Proposition 16.9.42

Proposition 16.9.42.FDEBDB1E8A22 Let $(\varphi _ i)_{i\in I}$ (resp. $(\psi _ i)_{i\in I}$) be a net in $\mathrm{PSH}(X,\theta )$ such that $\varphi _ i \xrightarrow {d_ S}\varphi \in \mathrm{PSH}(X,\theta )$ (resp. $\psi _ i \xrightarrow {d_ S}\psi \in \mathrm{PSH}(X,\theta )$). Then

\[ \varphi _ i\lor \psi _ i\xrightarrow {d_ S}\varphi \lor \psi . \]

Proof. Since $d_ S$ is a pseudometric, we may assume that both nets are actually sequences and $I=\mathbb {Z}_{>0}$. By corollary 16.9.36, we may assume that the masses $\int _ X \theta _{\varphi }^ n>0$, $\int _ X \theta _{\psi }^ n>0$.

By proposition 16.9.6 and proposition 16.8.28, replacing all potentials by their $P$-envelopes changes neither the hypotheses nor the conclusion. Thus we may assume that the $\varphi _ j$'s, the $\psi _ j$'s, $\varphi $ and $\psi $ are all model. After discarding finitely many terms, the masses of the two sequences are bounded below by a positive constant. Applying proposition 16.9.17 to both sequences and using the sandwich criterion corollary 16.9.39, it is enough to treat the case where each of $(\varphi _ j)_ j$ and $(\psi _ j)_ j$ is monotone. In this reduced situation, $(\varphi _ j)_ j$ (resp. $(\psi _ j)_ j$) converges to $\varphi $ (resp. $\psi $) almost everywhere due to proposition 16.9.47, corollary 16.9.12 and proposition 16.4.9.

We handle three cases separately.

Step 1. Assume that both sequences are increasing.

In this case, we have $\varphi _ j\lor \psi _ j\nearrow \varphi \lor \psi $ almost everywhere. Therefore, $\varphi _ j\lor \psi _ j\xrightarrow {d_ S} \varphi \lor \psi $ by corollary 16.9.12.

Step 2. Assume that one sequence, say $(\varphi _ j)_ j$ is increasing while the other is decreasing. Then we have

\[ \varphi _ j \lor \psi \leq \varphi _ j \lor \psi _ j \leq \varphi \lor \psi _ j. \]

Thanks to corollary 16.9.39, it suffices to show that both sides converge to $\varphi \lor \psi $ with respect to $d_ S$. So we reduce to the case where both sequences are decreasing.

Step 3. Assume that both sequences are decreasing.

In this case, due to corollary 16.9.14, it suffices to show that

76B405D47467 16.9.43

\begin{equation} \lim _{j\to \infty } \int _ X \theta _{\varphi _ j\lor \psi _ j}^ n=\int _ X \theta _{\varphi \lor \psi }^ n. \end{equation}

The $\geq $ direction follows from theorem 13.13.11; it remains to argue the $\leq $ direction.

Thanks to 13.13.30, we may find a sequence $(\epsilon _ j)_ j$ in $(0,1)$ with limit $0$ and a sequence $(\eta _ j)_ j$ in $\mathrm{PSH}(X,\theta )_{>0}$ such that

\[ (1-\epsilon _ j)\varphi _ j+\epsilon _ j \eta _ j \leq \varphi ,\quad \eta _ j \leq \varphi _ j. \]

It follows that for each $j\geq 1$, we have

\[ (1-\epsilon _ j)(\varphi _ j\lor \psi _ j)+\epsilon _ j \eta _ j \leq \varphi \lor \psi _ j. \]

Therefore by theorem 13.13.11,

\[ (1-\epsilon _ j)^ n \int _ X \theta _{\varphi _ j\lor \psi _ j}^ n\leq \int _ X \theta _{\varphi \lor \psi _ j}^ n. \]

Letting $j\to \infty $, we find that

\[ \lim _{j\to \infty }\int _ X \theta _{\varphi _ j\lor \psi _ j}^ n\leq \lim _{j\to \infty }\int _ X \theta _{\varphi \lor \psi _ j}^ n. \]

Therefore, in order to prove (16.9.43), we may assume that one of the sequences is constant; let us say $\psi _ j=\psi $ for all $j$. Repeating the same argument as before and constructing $(\epsilon _ j)_ j$, $(\eta _ j)_ j$ as above, we get

\[ (1-\epsilon _ j)^ n \int _ X \theta _{\varphi _ j\lor \psi }^ n\leq \int _ X \theta _{\varphi \lor \psi }^ n. \]

Letting $j\to \infty $, we conclude (16.9.43). $\square$

Proposition 16.9.42`FDEBDB1E8A22`

Comments

Proposition 16.9.42FDEBDB1E8A22

Comments

Proposition 16.9.42`FDEBDB1E8A22`