PomeProposition 5.44.4.A89FF7CB0129 Let $\varphi \in \mathrm{PSH}(X,\theta )$ be unbounded below, and for each $\lambda >0$ let $u_\lambda $ be a quasi-plurisubharmonic function with $u_\lambda \sim \varphi $. Then, for each $\lambda >0$, the function $\lambda u_\lambda $ has infinite distance in $L^\infty (\mathrm{d}V)$ from every constant. In particular, $\lambda u_\lambda $ cannot converge uniformly on $X$ to a finite constant as $\lambda \downarrow 0$.
Proposition 5.44.4A89FF7CB0129
Proof. Since $u_\lambda \leq \varphi +C$ for a constant $C$, the function $u_\lambda $ is unbounded below. Every nonempty strict sublevel set $\{ u_\lambda <t\} $ of the upper semicontinuous function $u_\lambda $ is open, hence of positive smooth volume. Thus $u_\lambda $, and with it $\lambda u_\lambda $, is essentially unbounded below, and both conclusions follow. $\square$
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