Metric currents and Alberti representations

We relate Ambrosio–Kirchheim metric currents to Alberti representations and Weaver derivations. In particular, given a metric current T  , we show that if the module X(‖T‖)X(‖T‖) of Weaver derivations is finitely generated, then T can be represented in terms of derivations; this extends previous results of Williams. Applications of this theory include an approximation of 1-dimensional metric currents in terms of normal currents and the construction of Alberti representations in the directions of vector fields.