Proposition 16.6.14.8DF15E9E1928 Let $\theta '=\theta +\mathrm{dd}^{\mathrm{c}}g$ for some $g\in C^{\infty }(X)$. Then for any $\varphi \in \mathrm{PSH}(X,\theta )$, we have $\varphi '=\varphi -g\in \mathrm{PSH}(X,\theta ')$ and
\[ \operatorname{vol}(\theta ,\varphi )= \operatorname{vol}\left(\theta ',\varphi '\right). \]
Proof.
By proposition 16.6.9, the currents $\theta +\mathrm{dd}^{\mathrm{c}}P_\theta [\varphi ]_{\mathcal I}$ and $\theta '+\mathrm{dd}^{\mathrm{c}}P_{\theta '}[\varphi ']_{\mathcal I}$ have the same singularity type in the same cohomology class. Applying the monotonicity theorem theorem 13.13.11 gives equality of their non-pluripolar masses.
$\square$
Pome
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