Let $(X,L)$ be a polarized complex projective variety of dimension $n$ and let $\mu$ be a positive measure on ${X^{\mathrm{an}}}$ of total mass $(L^{n})$.

• (i)

  If $\mu$ is a volume form, then there exists a smooth positive metric $\phi$ on ${L^{\mathrm{an}}}$ such that $\operatorname{MA}(\phi)=\mu$.

• (ii)

  If $\mu$ is absolutely continuous with respect to Lebesgue measure, with density in $L^{p}$ for some $p>1$, then there exists a (Hölder) continuous metric $\phi$ on ${L^{\mathrm{an}}}$ such that $\operatorname{MA}(\phi)=\mu$.

• (iii)

  The metrics in (i) and (ii) are unique up to additive constants.