Let $K$ be either ${\mathbf{C}}$ or a discretely valued field of residue characteristic zero, and let $(X,L)$ be a smooth projective polarized variety over $K$. Then there exists a unique class $\operatorname{PSH}({L^{\mathrm{an}}})$, the set of singular semipositive metrics, with the following properties:

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  $\operatorname{PSH}({L^{\mathrm{an}}})$ is a convex set which is closed under maxima and addition of constants;

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  $\operatorname{PSH}({L^{\mathrm{an}}})\cap C^{0}({L^{\mathrm{an}}})=\operatorname{PSH}^{0}({L^{\mathrm{an}}})$;

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  if $s_{i}$, $1\leq i\leq p$, are nonzero global sections of $mL$ for some $m\geq 1$, then $\phi:=\frac{1}{m}\max_{i}\log|s_{i}|\in\operatorname{PSH}({L^{\mathrm{an}}})$; further, $\phi$ is continuous iff the sections $s_{i}$ have no common zero.

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  if $(\phi_{j})$ is an arbitrary family in $\operatorname{PSH}({L^{\mathrm{an}}})$ that is uniformly bounded from above, then the usc regularization of $\sup_{j}\phi_{j}$ belongs to $\operatorname{PSH}({L^{\mathrm{an}}})$;

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  if $(\phi_{j})$ is a decreasing net in $\operatorname{PSH}({L^{\mathrm{an}}})$, then either $\phi_{j}\to-\infty$ uniformly on ${X^{\mathrm{an}}}$, or $\phi_{j}\to\phi$ pointwise on ${X^{\mathrm{an}}}$ for some $\phi\in\operatorname{PSH}({L^{\mathrm{an}}})$;

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  Regularization: for every $\phi\in\operatorname{PSH}({L^{\mathrm{an}}})$ there exists a decreasing sequence $(\phi_{m})_{m=1}^{\infty}$ of smooth/model metrics such that $\phi_{m}$ converges pointwise to $\phi$ on ${X^{\mathrm{an}}}$ as $m\to\infty$;

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  Compactness: the space $\operatorname{PSH}({L^{\mathrm{an}}})/{\mathbf{R}}$ is compact.