Pome

Remark 13.13.46B80857C56A45

Remark 13.13.46.B80857C56A45 Using the terminologies to be introduced in section 16.1, we can reformulate a special case of theorem 13.13.43 as follows: Suppose that $\phi \in \mathrm{PSH}(X,\theta )_{>0}$ is a model potential and $\varphi ,\psi \in \mathcal{E}(X,\theta ;\phi )$. Assume that (13.13.45) holds then $\varphi \geq \psi $.

Equivalently, using the terminologies of section 16.7, the assumption (13.13.44) can be reformulated as $\varphi \sim _ P \psi $.

Proof. Thanks to the monotonicity theorem theorem 13.13.11,

\[ \int _ X \theta _{\varphi \lor \psi }^ n=\int _ X \theta _{\phi }^ n. \]

We may replace $\psi $ by $\varphi \lor \psi $ and assume that $\varphi \leq \psi $. Without loss of generality, we may also assume that $\psi \leq 0$.

Now fix $a>1$. We define

\[ \eta _ a\coloneqq P_{\theta }\Bigl ( a\varphi +(1-a)\psi \Bigr ). \]

Note that $\eta _ a\in \mathrm{PSH}(X,\theta )$ by corollary 13.13.42 and

\[ \int _ X \theta _{\eta _ a}^ n=\int _ X \theta _{\phi }^ n. \]

Furthermore, $\eta _ a\leq P_{\theta }(a\varphi +(1-a)\varphi )=\varphi $.

Define

\[ \gamma _ a\coloneqq a^{-1}\eta _ a+(1-a^{-1})\psi ,\quad D_ a\coloneqq \{ \gamma _ a=\varphi \} . \]

Then $\gamma _ a\leq \varphi $ with equality on $D_ a$. Therefore,

\[ \begin{aligned} a^{-n}\, \theta _{\eta _ a}^ n\leq & a^{-n}\, \theta _{\eta _ a}^ n+\mathds {1}_{D_ a} (1-a^{-1})^ n \, \theta _{\psi }^ n& \\ =& \mathds {1}_{D_ a} a^{-n}\, \theta _{\eta _ a}^ n+\mathds {1}_{D_ a} (1-a^{-1})^ n \, \theta _{\psi }^ n & \textup{ by theorem \href{../../tag/A60C61FA5696/}{13.13.24}}\\ \leq & \mathds {1}_{D_ a} \theta _{\gamma _ a}^ n & \\ \leq & \mathds {1}_{D_ a} \theta _{\varphi }^ n & \textup{ by \href{../../tag/F56651914DE1/}{13.13.14}}\\ =& \mathds {1}_{D_ a\cap \{ \varphi =\psi \} }\, \theta _{\varphi }^ n & \textup{ by (\href{../../tag/99773FAEB14E/}{13.13.45})}. \end{aligned} \]

Note that on $D_ a\cap \{ \varphi =\psi \} $, we have $\eta _ a=\varphi $. We deduce that

\[ \theta _{\eta _ a}^ n=\mathds {1}_{\{ \eta _ a=\varphi \} }\, \theta _{\eta _ a}^ n \leq \theta _{\varphi }^ n, \]

where the inequality follows from 13.13.14.

But the two ends have the same mass, and hence

\[ \theta _{\eta _ a}^ n=\theta _{\varphi }^ n=\mathds {1}_{\{ \eta _ a=\varphi \} }\, \theta _{\varphi }^ n. \]

Therefore,

\[ \int _ X \mathrm{e}^{\eta _ a}\, \theta _{\varphi }^ n=\int _ X \mathrm{e}^{\varphi }\, \theta _{\varphi }^ n>0. \]

Note that $\eta _ a$ is decreasing in $a$. The above equation shows that

\[ \eta \coloneqq \inf _{a>1} \eta _ a\not\equiv -\infty . \]

On the other hand, if $x\in X$ is such that $\varphi (x)<\psi (x)$, we then have

\[ \eta _ a(x)\leq a\varphi (x)-(a-1)\psi (x)= \psi (x)+a\Bigl ( \varphi (x)-\psi (x) \Bigr ). \]

Letting $a\to \infty $, we find that $\eta (x)=-\infty $. Therefore, $\{ \varphi \neq \psi \} $ is pluripolar, and hence empty by proposition 5.14.6. Our assertion follows. $\square$

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