Lemma 13.4.54AD0BFC1BFCD

Proof. If $r=0$ or $m=0$, one factor of $Q\times D$ is a point, carrying counting measure, and the corresponding product is the empty product $1$; then the two sides of (13.4.6) are the same integral. Let $r,m\geq 1$. For smooth potentials the formula is Fubini's theorem; we pass to the limit for continuous weights $h$, and then extend the formula to Borel weights by the uniqueness of measures.

Approximation. Let $Q'\Subset Q$ and $D'\Subset D$ be open. For $\varepsilon _ l\downarrow 0$, the mollifications $u_{j,l}=u_ j*\rho _{\varepsilon _ l}$ are smooth and plurisubharmonic on a neighborhood of $\overline{Q'\times D'}$ for large $l$, they decrease to $u_ j$ (proposition 5.9.2(2)), and they are bounded there by a constant $B$, since $u_ j\leq u_{j,l}\leq u_{j,l_0}$. Let $M^{(l)}$ and $M_ z^{(l)}$ be the corresponding products. By theorem 7.3.7, $M^{(l)}\to M$ weakly on $Q'\times D'$, and for every $z\in D'$, $M_ z^{(l)}\to M_ z$ weakly on $Q'$, since $u_{j,l}(\, \cdot \, ,z)$ decreases to $u_ j(\, \cdot \, ,z)$. By theorem 7.2.5, for every compact set $A\subset Q'$ the masses $M_ z^{(l)}(A)$ and $M_ z(A)$ are bounded by a constant depending only on $A$, $Q'$ and $B$, uniformly in $l$ and $z\in D'$; similarly for $M^{(l)}$ and $M$ on compact subsets of $Q'\times D'$. For positive measures with locally uniformly bounded masses, weak convergence implies convergence of the integrals of continuous functions with compact support, because such functions are uniform limits of test functions supported in a fixed compact set (lemma 2.5.1).

Continuous weights. Let $\eta =f\, \mathrm dV_ z\geq 0$, where $\mathrm dV_ z$ is the Euclidean volume form of $\mathbb C^ m$ and $f\geq 0$ is smooth, and let $h$ be continuous with compact support in $Q'\times D'$. In the smooth exterior product $M^{(l)}$, wedging with the form $p_ D^*\eta $ of top degree in $z$ kills every term containing a differential $\mathrm dz_ i$ or $\mathrm d\bar z_ i$, so that $p_ D^*\eta \wedge M^{(l)}=f(z)\, M_ z^{(l)}(x)\wedge \mathrm dV_ z$, and Fubini's theorem (proposition 2.2.7) gives (13.4.6) for the smooth approximations. As $l\to \infty $, the left side converges to $\int h\, p_ D^*\eta \wedge M$, by the approximation step applied to the positive measures $p_ D^*\eta \wedge M^{(l)}$. On the right, the inner integrals are continuous functions of $z$, they converge to $\int _ Qh(x,z)\, M_ z(x)$ for every $z$, and they are bounded by $\sup |h|$ times a constant, uniformly in $l$ and $z$; hence the limit is a Borel function of $z$, and dominated convergence proves (13.4.6). A finite smooth partition of unity (lemma 2.5.2), subordinate to a cover of $\operatorname{Supp}h$ by such products $Q'\times D'$, treats arbitrary continuous $h$ with compact support.

Measurability. For a continuous function $a$ with compact support in $Q$, the function $z\mapsto \int a\, M_ z$ is Borel on $D$: if the support of $a$ lies in $Q'$, then on $D'$ it is the pointwise limit of the continuous functions $z\mapsto \int a\, M_ z^{(l)}$, and $D$ is covered by countably many such sets $D'$. Hence lemma 13.4.4, applied with $G=D$ and $\mu _ z=M_ z$, shows that $z\mapsto \int _ Qh(x,z)\, M_ z(x)$ is Borel for every nonnegative Borel function $h$, and hence for every bounded Borel function $h$ with compact support.

Borel weights. Let $\eta =f\, \mathrm dV_ z\geq 0$ as before, and let $\lambda $ be the measure of lemma 13.4.4 for $\mu _ z=M_ z$ and this function $f$, so that

for Borel sets $E\subset Q\times D$. The uniform local bound for the fiber masses makes $\lambda $ finite on compact sets, so it is a Radon measure by proposition 2.2.2. By lemma 13.4.4, $\int h\, \mathrm d\lambda =\int _ D(\int _ Qh(x,z)\, M_ z(x))\, \eta (z)$ for every nonnegative Borel function $h$, hence for every continuous $h$ with compact support. By the step on continuous weights, $\lambda $ and $p_ D^*\eta \wedge M$ therefore have the same integral against every test function, so they are equal by the uniqueness in theorem 2.5.6. This proves the formula for nonnegative Borel $h$, and then for bounded real Borel $h$ with compact support, by splitting $h$ into its positive and negative parts.

Finally, let $\eta =f\, \mathrm dV_ z$ be a smooth real form and $h$ a bounded Borel function supported in $K\times K_ D$, with $K\subset Q$ and $K_ D\subset D$ compact. Choose $\kappa \in C_ c^\infty (D)$ with $0\leq \kappa \leq 1$ and $\kappa =1$ on $K_ D$, and put $c=\sup _{\operatorname{Supp}\kappa }|f|$. Replacing $\eta $ by $\kappa \eta $ changes neither side of (13.4.6), and $\kappa \eta =\kappa (f+c)\, \mathrm dV_ z-c\kappa \, \mathrm dV_ z$ is the difference of two smooth nonnegative forms. This proves the formula for signed $\eta $. $\square$