Let $(X,L)$ be a polarized complex projective variety of dimension $n$ over $K$. Assume $X$ is defined over a smooth $k$-curve. Let $\mu$ be a positive measure on ${X^{\mathrm{an}}}$ of total mass $(L^{n})$, supported on the dual complex of some SNC model.

• (i)

  There exists a continuous metric $\phi$ on ${L^{\mathrm{an}}}$ such that $\operatorname{MA}(\phi)=\mu$.

• (ii)

  The metric in (i) is unique up to an additive constant.