Geodesic
Integration of differential forms on manifolds with locally finite variations. Part II
Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 34, Tome 333 (2006), pp. 66-85 
A. V. Potepun. Integration of differential forms on manifolds with locally finite variations. Part II. Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 34, Tome 333 (2006), pp. 66-85. http://geodesic.mathdoc.fr/item/ZNSL_2006_333_a6/
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     author = {A. V. Potepun},
     title = {Integration of differential forms on manifolds with locally finite variations. {Part~II}},
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     year = {2006},
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     url = {http://geodesic.mathdoc.fr/item/ZNSL_2006_333_a6/}
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Voir la notice du chapitre de livre provenant de la source Math-Net.Ru

In the part I of the paper the $n$-dimensional $C^0$-manifolds in $\mathbb R^n$ $(m\ge n)$ with locally finite $n$-dimensional variations (a generalization of locally rectifiable curves to dimension $n>1$) and integration of measurable differential $n$-forms over such manifolds were defined. The main result of part II states that an $n$-dimensional manifold $C^1$-embedded in $\mathbb R^m$ has locally finite variations and the integral of measurable differential $n$-form defined in part I can be calculated by well-known formula.

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[2] B. Z. Vulikh, Kratkii kurs teorii funktsii veschestvennoi peremennoi, M., 1973

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[4] N. Dinculeanu, Vector measures, Berlin, 1966 | MR

[5] A. V. Potepun, “Integrirovanie differentsialnykh form na mnogoobraziyakh s lokalno konechnymi variatsiyami, chast, I”, Zap. nauchn. semin. POMI, 327, 2005, 168–206 | MR | Zbl