Lemma 13.14.122CA81432B327

Proof. For bounded $u$, lemma 5.28.7, applied in local coordinates with $T_1=\omega +\mathrm{dd}^{\mathrm{c}}u$, $T_2=\gamma $ and $k=1$, gives the asserted inequality.

In general, subtract a constant so that $u\leq 0$, and put $u_ k=\max (u,-k)$. Positivity and locality give

so the bounded case applies to $u_ k$ with the density $\mathbf1_{\{ u>-k\} }g$. Since $u_ k\to u$ in $L^1$, the positive measures $(\omega +\mathrm{dd}^{\mathrm{c}}u_ k)\wedge \gamma ^{n-1}$ converge weakly to $(\omega +\mathrm{dd}^{\mathrm{c}}u)\wedge \gamma ^{n-1}$. The densities on the right converge in $L^1$, because $g^{1/n}$ is integrable and the pole set has zero smooth volume. Passing to the limit proves the assertion. $\square$