Let us say that a continuous metrized line bundle is admissible if it can be written as $\overline{L}\otimes\overline{M}^{\vee}$, where $\overline{L}$ and $\overline{M}$ are metrized line bundles whose metrics are continuous and semi-positive. Admissible metrized line bundles form a subgroup $\overline{\operatorname{Pic}}_{\text{ad}}(\mathrm{X})$ of $\overline{\operatorname{Pic}}(\mathrm{X})$ which maps surjectively onto $\operatorname{Pic}(\mathrm{X})$ if $\mathrm{X}$ is projective.

The curvature current $c_{1}(\overline{L})$ of an admissible metrized line bundle $\overline{L}$ is a differential form of type $(1,1)$ whose coefficients are signed measures. Its $n$th product $c_{1}(\overline{L})^{n}$ is well-defined as a signed measure on $\mathrm{X}$.