Proof.
Step 1. We first prove the right-hand part of (17.10.49).
Thanks to proposition 16.5.36 and 16.3.3, we have
\[ \begin{aligned} E_{\theta }^{\phi }(\varphi )-E_{\theta }^{\phi }(\varphi \land \psi )\leq & \int _ X (\varphi -\varphi \land \psi )\, \theta _{\varphi \land \psi }^ n \\ \leq & \int _{\{ \psi =\varphi \land \psi \} } (\varphi -\psi )\, \theta _{\psi }^ n\\ \leq & \int _ X |\varphi -\psi |\, \theta _{\psi }^ n. \end{aligned} \]
Similarly,
\[ E_{\theta }^{\phi }(\psi )-E_{\theta }^{\phi }(\varphi \land \psi )\leq \int _ X |\varphi -\psi |\, \theta _{\varphi }^ n. \]
Adding these inequalities up, we find
\[ d_1(\varphi ,\psi )\leq I_{\theta }(\varphi ,\psi ). \]
Step 2. We prove the left-hand part of (17.10.49).
We claim that
C3DBA22EBCCD
17.10.50
\begin{equation} d_1\left(\varphi ,\frac{\varphi +\psi }{2}\right)\leq \frac{3(n+1)}{2} d_1(\varphi ,\psi ). \end{equation}
For this purpose, we compute directly
\[ \begin{aligned} & d_1\left(\varphi ,\frac{\varphi +\psi }{2}\right)\\ =& d_1\left(\varphi ,\varphi \land \frac{\varphi +\psi }{2}\right)+d_1\left(\frac{\varphi +\psi }{2},\varphi \land \frac{\varphi +\psi }{2}\right)\\ \leq & d_1\left(\varphi ,\varphi \land \psi \right)+d_1\left(\frac{\varphi +\psi }{2},\varphi \land \psi \right)& \textup{by \href{../../tag/93210731F7FD/}{16.5.14}}\\ \leq & \int _ X \left( \varphi -\varphi \land \psi \right)\, \theta _{\varphi \land \psi }^ n+ \int _ X \left( \frac{\varphi +\psi }{2}-\varphi \land \psi \right)\, \theta _{\varphi \land \psi }^ n & \textup{by proposition \href{../../tag/51CFFEE97C34/}{16.5.36}}\\ =& \frac{3}{2}\int _ X \left( \varphi -\varphi \land \psi \right)\, \theta _{\varphi \land \psi }^ n+\frac{1}{2}\int _ X \left( \psi -\varphi \land \psi \right)\, \theta _{\varphi \land \psi }^ n \\ \leq & \frac{3(n+1)}{2}d_1\left(\varphi ,\varphi \land \psi \right) + \frac{n+1}{2}d_1(\psi ,\varphi \land \psi )& \textup{by (\href{../../tag/F8225060CBAC/}{16.5.31})}\\ \leq & \frac{3(n+1)}{2}d_1(\varphi ,\psi ), \end{aligned} \]
and (17.10.50) follows.
We now estimate the left-hand side of (17.10.50):
\[ \begin{aligned} d_1\left(\varphi ,\frac{\varphi +\psi }{2}\right)\geq & d_1\left(\varphi ,\varphi \land \frac{\varphi +\psi }{2}\right)\\ \geq & \int _ X \left(\varphi -\varphi \land \frac{\varphi +\psi }{2}\right)\, \theta _{\varphi }^ n\\ \geq & \int _{\{ \varphi >\psi \} }\left(\varphi -\varphi \land \frac{\varphi +\psi }{2}\right)\, \theta _{\varphi }^ n \\ \geq & \int _{\{ \varphi >\psi \} }\left(\varphi -\frac{\varphi +\psi }{2}\right)\, \theta _{\varphi }^ n\\ =& \frac{1}{2}\int _{\{ \varphi >\psi \} }\left(\varphi -\psi \right)\, \theta _{\varphi }^ n, \end{aligned} \]
where the second inequality follows from proposition 16.5.36.
Similarly,
\[ \begin{aligned} d_1\left(\varphi ,\frac{\varphi +\psi }{2}\right)\geq & d_1\left(\frac{\varphi +\psi }{2},\varphi \land \frac{\varphi +\psi }{2}\right)\\ \geq & \int _ X \left(\frac{\varphi +\psi }{2}-\varphi \land \frac{\varphi +\psi }{2}\right)\, \theta _{(\varphi +\psi )/2}^ n\\ \geq & 2^{-n} \int _ X \left(\frac{\varphi +\psi }{2}-\varphi \land \frac{\varphi +\psi }{2}\right)\, \theta _{\varphi }^ n\\ \geq & 2^{-n} \int _{\{ \varphi <\psi \} } \left(\frac{\varphi +\psi }{2}-\varphi \land \frac{\varphi +\psi }{2}\right)\, \theta _{\varphi }^ n\\ \geq & 2^{-n} \int _{\{ \varphi <\psi \} } \left(\frac{\varphi +\psi }{2}-\varphi \right)\, \theta _{\varphi }^ n\\ =& 2^{-n-1} \int _{\{ \varphi <\psi \} } \left(\psi -\varphi \right)\, \theta _{\varphi }^ n, \end{aligned} \]
where the third inequality follows from the multi-linearity of the non-pluripolar product proposition 13.11.4 6.
Adding these estimates up, we find
\[ 2^{n+2} d_1\left(\varphi ,\frac{\varphi +\psi }{2}\right)\geq \left(2^{n+1}+2\right) d_1\left(\varphi ,\frac{\varphi +\psi }{2}\right)\geq \int _ X |\varphi -\psi |\, \theta _{\varphi }^ n. \]
Therefore, combined with (17.10.50) we conclude
\[ 3\cdot 2^{n+1}(n+1)d_1(\varphi ,\psi )\geq \int _ X |\varphi -\psi |\, \theta _{\varphi }^ n. \]
By symmetry, we also get a similar expression after exchanging $\varphi $ and $\psi $. Adding these inequalities together, we find
\[ 3\cdot 2^{n+2}(n+1)d_1(\varphi ,\psi )\geq I_{\theta }(\varphi ,\psi ). \]
The left-hand part of (17.10.49) then follows.
$\square$
Pome
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