Corollary 16.9.39300ED38851D0

Proof. By corollary 16.9.36, we may replace $\theta $ by $\theta +\omega $, where $\omega $ is a Kähler form on $X$. In particular, we may assume that $\varphi _ i,\psi _ i,\eta _ i\in \mathrm{PSH}(X,\theta )_{>0}$ for all $i\in I$. By proposition 16.9.6, we may assume that $\varphi _ i,\psi _ i,\eta _ i$ are model potentials for all $i\in I$ and hence $\psi _ i \leq \varphi _ i\leq \eta _ i$ for all $i\in I$.

It follows from theorem 13.13.11 that for each $k=0,\ldots ,n$, we have

for all $i\in I$. By theorem 16.9.30, the limits, with respect to $i\in I$, of the two outer terms are $\int _ X \theta _{\varphi }^ k\wedge \theta _{V_{\theta }}^{n-k}$. It follows that

By corollary 16.9.33, it remains to prove that $\varphi _ i \lor \varphi \xrightarrow {d_ S}\varphi $. By corollary 16.9.33 and proposition 16.8.28, up to replacing $\psi _ i$ (resp. $\varphi _ i$, $\eta _ i$) by $\psi _ i\lor \varphi $ (resp. $\varphi _ i\lor \varphi $, $\eta _ i\lor \varphi $), we may assume from the beginning that $\psi _ i,\varphi _ i,\eta _ i\geq \varphi $. Now $\varphi _ i\xrightarrow {d_ S}\varphi $ by (16.9.40) and 16.9.7. $\square$