Proof.
Take a closed smooth real $(1,1)$-form $\theta $ on $X$ such that $\theta _{\varphi }$, $\theta _{\psi }$ are both Kähler currents. Let $(\varphi _ j)_ j$ and $(\psi _ j)_ j$ be quasi-equisingular approximations of $\varphi $ and $\psi $ in $\mathrm{PSH}(X,\theta )$ respectively. We may assume that $\varphi _ j\leq 0$ and $\psi _ j\leq 0$ for all $j\geq 1$.
1 We choose the quasi-equisingular approximations compatibly. By theorem 5.18.11, we may take $\varphi _ j$ and $\psi _ j$ with analytic singularities of types
\[ \left(2^{-j},\mathcal I(2^ j\varphi )\right), \quad \left(2^{-j},\mathcal I(2^ j\psi )\right), \]
respectively. Since $\varphi \preceq _{\mathcal I}\psi $, we have
\[ \mathcal I(2^ j\varphi )\subseteq \mathcal I(2^ j\psi ) \]
for every $j$. Hence $\varphi _ j\preceq \psi _ j$ for every $j$. Restricting to $\widetilde{Y}$, we get $\varphi _ j|_{\widetilde{Y}}\preceq \psi _ j|_{\widetilde{Y}}$. Passing to the $d_ S$-limits by proposition 16.9.44, we obtain
\[ \operatorname{Tr}_ Y(\varphi )\preceq _ P\operatorname{Tr}_ Y(\psi ). \]
Restricting to $\widetilde{Y}$ and passing to the $d_ S$-limits by proposition 16.9.44, we obtain
\[ \operatorname{Tr}_ Y(\varphi )\preceq _ P \operatorname{Tr}_ Y(\psi ). \]
2 It follows from theorem 16.9.45 that $\varphi _ j+\psi _ j\xrightarrow {d_ S}P_{\theta }[\varphi ]_{\mathcal{I}}+P_{\theta }[\psi ]_{\mathcal{I}}$. However, by proposition 16.6.27 and proposition 18.1.17, we have
\[ P_{\theta }[\varphi ]_{\mathcal{I}}+P_{\theta }[\psi ]_{\mathcal{I}}\sim _ P P_{2\theta }[\varphi +\psi ]_{\mathcal{I}}. \]
Therefore, by proposition 16.9.6, corollary 18.1.14 and proposition 5.18.2, $(\varphi _ j+\psi _ j)_ j$ is a quasi-equisingular approximation of $\varphi +\psi $. We conclude using theorem 16.9.45.
3 Let $(\lambda _ j)_ j$ be an increasing sequence of positive rational numbers with limit $\lambda $. Then $(\lambda _ j\varphi _ j)_ j$ is a quasi-equisingular approximation of $\lambda \varphi $. Since $\lambda _ j/\lambda \to 1$, 16.9.25 gives
\[ d_ S\left((\lambda _ j/\lambda )\varphi _ j|_{\widetilde{Y}},\varphi _ j|_{\widetilde{Y}}\right)\to 0. \]
After scaling the background class, 16.9.41 shows that $\lambda _ j\varphi _ j|_{\widetilde{Y}}$ and $\lambda \varphi _ j|_{\widetilde{Y}}$ have the same $d_ S$-limit. The assertion follows.
4 By proposition 16.9.42, we have
\[ \varphi _ j\lor \psi _ j\xrightarrow {d_ S} P_{\theta }[\varphi ]_{\mathcal{I}}\lor P_{\theta }[\psi ]_{\mathcal{I}}. \]
By proposition 16.6.27 and proposition 18.1.17, we have
\[ P_{\theta }[\varphi ]_{\mathcal{I}}\lor P_{\theta }[\psi ]_{\mathcal{I}}\sim _ P P_{\theta }[\varphi \lor \psi ]_{\mathcal{I}}. \]
Therefore, our assertion follows exactly as in the proof of 2.
$\square$
Pome
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