  Based on Smirnov's decomposition theorem we prove that every rectifiable
$1$-current $T$ with finite mass $\mathbb{M}(T)$ and finite mass
$\mathbb{M}(\partial T)$ of its boundary $\partial T$ can be approximated in
mass by a sequence of rectifiable $1$-currents $T_n$ with polyhedral boundary
$\partial T_n$ and $\mathbb{M}(\partial T_n)$ no larger than
$\mathbb{M}(\partial T)$. Using this result we can compute the relaxation of
the $h$-mass for polyhedral $1$-currents with respect to the joint weak-$\ast$
convergence of currents and their boundaries. We obtain that this relaxation
coincides with the usual $h$-mass for normal currents. This shows that the
concepts of so-called generalized branched transport and the $h$-mass are
equivalent.
