Theorem 5.47.22C1A6B45ED829

Proof. The identity and the measurability of $W$ are local properties, so we may assume that $X$ is an open ball in $\mathbb C^ n$; by proposition 5.47.15(3), the intersection of $W$ with this ball is plurifine open in the ball. Write $S_ u=\mathrm{dd}^{\mathrm{c}}u_1\wedge \cdots \wedge \mathrm{dd}^{\mathrm{c}}u_ q$ and $S_ v=\mathrm{dd}^{\mathrm{c}}v_1\wedge \cdots \wedge \mathrm{dd}^{\mathrm{c}}v_ q$. We first prove the identity on sets of the form $Q\cap \{ \psi >0\} $, and then cover $W$ by countably many such sets up to a pluripolar set, which the products do not charge.

Step 1: basic sets. We show that $\mathbf1_ AS_ u=\mathbf1_ AS_ v$ whenever $Q\subset X$ is open, $\psi \in \mathrm{PSH}(Q)$ and $A=Q\cap \{ \psi >0\} \subset W$. For every nonempty set $I\subset \{ 1,\ldots ,q\} $, the functions $u_ I=\sum _{l\in I}u_ l$ and $v_ I=\sum _{l\in I}v_ l$ are locally bounded and plurisubharmonic on $Q$, and they coincide on $A$, because $u_ l=v_ l$ on $W$. By lemma 5.47.21, applied on $Q$ with $t=\psi $, $T=1$ and $p=q$, we have $\mathbf1_ A(\mathrm{dd}^{\mathrm{c}}u_ I)^ q=\mathbf1_ A(\mathrm{dd}^{\mathrm{c}}v_ I)^ q$.

We now recover $S_ u$ from the products $(\mathrm{dd}^{\mathrm{c}}u_ I)^ q$ by polarization. For a multi-index $\alpha \in \mathbb N^ q$ with $|\alpha |=q$, let $S_ u^\alpha $ be the product of $q$ factors in which $\mathrm{dd}^{\mathrm{c}}u_ l$ occurs $\alpha _ l$ times; it does not depend on the order of the factors, because products of locally bounded plurisubharmonic functions are symmetric (theorem 7.3.7). Since these products are also additive in each factor (proposition 7.2.2(3)), expanding $(\mathrm{dd}^{\mathrm{c}}u_ I)^ q$ with $\mathrm{dd}^{\mathrm{c}}u_ I=\sum _{l\in I}\mathrm{dd}^{\mathrm{c}}u_ l$ gives, exactly as in the multinomial expansion of $(\sum _{l\in I}x_ l)^ q$,

where $\alpha $ ranges over the multi-indices with $|\alpha |=q$ and $\alpha _ l=0$ for $l\notin I$. Multiplying by $(-1)^{q-|I|}$ and summing over the nonempty sets $I\subset \{ 1,\ldots ,q\} $, we obtain $\sum _\alpha c_\alpha S_ u^\alpha $, summed over all $\alpha $ with $|\alpha |=q$, where $c_\alpha $ is the coefficient of the monomial $x^\alpha $ on the right-hand side of the polarization identity (7.3.9) with $N=q$ (in which the term with $I=\varnothing $ vanishes). Both sides of that identity are polynomials in $x_1,\ldots ,x_ q$ that agree on $\mathbb R^ q$, so they have the same coefficients: $c_\alpha =q!$ if $\alpha =(1,\ldots ,1)$, and $c_\alpha =0$ otherwise. Since $S_ u^{(1,\ldots ,1)}=S_ u$, we obtain

The same identity holds for the $v_ l$. Multiplying both identities by $\mathbf1_ A$ and using $\mathbf1_ A(\mathrm{dd}^{\mathrm{c}}u_ I)^ q=\mathbf1_ A(\mathrm{dd}^{\mathrm{c}}v_ I)^ q$, we obtain $\mathbf1_ AS_ u=\mathbf1_ AS_ v$.

Step 2: countable selection. By theorem 5.47.20(2), applied to the ball $X$ and its plurifine open subset $W$, there are countably many sets $A_ k=Q_ k\cap \{ \psi _ k>0\} \subset W$, with $Q_ k\subset X$ open and $\psi _ k\in \mathrm{PSH}(Q_ k)$, whose union $V$ is Borel and such that $W\setminus V$ is pluripolar. By corollary 5.47.11(3), every pluripolar subset of $X$ has outer measure zero for the total variations of the coefficient measures of $S_ u$ and $S_ v$. Hence, by the last assertion of theorem 5.47.20(2), $W$ is measurable for their completions; moreover $W\setminus V$ is a null set for them, so $\mathbf1_ WS_ u=\mathbf1_ VS_ u$ and $\mathbf1_ WS_ v=\mathbf1_ VS_ v$. The Borel sets $A_ k'=A_ k\setminus (A_1\cup \cdots \cup A_{k-1})$ partition $V$, and Step 1 gives $\mathbf1_{A_ k'}S_ u=\mathbf1_{A_ k'}\mathbf1_{A_ k}S_ u=\mathbf1_{A_ k'}\mathbf1_{A_ k}S_ v=\mathbf1_{A_ k'}S_ v$. Summing over $k$, by countable additivity of the coefficient measures, we obtain $\mathbf1_ VS_ u=\mathbf1_ VS_ v$, which proves the theorem. $\square$