Let ${\overline{L}}=(L,\|\cdot\|)$ be a metrized line bundle on $X$. The metric $\|\cdot\|$ is approachable if there exists a sequence of semipositive smooth (in the Archimedean case) or semipositive algebraic (in the non-Archimedean case) metrics $(\|\cdot\|_{l})_{l\geq 0}$ on $L^{{\text{\rm an}}}$ such that

$$\lim_{l\to\infty}\operatorname{dist}(\|\cdot\|,\|\cdot\|_{l})=0.$$

If this is the case, we say that ${\overline{L}}$ is approachable. This metrized line bundle is integrable if there are approachable line bundles ${\overline{M}}$, ${\overline{N}}$ such that ${\overline{L}}={\overline{M}}\otimes{\overline{N}}^{-1}$.