Every metric 1-current in a complete quasiconvex metric space is a mass limit of normal 1-currents, and in complete separable spaces it is an integral of curve fragments.
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Structure of Metric $1$-currents: approximation by normal currents and representation results
Every metric 1-current in a complete quasiconvex metric space is a mass limit of normal 1-currents, and in complete separable spaces it is an integral of curve fragments.