  We propose a slight variant of Ambrosio and Kirchheim's definition of a
metric current. We show that with this new definition it is possible to obtain
certain volume functionals from Finsler geometry as mass measures of currents.
As an application, we obtain a whole family of extendibly convex $n$-volume
densities. This family includes the mass* and the circumscribed Riemannian
volume densities.
