More generally, both the curvature and the Poincaré–Lelong equation make sense for metrized line bundles with arbitrary (continuous) metrics, except that $c_{1}(\overline{L})$ has to be considered as a current. The notion a semi-positivity can even be extended to this more general case, because it can be tested by duality : a current is positive if its evaluation on any nonnegative differential form is nonnegative. Alternatively, semi-positive (continuous) metrized line bundles are characterized by the fact that for any local frame $s$ of $\overline{L}$ over an open set $\mathrm{U}$, the continuous function $\log\left\|{s}\right\|^{-1}$ is plurisubharmonic on $\mathrm{U}$. In turn, this means that for any morphism $\varphi\colon\overline{D}\rightarrow\mathrm{U}$, where $\overline{D}=\overline{D}(0,1)$ is the closed unit disk in ${\mathbf{C}}$,

$$\log\left\|{s}\right\|^{-1}(\varphi(0))\leq\frac{1}{2\pi}\int_{0}^{2\pi}\log\left\|{s}\right\|^{-1}(\varphi(e^{i\theta}))\mathrm{d}\theta.$$

Assume that $\overline{L}$ is semi-positive. Although products of currents are not defined in general (not more than products of distributions), the theory of Bedford–Taylor [8, 7] and Demailly [24, 25] defines a current $c_{1}(\overline{L})^{n}$ which then is a positive measure on $\mathrm{X}$. There are two ways to define this current. The first one works locally and proceeds by induction : if $u=\log\left\|{s}\right\|^{-1}$, for a local non-vanishing section $s$ of $L$, one defines a sequence $(T_{k})$ of closed positive currents by the formulae $T_{0}=1$, $T_{1}=\mathop{\mathrm{d}\mathrm{d}^{c}}u$,…, $T_{k+1}=\mathop{\mathrm{d}\mathrm{d}^{c}}(uT_{k})$ and $c_{1}(\overline{L})^{n}=\mathop{\mathrm{d}\mathrm{d}^{c}}(u)^{n}$ is defined to be $T_{n}$. What makes this construction work is the fact that at each step, $uT_{k}$ is a well defined current (product of a continuous function and of a positive current), and one has to prove that $T_{k+1}$ is again a closed positive current. The other way, which shall be the one akin to a generalization in the ultrametric framework, consists in observing that if $L$ is a line bundle with a continuous semi-positive metric $\left\|{\cdot}\right\|$, then there exists a sequence of smooth semi-positive metrics $\left\|{\cdot}\right\|_{k}$ on the line bundle $L$ which converges uniformly to the initial metric : for any local section $s$, $\left\|{s}\right\|_{k}$ converges uniformly to $\left\|{s}\right\|$ on compact sets. The curvature current $c_{1}(\overline{L})$ is then the limit of the positive currents $c_{1}(\overline{L}_{k})$, and the measure $c_{1}(\overline{L})^{n}$ is the limit of the measures $c_{1}(\overline{L}_{k})^{n}$. (We refer to [43] for the global statement ; to construct the currents, one can in fact work locally in which case a simple convolution argument establishes the claim.)

An important example of semi-positive metric which is continuous, but not smoth, is furnished by the Weil metric on the line bundle $\mathscr{O}(1)$ on ${\mathbf{P}}^{n}({\mathbf{C}})$. This metric is defined as follows : if $\mathrm{U}\subset{\mathbf{P}}^{n}({\mathbf{C}})$ is an open set, and $s$ is a section of $\mathscr{O}(1)$ on $U$ corresponding to an analytic function $F_{s}$ on $\pi^{-1}(\mathrm{U})\subset{\mathbf{C}}^{n+1}_{*}$ which is homogeneous of degree $1$, then for any $(x_{0},\dots,x_{n})\in\pi^{-1}(\mathrm{U})$, one has

$$\left\|{s}\right\|_{\mathrm{W}}=\frac{\left|{F_{s}(x_{0},\dots,x_{n})}\right|}{\max(\left|{x_{0}}\right|,\dots,\left|{x_{n}}\right|}.$$

The associated measure $c_{1}(\overline{\mathscr{O}(1)}_{\mathrm{W}})^{n}$ on ${\mathbf{P}}^{n}({\mathbf{C}})$ is as follows, cf. [58, 43] : the subset of all points $[x_{0}:\dots:x_{n}]\in{\mathbf{P}}^{n}({\mathbf{C}})$ such that $\left|{x_{j}}\right|=\left|{x_{k}}\right|$ for all $j,k$ is naturally identified with the polycircle $\mathbf{S}_{1}^{n}$ (map $[x_{0}:\dots:x_{n}]$ to $(x_{1}/x_{0},\dots,x_{n}/x_{0})$) ; take the normalized Haar measure of this compact group and push it onto ${\mathbf{P}}^{n}({\mathbf{C}})$.