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

Based on Smirnov’s decomposition theorem we prove that every rectifiable $1$-current $T$ with finite mass $\mathbb{M}(T)$ and finite mass $\mathbb{M}(\mathrm{\partial }T)$ of its boundary $\mathrm{\partial }T$ can be approximated in mass by a sequence of rectifiable $1$-currents $T_n$ with polyhedral boundary $\mathrm{\partial }{T}_n$ and $\mathbb{M}(\mathrm{\partial }{T}_n)$ no larger than $\mathbb{M}(\mathrm{\partial }T)$. 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.