Proposition 13.4.390E0E28D9A95

Proof. For $q=0$ there is nothing to prove. Inserting the bounded factors $u_ q,u_{q-1},\ldots ,u_1$ one at a time, we reduce to the case $q=1$. So let $v=u_1$ be locally bounded and $S=\langle \mathrm{dd}^{\mathrm{c}}u_2\wedge \cdots \wedge \mathrm{dd}^{\mathrm{c}}u_ p\rangle $ locally finite. By theorem 13.3.1, $S$ is a closed positive current, so $R=\mathrm{dd}^{\mathrm{c}}v\wedge S$ is defined; it is closed and positive, and by lemma 13.4.2 it gives zero mass to pluripolar sets, because $S$ does. If $p=1$, then $S=1$ and the assertion is that of bounded Bedford–Taylor theory. Let $p\geq 2$. The assertion is local, so we may assume $u_ j\leq 0$ for $j\geq 2$ and put

The functions $w_ k$ are plurisubharmonic by proposition 5.9.5, and they take values in $[0,1]$ because all $u_ j\leq 0$. They vanish outside $O_ k$: if $u_ i\leq -k$ for some $i\geq 2$, then $\sum _{j\geq 2}u_ j\leq u_ i\leq -k$. The sets $\{ w_ k>0\} =\{ \sum _{j\geq 2}u_ j>-k\} $ increase to the set where all $u_ j$, $j\geq 2$, are finite.

Take smooth plurisubharmonic functions $v_ l$ decreasing to $v$ near a given compact set. By lemma 13.2.2, $S$ and $\Theta _ k$ have the same restriction to $O_ k$; since $w_ k$ vanishes outside $O_ k$ and $\mathrm{dd}^{\mathrm{c}}v_ l$ is smooth,

The weighted form of theorem 7.3.7, applied with the constant sequence of weights $w_ k$ and the fixed base currents $S$ on the left and $\Theta _ k$ on the right, gives $w_ kR=w_ k\mathrm{dd}^{\mathrm{c}}v\wedge \Theta _ k$ in the limit $l\to \infty $. Multiplying by the bounded Borel functions $\mathbf1_{\{ w_ k>\varepsilon \} }/w_ k$ and letting $\varepsilon \downarrow 0$, we see that $R$ and $\mathrm{dd}^{\mathrm{c}}v\wedge \Theta _ k$ have the same restriction to $\{ w_ k>0\} $. Now fix an integer $m\geq 1$. For $k\geq m$, the bounded functions $\max \{ u_ i,-k\} $ and $\max \{ u_ i,-m\} $ coincide on $O_ m$, so $\mathrm{dd}^{\mathrm{c}}v\wedge \Theta _ k=\mathrm{dd}^{\mathrm{c}}v\wedge \Theta _ m$ on $O_ m$ by theorem 5.47.22. Since the sets $\{ w_ k>0\} \cap O_ m$, $k\geq m$, exhaust $O_ m$, we conclude that $\mathbf1_{O_ m}R=\mathbf1_{O_ m}\mathrm{dd}^{\mathrm{c}}v\wedge \Theta _ m$.

Consider finally the full tuple $v,u_2,\ldots ,u_ p$ on a relatively compact open set on which $|v|<m_0$. For $m\geq m_0$, its level-$m$ set is $O_ m$ and $v$ needs no truncation, so its truncated product at level $m$ is $\mathbf1_{O_ m}\mathrm{dd}^{\mathrm{c}}v\wedge \Theta _ m=\mathbf1_{O_ m}R$. These restrictions are bounded by $R$, hence have locally bounded mass, and they increase to $R$ because $R$ gives the pole sets of $u_2,\ldots ,u_ p$ zero mass. Therefore the non-pluripolar product of $v,u_2,\ldots ,u_ p$ is locally finite and equals $R=\mathrm{dd}^{\mathrm{c}}v\wedge S$. $\square$