PomeA1D8E7565DF1Proposition 5.17.7.A1D8E7565DF1 Assume that $X$ is compact. Let $\theta $ be a closed real smooth $(1,1)$-form on $X$ and $E$ be a prime divisor over $X$. Then
Proof. It follows from the proof of [Exercise 2.7, GZ17] that $\nu (\bullet ,E)$ is upper semicontinuous with respect to the $L^1$-topology on $\mathrm{PSH}(X,\theta )$. Since adding a constant does not change Lelong numbers, we may restrict to the compact set $\{ \varphi \in \mathrm{PSH}(X,\theta ):\sup _ X\varphi =0\} $ from proposition 5.17.3. The desired upper bound follows. $\square$
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