  We prove that if $\Phi:X\to Y$ a mapping of weak bounded length distortion
from a quasiconvex and complete metric space $X$ to any metric space $Y$, then
for any Lipschitz mapping $f:\mathbb{R}^k\supset E\to X$ we have that
${\mathcal H}^k(f(E))=0$ in $X$ if and only if ${\mathcal H}^k(\Phi(f(E)))=0$
in $Y$. This generalizes an earlier result of Haj\l{}asz and Malekzadeh where
the target space $Y$ was a Euclidean space $Y=\mathbb{R}^N$.
