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Lemma 16.5.404C2A27A8F97A

Lemma 16.5.40.4C2A27A8F97A Let $\varphi ,\psi \in \mathcal{E}^{\infty }(X,\theta ;\phi )$. Then there is $\gamma \in \mathcal{E}^{\infty }(X,\theta ;\phi )$ such that

4E22C768C562 16.5.41
\begin{equation} \theta _{\gamma }^ n=\mathrm{e}^{\gamma -\varphi }\, \theta _{\varphi }^ n+\mathrm{e}^{\gamma -\psi }\, \theta _{\psi }^ n. \end{equation}

Proof. For each $j\geq 1$, let $\varphi _ j\coloneqq \varphi \lor (-j)$, $\psi _ j\coloneqq \psi \lor (-j)$. These functions are not necessarily $\theta $-psh. Let

\[ \mu _ j=\mathrm{e}^{-\varphi _ j}\theta _{\varphi }^ n+\mathrm{e}^{-\psi _ j}\theta _{\psi }^ n. \]

By theorem 16.5.9, we can find $\gamma _ j\in \mathcal{E}(X,\theta ;\phi )$ such that

5B35DF529607 16.5.42
\begin{equation} \theta _{\gamma _ j}^ n=\mathrm{e}^{\gamma _ j}\mu _ j=\mathrm{e}^{\gamma _ j-\varphi _ j}\theta _{\varphi }^ n+\mathrm{e}^{\gamma _ j-\psi _ j}\theta _{\psi }^ n. \end{equation}

Take a constant $C>0$ so that $\psi -2C\leq \varphi \leq \psi +2C$. Let

\[ \eta \coloneqq \frac{\varphi +\psi }{2}-C-n\log 2. \]

Then $\eta \in \mathcal{E}^{\infty }(X,\theta ;\phi )$ and we have the following inequalities:

\[ \begin{aligned} \theta _{\eta }^ n =& \theta _{(\varphi +\psi )/2}^ n\\ \geq & 2^{-n}\theta _{\varphi }^ n+2^{-n}\theta _{\psi }^ n\\ \geq & 2^{-n}\mathrm{e}^{(\varphi +\psi )/2}\mathrm{e}^{-C}\left( \mathrm{e}^{-\varphi }\theta _{\varphi }^ n+\mathrm{e}^{-\psi }\theta _{\psi }^ n \right)\\ \geq & 2^{-n}\mathrm{e}^{(\varphi +\psi )/2}\mathrm{e}^{-C}\left( \mathrm{e}^{-\varphi _ j}\theta _{\varphi }^ n+\mathrm{e}^{-\psi _ j}\theta _{\psi }^ n \right)\\ = & \mathrm{e}^{\eta }\mu _ j. \end{aligned} \]

Hence, $\gamma _ j\geq \eta $ by theorem 16.5.9. By theorem 16.5.9 again, $\gamma _ j$ is decreasing in $j$; let $\gamma =\inf _{j>0}\gamma _ j$. Then $\gamma \geq \eta $, hence $\gamma \in \mathcal{E}^{\infty }(X,\theta ;\phi )$.

Now observe that

\[ \theta _{\gamma _ j}^ n\rightharpoonup \theta _{\gamma }^ n \]

as $j\to \infty $, by theorem 13.13.9. Finally observe that there is a constant $C'>0$ so that

\[ \gamma _1\leq \varphi +C',\quad \gamma _1\leq \psi +C', \]

since $\varphi ,\psi \in \mathcal{E}^{\infty }(X,\theta ;\phi )$. Therefore, for each $j>0$,

\[ \gamma _ j\leq \varphi _ j+C',\quad \gamma _ j\leq \psi _ j+C'. \]

Hence, (16.5.41) follows from (16.5.42) and the dominated convergence theorem by letting $j\to \infty $. $\square$

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