One has to show that for any continuous function $\varphi$ with compact support contained in $\mathrm{U}$

$$\int_{\mathrm{X}}\varphi\,c_{1}(\overline{L}_{1})\dots c_{1}(\overline{L}_{k})\delta_{\mathrm{Z}}=0.$$

By Gubler’s theorem, the space of smooth functions is dense in the space of smooth functions on $\mathrm{X}$. Using the fact that the maximum and the minimum of smooth functions are still smooth, one proves that the space of smooth functions with compact support contained in $\mathrm{U}$ is dense in the space of continuous functions with compact support contained in $\mathrm{U}$, for the topology of uniform convergence. We thus may assume that $\varphi$ is smooth, with compact support contained in $\mathrm{U}$. Finally, we may also assume that the metric on the line bundles $\overline{L}_{2},\dots,\overline{L}_{k}$ are smooth.

We may argue locally and assume that $L_{1}$ has a meromorphic section $s$ whose divisor $\operatorname{div}(s)$ is disjoint from $\mathrm{U}$. Up to shrinking $\mathrm{U}$ again, we may assume that there exists a sequence $(u_{n})$ of rational functions without zeroes nor poles on $\mathrm{U}$ such that $\log\left\|{s}\right\|=\lim\log\left|{u_{n}}\right|^{1/n}$.

According to Prop. 1.3.2, one has

$$\int_{\mathrm{X}}\varphi\,c_{1}(\overline{L}_{1})\dots c_{1}(\overline{L}_{k})\delta_{\mathrm{Z}}\\ =\int_{\mathrm{X}}\varphi c_{1}(\overline{L}_{2})\dots c_{1}(\overline{L}_{k})\delta_{\operatorname{div}(s|_{\mathrm{Z}})}+\int_{\mathrm{X}}\log\left\|{s}\right\|^{-1}\mathop{\mathrm{d}\mathrm{d}^{c}}\varphi c_{1}(\overline{L}_{2})\dots c_{1}(\overline{L}_{k})\delta_{\mathrm{Z}},$$

The first term vanishes because $\operatorname{div}(s|_{\mathrm{Z}})$ and the support of $\varphi$ are disjoint. The second is the limit of

$$\int_{\mathrm{X}}\log\left|{u_{n}}\right|^{-1/n}\mathop{\mathrm{d}\mathrm{d}^{c}}\varphi c_{1}(\overline{L})^{k-1}\delta_{\mathrm{Z}}.$$

Using the fact that $\operatorname{div}(u_{n})\cap\mathrm{U}$ is empty and applying the same computation, the term of index $n$ equals

$$\frac{1}{n}\int_{\mathrm{X}}\varphi\,c_{1}(\overline{M}_{n})c_{1}(\overline{L}_{2})\dots c_{1}(\overline{L}_{k})\delta_{\mathrm{Z}}.$$

where $\overline{M}_{n}$ is the trivial metrized line bundle $\mathscr{O}_{X}$, and its meromorphic section $u_{n}$ replacing $s$. But this integral is zero, by definition of the measures associated to smooth metrized line bundles. ∎