Inserted: 4 jul 2011
The rich structure of the real line or of the Euclidean space allows to assign a kind of integral to some highly oscillating functions which are not absolutely (Lebesgue) integrable; such integrals have been introduced by Denjoy, Perron, Luzin, Henstock, Kurzweil and others. We show that already the structure on metric space is enough to establish a natural concept of a nonabsolutely convergent integral. We integrate functions with respect to ``distributions'', which (in this metric space setting) are dual objects to Lipschitz test functions. We apply the new integral in the Ambrosio-Kirchheim theory of metric currents in connection with the Stokes formula.