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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 inmass 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 ofthe $h$-mass for polyhedral $1$-currents with respect to the joint weak-$\ast$convergence of currents and their boundaries. We obtain that this relaxationcoincides with the usual $h$-mass for normal currents. This shows that theconcepts of so-called generalized branched transport and the $h$-mass areequivalent.

Approximation of rectifiable $1$-currents and weak-$\ast$ relaxation of the $h$-mass