Let us now return to semi-positive metrized line bundles $\overline{L}_{0},\dots,\overline{L}_{k-1}$, approximated by smooth semi-positive metrized line bundles $\overline{L}_{j}^{(m)}$. I claim that for any $k$-dimensional variety $\mathrm{Z}\subset\mathrm{X}$, the sequence of measures $c_{1}(\overline{L}_{0}^{(m)})\dots c_{1}(\overline{L}_{k-1}^{(m)})\delta_{\mathrm{Z}}$ converges to a measure on $\mathrm{X}$.

To prove the claim, we may assume that $\overline{L}_{0},\dots,\overline{L}_{k-1}$ have sections $s_{0},\dots,s_{k-1}$ without common zeroes on $\mathrm{Z}$. Let also consider a smooth function $\varphi$ on $\mathrm{X}$ ; let $\overline{L}_{k}$ be the trivial line bundle with the section $s_{k}=1$, metrized in such a way that $\left\|{s_{k}}\right\|=e^{-\varphi}$. Then, one has

$$\int_{\mathrm{X}}\varphi c_{1}(\overline{L}^{(m)}_{0})\dots c_{1}(\overline{L}^{(m)}_{k-1})\delta_{\mathrm{Z}}=(\mathop{\widehat{\operatorname{div}}}(s_{0})^{(m)}\dots\mathop{\widehat{\operatorname{div}}}(s_{k-1})^{(m)}\mathop{\widehat{\operatorname{div}}}(s_{k})|\mathrm{Z});$$

writing $\overline{L}_{k}$ has the quotient of two ample metrized line bundles, we deduce from the existence of the local height pairing for admissible metrics that these integrals converge when $m\rightarrow\infty$. Consequently, the sequence of measures $(c_{1}(\overline{L}^{(m)}_{0})\dots c_{1}(\overline{L}^{(m)}_{k-1})\delta_{\mathrm{Z}})_{m}$ converges to a positive linear form on the space of smooth functions. By a theorem of Gubler ([34], Theorem 7.12), which builds on the Stone-Weierstraß theorem and the compactness of the Berkovich space $\mathrm{X}$, the space of smooth functions is dense in the space of continuous complex functions on $\mathrm{X}$. A positivity argument, analogous to the proof that positive distributions are measures, then implies that our linear form is actually a positive measure which deserves the notation

$$c_{1}(\overline{L}_{0})\dots c_{1}(\overline{L}_{k-1})\delta_{\mathrm{Z}}.$$

We then extend this definition by linearity to the case of arbitrary admissible line bundles. The total mass of this measure is again the multidegree of ${\mathrm{Z}}$ with respect to the line bundles $L_{j}$ (for $0\leq j\leq k-1$).