Let $X$ be an algebraizable smooth projective $K$-variety as in Theorem A. Let $\omega\in\mathcal{Z}^{1,1}(X)$ be a closed semipositive $(1,1)$-form such that $\{\omega\}\in N^{1}(X)$ is ample and let $\mu$ be a positive Radon measure on $X$ of mass $\{\omega\}^{n}$. If $\mu$ is supported in a dual complex then there exists a continuous $\omega$-psh function $\varphi$ such that

(1.2)

$$(\omega+dd^{c}\varphi)^{n}=\mu.$$

The function $\varphi$ is furthermore unique up to an additive constant.