Integration of differential forms on manifolds with locally-finite variations

Abstract

It is well known that one can integrate any compactly supported, continuous, differential n-form over n-dimensional C¹-manifolds in ℝ^m (m ≥ n). For n = 1, the integral may be defined over any locally rectifiable curve. Another generalization is the theory of currents (linear functionals on the space of compactly supported C^∞-differential forms). The theme of the article is integration of measurable differential n-forms over n-dimensional n C⁰-manifolds in ℝ^m with locally-finite n-dimensional variations (a generalization of locally rectifiable curves to dimension n > 1). The main result states that any such manifold generates an n-dimensional current in ℝ^m (i.e., any compactly supported C^∞ n-form can be integrated over a manifold with the properties mentioned above). Bibliography: 8 titles.