PomeLemma 16.5.14.93210731F7FD The functional $E_{\theta }^{\phi }\colon \mathcal{E}^{\infty }(X,\theta ;\phi )\rightarrow \mathbb {R}$ is increasing. In particular, the extended definition (16.5.12) agrees with (16.5.11) on $\mathcal{E}^{\infty }(X,\theta ;\phi )$, and is increasing as well.
Lemma 16.5.1493210731F7FD
Proof. Let $\varphi ,\psi \in \mathcal{E}^{\infty }(X,\theta ;\phi )$, we have
\[ \begin{aligned} & (n+1)E^{\phi }_{\theta }(\varphi )-(n+1)E^{\phi }_{\theta }(\psi )-\sum _{j=0}^ n \int _ X (\varphi -\psi )\, \theta _{\varphi }^ j\wedge \theta _{\psi }^{n-j} \\ =& \sum _{j=0}^ n \int _ X (\varphi -\phi )\, \theta _{\varphi }^ j\wedge \theta _{\phi }^{n-j}-\sum _{j=0}^ n \int _ X (\psi -\phi )\, \theta _{\psi }^ j\wedge \theta _{\phi }^{n-j}-\sum _{j=0}^ n \int _ X (\varphi -\psi )\, \theta _{\varphi }^ j\wedge \theta _{\psi }^{n-j}\\ =& \sum _{j=0}^ n \int _ X (\varphi -\psi )\, \left( \theta _{\varphi }^ j\wedge \theta _{\phi }^{n-j}-\theta _{\varphi }^ j\wedge \theta _{\psi }^{n-j} \right)\\ & +\sum _{j=0}^ n \int _ X (\psi -\phi )\, \left( \theta _{\varphi }^ j\wedge \theta _{\phi }^{n-j}-\theta _{\psi }^ j\wedge \theta _{\phi }^{n-j} \right)\\ =& \sum _{\substack {j+a+b=n-1,\\ { j,a,b\geq 0
}}} (n-j)\int _ X (\psi -\phi )\, \left( \theta _{\phi }^ a\wedge \theta _{\psi }^{b+1}\wedge \theta _{\varphi }^ j- \theta _{\phi }^ a\wedge \theta _{\psi }^{b}\wedge \theta _{\varphi }^{j+1}\right)\\ & +\sum _{\substack {j+a+b=n-1,\\ { j,a,b\geq 0
}}} (n-j)\int _ X (\psi -\phi )\, \left( \theta _{\phi }^ j\wedge \theta _{\varphi }^{a+1}\wedge \theta _{\psi }^ b- \theta _{\phi }^ j\wedge \theta _{\varphi }^{a}\wedge \theta _{\psi }^{b+1}\right)\\ =& 0, \end{aligned} \]
where the third equality follows from the integration by parts formula. See [Xia19b, Lu21] for the proof in the context of non-pluripolar products.
In other words,
F7FEEE32D362
16.5.15
\begin{equation} E^{\phi }_{\theta }(\varphi )-E^{\phi }_{\theta }(\psi )=\frac{1}{n+1}\sum _{j=0}^ n \int _ X(\varphi -\psi )\, \theta _{\varphi }^ j\wedge \theta _{\psi }^{n-j}. \end{equation}
The monotonicity follows. $\square$
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