PomeDefinition 13.11.1.1A52175C3D64 Let $\varphi _1,\ldots ,\varphi _ p\in \mathrm{PSH}(X)$. We set
\[ O_ k \coloneqq \bigcap _{j=1}^ p \{ \varphi _ j>-k\} ,\quad k\in \mathbb {N}. \]
We say that $\mathrm{dd}^{\mathrm{c}}\varphi _1\wedge \cdots \wedge \mathrm{dd}^{\mathrm{c}}\varphi _ p$ is well-defined if for each connected open subset $U\subseteq X$, any smooth positive $(1,1)$-form $\omega $ on $U$, and each compact subset $K\subseteq U$, we have
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13.11.2
\begin{equation} \sup _{k\geq 0} \int _{K\cap O_ k}\left.\left(\bigwedge _{j=1}^ p \mathrm{dd}^{\mathrm{c}}(\varphi _ j\lor (-k))\right)\right|_ U\wedge \omega ^{\dim U-p}<\infty . \end{equation}
In this case, we define the non-pluripolar product $\mathrm{dd}^{\mathrm{c}}\varphi _1\wedge \cdots \wedge \mathrm{dd}^{\mathrm{c}}\varphi _ p$ by
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13.11.3
\begin{equation} \mathds {1}_{O_ k}\, \mathrm{dd}^{\mathrm{c}}\varphi _1\wedge \cdots \wedge \mathrm{dd}^{\mathrm{c}}\varphi _ p=\mathds {1}_{O_ k}\bigwedge _{j=1}^ p \mathrm{dd}^{\mathrm{c}}\left( \varphi _ j\lor (-k)\right) \end{equation}
on $\bigcup _{k\geq 0} O_ k$ and extend it by zero to $X$.
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