Assume that $\omega$ is a form as in §4 with $\{\omega\}^{n}=1$ that satisfies the orthogonality property (see Definition 7.1). Let $\mu$ be a probability measure on $X$ supported on the dual complex of some SNC model of $X$. Then there exists a unique, continuous $\omega$-psh function $\varphi\in\PSH(X,\omega)$ such that $\MA(\varphi)=\mu$, and $\sup\varphi=0$.