Lemma 13.10.510BB181E99E0

Proof. By linearity, it suffices to consider $\eta \geq 0$. We first prove (13.10.6) when $h$ is continuous. On a product neighborhood of $\operatorname{Supp}h$, choose smooth psh functions $u_ r^{(l)}\searrow u_ r$, $1\leq r\leq n$. For the smooth approximants, decompose each $\mathrm{dd}^{\mathrm{c}}u_ r^{(l)}$ according to the $x$- and $z$-directions. After wedging with $p_2^*\eta $, every term containing a factor $\mathrm{d}z_\alpha $ or $\mathrm{d}\bar z_\beta $ vanishes; hence only the factors $\mathrm{dd}^{\mathrm{c}}_ x(u_ r^{(l)})_ z$ remain, and (13.10.6) follows from the usual Fubini theorem. As $l\to \infty $, the Bedford–Taylor monotone continuity theorem theorem 13.10.4 gives convergence on $Q\times D$ and on every fibre. Fix $K\Subset Q$ and $L\Subset D$ containing the projections of $\operatorname{Supp}h$. The Chern–Levine–Nirenberg estimate [Theorem 3.9, GZ17], together with the local uniform bounds for $u_ r^{(l)}$, bounds the masses of the fibre products on $K$ uniformly for $z\in L$ and $l$. Since $h_ z=0$ for $z\notin L$, dominated convergence proves (13.10.6).

It remains to pass from continuous to bounded Borel functions. Fix relatively compact open sets $U\Subset Q$ and $V\Subset D$. If $\chi \in C_ c(U)$, then

is Borel measurable. Indeed, choose the smooth decreasing approximations on a neighborhood of $\overline U\times \overline V$ and set

The functions $z\mapsto \int _ U\chi \, M_ z^{(l)}$ are continuous on $V$ and converge pointwise to $F_\chi $ by theorem 13.10.4.

Let $\mathcal{A}_{U,V}$ be the vector space of bounded Borel functions $\chi $ on $U$ for which $F_\chi $ is Borel measurable on $V$. It is closed under bounded increasing pointwise limits of non-negative functions by the monotone convergence theorem. Taking compactly supported continuous cut-off functions increasing to $1$ on $U$ shows that $1\in \mathcal{A}_{U,V}$. Thus $\mathcal{A}_{U,V}$ contains the algebra $\mathbb {R}+C_ c(U)$, which generates the Borel $\sigma $-algebra of $U$. The functional monotone class theorem therefore shows that $\mathcal{A}_{U,V}$ contains every bounded Borel function on $U$.

Next, let $\mathcal{B}_{U,V}$ be the vector space of bounded Borel functions $g$ on $U\times V$ such that

is Borel measurable on $V$. This space is again closed under bounded increasing pointwise limits of non-negative functions. It contains every product $g(x,z)=\chi (x)\psi (z)$ of bounded Borel functions, by the preceding paragraph. Finite sums of such products form an algebra containing the indicators of all Borel rectangles $A\times B\subseteq U\times V$; hence this algebra generates the Borel $\sigma $-algebra of $U\times V$. A second application of the functional monotone class theorem shows that $\mathcal{B}_{U,V}$ contains every bounded Borel function on $U\times V$.

To pass from $U\times V$ to $Q\times D$, choose exhaustions $U_ j\Subset Q$ and $V_ j\Subset D$ by relatively compact open sets. For a Borel set $E\subseteq Q\times D$, the preceding paragraph, applied to the restriction of $\mathds {1}_ E$ to $U_ j\times V_ j$, shows that

is Borel measurable on $D$. We may take both exhaustions increasing, so $G_ j\nearrow M_ z(E_ z)$ pointwise, where

Thus $z\mapsto M_ z(E_ z)$ is Borel measurable on $D$. We may therefore define

The monotone convergence theorem, first on each fibre and then on $D$, shows that $\lambda $ is a Borel measure on $Q\times D$. The local uniform mass bound above shows that $\lambda $ is locally finite and hence Radon. Approximation by non-negative simple functions gives

for every non-negative Borel function $h$. For continuous compactly supported $h$, the first part of the proof shows that $\lambda $ agrees with $p_2^*\eta \wedge M$. The uniqueness of Radon measures then gives the same equality for every bounded Borel function $h$ with compact support. $\square$