Pome917763AE5E35Theorem 5.16.21 (Guan–Zhou).917763AE5E35 Let $\varphi ,\psi _ j\in \mathrm{PSH}(X)$ ($j\in \mathbb {Z}_{>0}$) such that $\psi _ j$ is an increasing sequence converging to $\varphi $ almost everywhere. Then for any $x\in X$, the germs satisfy
when $j$ is large enough.
Proof. The multiplier ideals $\mathcal I(\psi_j)_x$ form an increasing sequence, since $\psi_j\leq\psi_{j+1}$. By monotone multiplier-ideal stability, their union is $\mathcal I(\varphi)_x$. Choose finitely many generators $f_1,\ldots,f_r$ of the latter ideal; finite generation follows from coherence. Each $f_l$ belongs to some $\mathcal I(\psi_{j_l})_x$. For $j\geq\max_l j_l$, all generators belong, so $\mathcal I(\varphi)_x\subseteq\mathcal I(\psi_j)_x$. The reverse inclusion follows from $\psi_j\leq\varphi$ almost everywhere. Thus equality holds for every sufficiently large $j$, with the index allowed to depend on $x$.
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