It was first introduced by Hassler Whitney in 1957, and then extended to integral currents by Federer and Fleming in 1960.
It forms a fundamental part of the field of geometric measure theory.
The notion was applied to find solutions to Plateau's problem.
In 2001 the notion of an integral current was extended to arbitrary metric spaces by Ambrosio and Kirchheim.
It is a deep theorem of Federer-Fleming that the boundary is then also an integral current.
whose supports lie in a compact set K with a uniform upper bound on
, then a subsequence converges in the flat sense to an integral current.
This theorem was applied to study sequences of submanifolds of fixed boundary whose volume approached the infimum over all volumes of submanifolds with the given boundary.
It produced a candidate weak solution to Plateau's problem.