PomeLemma 17.10.47.CB87AA08E67C Let $\varphi ,\psi \in \mathcal{E}^1(X,\theta ;\phi )$. Then
\[ I_{\theta }(\varphi ,\psi )=I_{\theta }(\varphi \lor \psi ,\varphi )+I_{\theta }(\varphi \lor \psi ,\psi ). \]
Lemma 17.10.47CB87AA08E67C
Proof. By the plurifine locality of non-pluripolar products proposition 13.11.4 1, we can write
\[ \begin{aligned} I_{\theta }(\varphi ,\psi )=& \int _{ \{ \varphi <\psi \} } (\psi -\varphi ) \left( \theta _{\varphi }^ n+\theta _{\psi }^ n \right)+\int _{ \{ \varphi >\psi \} } (\varphi -\psi ) \left( \theta _{\varphi }^ n+\theta _{\psi }^ n \right),\\ I_{\theta }(\varphi \lor \psi ,\varphi )=& \int _{\{ \psi >\varphi \} } (\psi -\varphi )\left(\theta _{\psi }^ n+\theta _{\varphi }^ n \right),\\ I_{\theta }(\varphi \lor \psi ,\psi )=& \int _{\{ \psi <\varphi \} } (\varphi -\psi )\left(\theta _{\psi }^ n+\theta _{\varphi }^ n \right). \end{aligned} \]
The desired equality follows by comparing these terms. $\square$
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