Proposition 20.3.5213817C91E7B4

Proof. Normalize $v\leq 0$ and put $v_ j=\max (v,-j)$. For bounded potentials, the horizontal current has rank at most $d=n-1$ and $\beta ^2=0$, so the bounded local product gives

This identity follows by local decreasing regularization on base charts and the bounded product convergence of section 13.14; it does not require $\beta $ to be closed. Restrict to $q^{-1}\{ v>-j\} $ and pass to the increasing non-pluripolar measures. Fiber integration gives (20.3.53), and (20.3.47) gives the two bounds. $\square$