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Integration of differential forms on manifolds with locally finite variations
Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 33, Tome 327 (2005), pp. 168-206 Cet article a éte moissonné depuis la source Math-Net.Ru

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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). 
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     author = {A. V. Potepun},
     title = {Integration of differential forms on manifolds with locally finite variations},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {168--206},
     year = {2005},
     volume = {327},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2005_327_a10/}
}
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A. V. Potepun. Integration of differential forms on manifolds with locally finite variations. Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 33, Tome 327 (2005), pp. 168-206. http://geodesic.mathdoc.fr/item/ZNSL_2005_327_a10/

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