PomeTheorem 17.10.60.C89C58354E49 The metric space $(\mathcal{E}^1(X,\theta ;\phi ),d_1)$ is complete.
Theorem 17.10.60C89C58354E49
Proof. As we have seen in 17.10.45 and proposition 17.10.43, the triple
\[ \left(\mathcal{E}^1(X,\theta ;\phi ),d_1,\land \right) \]
is a rooftop metric space. Hence thanks to proposition 17.10.59, it remains to show that each increasing or decreasing Cauchy sequence in $\mathcal{E}^1(X,\theta ;\phi )$ converges.
We first consider an increasing sequence $(\varphi _ i)_{i>0}$ in $\mathcal{E}^1(X,\theta ;\phi )$. The Cauchy property simply means that
\[ \sup _{i>0} E_{\theta }^{\phi }(\varphi _ i)<\infty . \]
We claim that
\[ \varphi \coloneqq \operatorname *{sup}_{i>0}{}\! {}^{*}\varphi _ i \]
is the $d_1$-limit of the sequence. We first observe that $(\sup _ X \varphi _ i)_{i>0}$ is bounded, as a consequence of 17.10.53. Therefore, proposition 16.5.53 guarantees that $\varphi \in \mathcal{E}^1(X,\theta ;\phi )$, and hence our assertion follows from proposition 16.5.37.
Next we consider a decreasing sequence $(\varphi _ i)_{i>0}$ in $\mathcal{E}^1(X,\theta ;\phi )$. The Cauchy property simply means that
\[ \inf _{i>0} E_{\theta }^{\phi }(\varphi _ i)>-\infty . \]
We claim that
\[ \varphi \coloneqq \inf _{i>0}\varphi _ i \]
is the $d_1$-limit of the sequence. Equivalently, we need to show that $\varphi \in \mathcal{E}^1(X,\theta ;\phi )$ and
25452B178A81
17.10.61
\begin{equation} E_{\theta }^{\phi }(\varphi )=\lim _{i\to \infty } E_{\theta }^{\phi }(\varphi _ i). \end{equation}
We first observe that $(\sup _ X \varphi _ i)_{i>0}$ is bounded, as a consequence of 17.10.53. Therefore, $\varphi \in \mathrm{PSH}(X,\theta ;\phi )$ and $\varphi _ i\xlongrightarrow {L^1} \varphi $, see corollary 5.17.5. We claim that $\varphi \in \mathcal{E}^1(X,\theta ;\phi )$. For this purpose, we may assume that $\varphi _1\leq 0$. Since $(\varphi _ i)_ i$ is decreasing, $\varphi _ i\leq \varphi _1\leq 0$ for all $i>0$. As $\varphi _ i\preceq \phi $ and $\phi $ is model, each $\varphi _ i$ is a candidate for $P_\theta [\phi ]=\phi $, so $\varphi _ i\leq \phi $. Thus, using the energy formula (16.5.11) and the fact that all mixed terms are non-positive, we get for each $i>0$,
\[ \frac{1}{n+1}\int _ X \left(\varphi _ i-\phi \right)\, \theta _{\varphi _ i}^ n\geq E_{\theta }^{\phi }(\varphi _ i)\geq \inf _{j>0} E_{\theta }^{\phi }(\varphi _ j)>-\infty . \]
We can then apply 16.5.45, with $\gamma =\phi $, to conclude that $\varphi \in \mathcal{E}^1(X,\theta ;\phi )$. Our assertion (17.10.61) then follows from proposition 16.5.20. We conclude the proof. $\square$
Comments
Corrections, questions, and mathematical discussion are welcome. Sign in with GitHub to comment. Comments are public.