Let $(\mathbb{K},\mathfrak{M}_{\mathbb{K}})$ be a global field and $X$ a proper variety over $\mathbb{K}$ of dimension $n$. Let $d\leq n$ and let ${\overline{L}}_{i}$, $i=0,\dots,d$, be a family of line bundles with quasi-algebraic integrable metrics. Then every $d$-dimensional cycle of $X$ is integrable with respect to ${\overline{L}}_{0},\dots,{\overline{L}}_{d}$.