PomeExample 13.11.8.00950EDCBA2C Let $\varphi \in \mathrm{PSH}(X)$, then
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\begin{equation} (\mathrm{dd}^{\mathrm{c}}\varphi )^1=\mathds {1}_{\{ \varphi >-\infty \} }\, \mathrm{dd}^{\mathrm{c}}\varphi . \end{equation}
Here $(\mathrm{dd}^{\mathrm{c}}\varphi )^1$ on the left-hand side denotes the non-pluripolar product, while $\mathrm{dd}^{\mathrm{c}}\varphi $ on the right-hand side denotes the usual derivative in the sense of current.
The problem is local, so we may assume that $X=\Delta ^ n$, the unit polydisk in $\mathbb {C}^ n$. Take $f\in C^{\infty }_ c(\Delta ^ n)$ with $f\geq 0$. We need to show that
\[ \lim _{k\to \infty } \int _{\Delta ^ n} \mathds {1}_{\{ \varphi >-k\} }f\, \mathrm{dd}^{\mathrm{c}}\bigl (\varphi \lor (-k)\bigr )\wedge \omega ^{n-1}=\int _{\Delta ^ n} \mathds {1}_{\{ \varphi >-\infty \} }f\, \mathrm{dd}^{\mathrm{c}}\varphi \wedge \omega ^{n-1}, \]
where $\omega $ is the standard Kähler form on $\mathbb {C}^ n$. By the dominated convergence theorem, this reduces immediately to the $1$-dimensional case:
\[ \lim _{k\to \infty } \int _{\Delta ^1} \mathds {1}_{\{ \varphi >-k\} }f\, \Delta \bigl (\varphi \lor (-k)\bigr )=\int _{\Delta ^1} \mathds {1}_{\{ \varphi >-\infty \} }f\, \Delta \varphi , \]
for any subharmonic function $\varphi $ on the unit disk $\Delta ^1$, which is a special case of [Proposition 3.3, EK24].
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