Representing Charges by Forms

A charge in a compact metric space is a continuous linear functional on the space of normal chains with real coefficients equipped with a modified flat norm topology. Charges form a cochain complex whose cohomology reflects metric, rather than topological, properties of metric spaces. Elaborating on an idea of De Giorgi, we construct a normed graded differential algebra of Lipschitz forms in a separable metric space. Each form in the completion of this algebra defines a mass norm continuous linear functional on the space of normal chains. In a compact metric space, these functionals are charges, and each charge can be represented by them. The wedge products of forms define cup products in the cohomology of charges.