Proof.
The decreasing case is already proved in proposition 16.5.20, so let us focus on the case where $(\varphi _ i)_ i$ is increasing.
We first handle the case where $I=\mathbb {Z}_{>0}$. Namely, $(\varphi _ i)_ i$ is an increasing sequence.
Step 1. We first prove the result when $\varphi _ j,\varphi \in \mathcal{E}^{\infty }(X,\theta ;\phi )$ for each $j\geq 1$.
Due to corollary 13.13.12, we have
\[ \lim _{j\to \infty } \int _ X \theta _{\varphi }^ k\wedge \theta _{\varphi _ j}^{n-k}=\int _ X \theta _{\varphi }^ n \]
for each $k=0,\ldots ,n$. By (16.5.15), we have
\[ \begin{aligned} 0\leq & E_{\theta }^{\phi }(\varphi )-E_{\theta }^{\phi }(\varphi _ j)\\ =& \frac{1}{n+1}\sum _{k=0}^ n \int _ X(\varphi -\varphi _ j)\, \theta _{\varphi }^ k\wedge \theta _{\varphi _ j}^{n-k}. \end{aligned} \]
We then apply theorem 13.13.9 to conclude that the right-hand side converges to $0$.
Step 2. We prove the general case. The $\leq $ direction in (16.5.38) follows from the monotonicity of $E_{\theta }^{\phi }$, as proved in 16.5.14. It remains to prove the $\geq $ direction.
After subtracting a constant from all $\varphi _ j$ and from $\varphi $, which shifts all the energies by the same constant in view of (16.5.13), we may assume that $\varphi \leq \phi $. For each $C>0$ and $j\geq 1$, we let
\[ \varphi _ C\coloneqq \varphi \lor (\phi -C),\quad \varphi _{j,C}\coloneqq \varphi _ j\lor (\phi -C). \]
By Step 1, for each $C>0$, we have
\[ \lim _{j\to \infty } E_{\theta }^{\phi }(\varphi _{j,C})= E_{\theta }^{\phi }(\varphi _{C})\geq E_{\theta }^{\phi }(\varphi ). \]
It suffices therefore to show that
7AAAE969D14B
16.5.39
\begin{equation} \lim _{C\to \infty } E_{\theta }^{\phi }(\varphi _{j,C})-E_{\theta }^{\phi }(\varphi _{j})=0\quad \textup{uniformly for $j$ in a tail}. \end{equation}
Fix $i\geq 1$. For $j\geq i$, we compute
\[ \begin{aligned} & E_{\theta }^{\phi }(\varphi _{j,C})-E_{\theta }^{\phi }(\varphi _{j})\\ \leq & \int _ X \left( \varphi _{j,C}-\varphi _{j}\right)\, \theta _{\varphi _{j}}^ n & \textup{by proposition \href{../../tag/51CFFEE97C34/}{16.5.36}}\\ \leq & \int _{\{ \varphi _ j\leq \phi -C\} } (\phi -C-\varphi _ j)\, \theta _{\varphi _ j}^ n\\ =& \int _ C^{\infty }\int _{\{ \varphi _ j\leq \phi -t\} } \theta _{\varphi _ j}^ n\, \mathrm{d}t\\ \leq & \int _ C^{\infty }\int _{\{ \varphi _ i\leq (\varphi _ j+\phi -t)/2\} } \theta _{\varphi _ j}^ n\, \mathrm{d}t\\ \leq & 2^ n\int _ C^{\infty }\int _{\{ \varphi _ i\leq (\varphi _ j+\phi -t)/2\} } \theta _{(\varphi _ j+\phi -t)/2}^ n\, \mathrm{d}t\\ \leq & 2^ n\int _ C^{\infty }\int _{\{ \varphi _ i\leq (\varphi _ j+\phi -t)/2\} } \theta _{\varphi _ i}^ n\, \mathrm{d}t & \textup{by proposition \href{../../tag/FDFD0A6F6287/}{16.5.5}}\\ \leq & 2^ n\int _ C^{\infty }\int _{\{ \varphi _ i\leq \phi -t/2\} } \theta _{\varphi _ i}^ n\, \mathrm{d}t\\ = & 2^{n+1}\int _{\{ \varphi _ i< \phi -C/2\} } (\phi -\varphi _ i-C/2)\, \theta _{\varphi _ i}^ n\\ = & 2^{n+1}\int _{\{ \varphi _ i< \phi -C/2\} } (\phi -\varphi _ i)\, \theta _{\varphi _ i}^ n-C 2^ n\int _{\{ \varphi _ i< \phi -C/2\} }\theta _{\varphi _ i}^ n. \end{aligned} \]
As $C\to \infty $, the first term converges to $0$ since $\varphi _ i\in \mathcal{E}^1(X,\theta ;\phi )$. As for the second term,
\[ \lim _{C\to \infty }C\int _{\{ \varphi _ i< \phi -C/2 \} } \theta _{\varphi _ i}^ n\leq 2 \lim _{C\to \infty }\int _{\{ \varphi _ i< \phi -C/2 \} }(\phi -\varphi _ i)\, \theta _{\varphi _ i}^ n = 0. \]
We conclude (16.5.39).
We now pass from sequences to nets. By 16.5.14, the net $(E_{\theta }^{\phi }(\varphi _ i))_{i\in I}$ is increasing and bounded above by $E_{\theta }^{\phi }(\varphi )$. Let
\[ L\coloneqq \lim _{i\in I}E_{\theta }^{\phi }(\varphi _ i). \]
Then $L\leq E_{\theta }^{\phi }(\varphi )$.
By proposition 5.14.6, we have
\[ \varphi =\operatorname *{sup}_{i\in I}{}\! {}^{*}\varphi _ i. \]
Moreover, $\varphi _ i\leq \varphi $ for all $i\in I$, so the family is locally uniformly bounded above. Applying Choquet's lemma proposition 5.14.3, we can find a countable subset $J\subseteq I$ such that
\[ \varphi =\operatorname *{sup}_{j\in J}{}\! {}^{*}\varphi _ j. \]
Enumerate $J=\{ j_1,j_2,\ldots \} $, repeating elements if necessary. Since $I$ is directed, we can choose an increasing sequence $(i_ k)_{k\geq 1}$ in $I$ such that $i_ k$ dominates $j_1,\ldots ,j_ k$. It follows that $(\varphi _{i_ k})_ k$ increases almost everywhere to $\varphi $. Applying the sequence case to this sequence gives
\[ \lim _{k\to \infty }E_{\theta }^{\phi }(\varphi _{i_ k})=E_{\theta }^{\phi }(\varphi ). \]
Since $E_{\theta }^{\phi }(\varphi _{i_ k})\leq L$ for every $k$, we conclude $E_{\theta }^{\phi }(\varphi )\leq L$.
$\square$
Pome
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