Integration of differential forms on manifolds with locally finite variations

It is well known that one can integrate any compactly supported continuous differential $n$-form over $n$-dimensional $C^1$-manifolds in $\mathbb R^m $ ($m\ge n$). 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 $C^\infty$-differential forms). The theme of the article is integration of measurable differential $n$-forms over $n$-dimensional $C^0$-manifolds in $\mathbb R^m$ 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 $\mathbb R^m$ (i.e., any compactly supported $C^\infty$ $n$-form may be integrated over a manifold with the properties mentioned above).