Proposition 7.3.100453AAB39932

Proof. The assertion is local, so it suffices to prove it on each $\Omega _{\varepsilon _0}$, $\varepsilon _0>0$. Let $u_ j=u*\rho _{\varepsilon _ j}$ with $\varepsilon _ j\downarrow 0$, $\varepsilon _ j<\varepsilon _0$; these are smooth plurisubharmonic functions on $\Omega _{\varepsilon _0}$ decreasing to $u$ (proposition 5.9.2). Since $u_ j$ is continuous and $v$ is upper semicontinuous, the set $\{ u_ j>v\} $ is open, and on it $w_ j=\max \{ u_ j,v\} $ equals $u_ j$; by locality, $(\mathrm{dd}^{\mathrm{c}}w_ j)^ p\wedge T=(\mathrm{dd}^{\mathrm{c}}u_ j)^ p\wedge T$ there. The nonnegative Borel function $f_ j=w_ j-v$ vanishes outside $\{ u_ j>v\} $, so

The functions $w_ j$ decrease to $w=\max \{ u,v\} $. Both sides of the last identity are differences of weighted products, for instance $f_ j(\mathrm{dd}^{\mathrm{c}}w_ j)^ p\wedge T=w_ j(\mathrm{dd}^{\mathrm{c}}w_ j)^ p\wedge T-v(\mathrm{dd}^{\mathrm{c}}w_ j)^ p\wedge T$. Applying theorem 7.3.7 to them, with the weights $w_ j$ and the constant weight $v$, and with the factors $w_ j$ on the left and $u_ j$ on the right, we obtain $f(\mathrm{dd}^{\mathrm{c}}w)^ p\wedge T=f(\mathrm{dd}^{\mathrm{c}}u)^ p\wedge T$, where $f=w-v\ge 0$. Multiply the coefficient measures of both sides by $1/(f+\varepsilon )$ and let $\varepsilon \downarrow 0$. Since $f/(f+\varepsilon )$ increases to the indicator of $\{ f>0\} =\{ u>v\} $, dominated convergence with respect to the locally finite total variations of the coefficient measures gives the assertion. $\square$