Corollary 10.2.195E8D4AB76812

Proof. Let $T\geq 0$ be closed with $\{ T\} =\{ \theta \} $. Write $T=H+\mathrm{dd}^{\mathrm{c}}f$ as in (10.2.16); since $\{ T\} =\{ \theta \} $, also $\theta =H+\mathrm{dd}^{\mathrm{c}}g$ with $g$ smooth. By theorem 10.2.15, $f$ has a canonical quasi-plurisubharmonic representative, and the difference $\varphi $ of this representative and $g$ satisfies $T=\theta +\mathrm{dd}^{\mathrm{c}}\varphi $; thus $\varphi \in \mathrm{PSH}(X,\theta )$. Conversely, if $\varphi \in \mathrm{PSH}(X,\theta )$, then $\theta +\mathrm{dd}^{\mathrm{c}}\varphi $ is a closed positive current, and it lies in $\{ \theta \} $ because adding $\mathrm{dd}^{\mathrm{c}}$ of a distribution does not change the class. If $\theta +\mathrm{dd}^{\mathrm{c}}\varphi =\theta +\mathrm{dd}^{\mathrm{c}}\varphi '$ with $\varphi ,\varphi '\in \mathrm{PSH}(X,\theta )$, then $\varphi -\varphi '$ is constant almost everywhere by the uniqueness part of theorem 10.2.15, say equal to $c$, and the quasi-plurisubharmonic functions $\varphi $ and $\varphi '+c$, which agree almost everywhere, agree everywhere by lemma 6.14.4(1). $\square$