.16
\begin{equation} P_{\theta }[\varphi ]=\inf _{j\in I}P_{\theta }[\varphi _ j]. \end{equation}
Proof.
1 $\iff $ 2. This follows immediately from 16.9.7.
Assume that $\int _ X \theta _{\varphi }^ n>0$.
2 $\implies $ 1. This is the case $k=n$ in (16.9.15).
1 $\implies $ 2. Assume (3). If for some $i\in I$, $\int _ X \theta _{\varphi _ i}^ n= \int _ X \theta _{\varphi }^ n$, then for all $j\geq i$ we have
\[ \varphi \preceq _ P\varphi _ j\preceq _ P\varphi _ i. \]
Hence $\int _ X\theta _{\varphi _ j}^ n=\int _ X\theta _{\varphi }^ n$ by proposition 16.8.22. It follows from proposition 16.8.7 that $\varphi _ j\sim _ P\varphi $, and (2) follows from proposition 16.3.10. So in the sequel, we shall assume that for all $i\in I$, we have $\int _ X \theta _{\varphi _ i}^ n>\int _ X \theta _{\varphi }^ n$. In particular, each $\varphi _ i$ has positive mass.
Let $(b_ j)_{j\in I}$ be a net converging to $\infty $ such that
\[ b_ j\in \left( 1, \left(\frac{\int _ X \theta _{\varphi _ j}^ n}{\int _ X \theta _{\varphi _ j}^ n-\int _ X \theta _{\varphi }^ n}\right)^{1/n} \right). \]
This can be achieved for example by taking $b_ j$ as the midpoint of this interval. By theorem 16.8.23, for each $j\in I$, we can find $\eta _ j\in \mathrm{PSH}(X,\theta )$ such that
\[ b_ j^{-1}\eta _ j +\left(1-b_ j^{-1}\right)\varphi _ j \leq \varphi . \]
It follows from the monotonicity theorem theorem 13.13.11 that for any $k=0,\ldots ,n$,
\[ \int _ X \theta _{\varphi }^ k \wedge \theta _{V_{\theta }}^{n-k}\geq \left(1-b_ j^{-1}\right)^ k\int _ X \theta _{\varphi _ j}^ k \wedge \theta _{V_{\theta }}^{n-k}. \]
Taking the limit, we conclude the $\leq $ direction in (16.9.15). The $\geq $ direction follows from proposition 16.8.22.
Finally, we argue (16.9.16). Let $\psi _ j=P_{\theta }[\varphi _ j]$. It follows from corollary 16.3.20 that $\psi _ j$ is a model potential. Since the net $(\varphi _ j)_{j\in I}$ is decreasing with respect to the $P$-preorder, $(\psi _ j)_{j\in I}$ is decreasing pointwise. Let
\[ \psi =\inf _{j\in I}\psi _ j\geq P_{\theta }[\varphi ]. \]
By proposition 16.4.3, $\psi $ is a model potential. Since $P_{\theta }[\varphi ]\leq \psi \leq \psi _ j$ for all $j\in I$, the monotonicity theorem theorem 13.13.11 and proposition 16.3.10 give
\[ \int _ X \theta _{\varphi }^ n=\int _ X\theta _{P_{\theta }[\varphi ]}^ n\leq \int _ X \theta _{\psi }^ n\leq \lim _{j\in I}\int _ X \theta _{\psi _ j}^ n=\lim _{j\in I}\int _ X \theta _{\varphi _ j}^ n=\int _ X \theta _{\varphi }^ n. \]
Hence $\psi $ has the same mass as $P_{\theta }[\varphi ]$. As $\psi $ is model and $\psi \geq P_{\theta }[\varphi ]$, theorem 16.3.15 gives $\psi =P_{\theta }[\varphi ]$.
$\square$
Pome
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