Proposition 13.14.1545F622BE862A

Proof. For another Hermitian form $\eta $, choose $C>0$ such that $\eta \leq C\omega $. If $v$ is a bounded $\eta $-plurisubharmonic function, then $v/C$ is $\omega $-plurisubharmonic and

By positivity of mixed bounded products, $\int _ X(\eta +\mathrm{dd}^{\mathrm{c}}v)^ n\leq C^ n v_+(\omega )$. Interchanging the two forms proves metric independence.

For the last assertion, approximate a bounded potential by theorem 5.32.1. The smooth approximants decrease and remain uniformly bounded, so the continuity of bounded products along decreasing sequences gives convergence of their total masses. The reverse inequality between the two suprema is immediate. $\square$