Let $X$ be a separated scheme of finite type over $K$ of dimension $n$. Let $M_{1},...,M_{n}$ be line bundles on $X^{\textup{an}}$, $V$ an open subset of $X^{\textup{an}}$ and for $i\in\{1,...,n\}$ let $(\|\cdot\|_{i,k})_{k\in\mathbb{N}}$ be piecewise $\mathbb{Q}$-linear metrics on $M_{1}\Big|_{V},...,M_{n}\Big|_{V}$ converging uniformly to a continuous metric $\|\cdot\|_{i}$ on $M_{i}\Big|_{V}$. Suppose that all $\|\cdot\|_{i,k}$ are semipositive in $V$. Write $\overline{M}_{i,k}:=\left(M_{i},\|\cdot\|_{i,k}\right)$ and let $\overline{L}_{1},...,\overline{L}_{n}$ be line bundles on $X^{\textup{an}}$ endowed with piecewise $\mathbb{Q}$-linear metrics on $V$. Then the measures $c_{1}\left(\overline{L}_{1}\otimes\overline{M}_{1,k}\right)\wedge...\wedge c_{1}\left(\overline{L}_{n}\otimes\overline{M}_{n,k}\right)$ converge weakly to a Radon measure on $V$ denoted by $c_{1}(\overline{L}_{1}\otimes(M_{1},\|\cdot\|_{1}))\wedge...\wedge c_{1}(\overline{L}_{n}\otimes(M_{n},\|\cdot\|_{n}))$ (as above this measure does not depend on the choice of the $\|\cdot\|_{i,k}$).