An m charge in the n dimensional Euclidean space is a linear functional acting on m dimensional polyhedral chains and satisfying the following continuity condition. The value of the linear functional approaches zero on chains whose normal masses are bounded and whose flat norms asymptotically vanish. Our main theorem relates m charges to pairs of continuous differential forms. Luzin's theorem states that every measurable function on the line is the derivative of a continuous, almost everywhere differentiable function. We show this can be improved in several dimensions. Finally we prove that a compact subset C of the n dimensional Euclidean space does not support the distributional divergence of a bounded measurable vector field if and only if C has vanishing (n-1) dimensional Hausdorff measure.

Les méthodes d’intégration de jauge, telle que l’intégrale de Pfeffer sur les ensembles bornés de périmètre fini sont particulièrement adaptées à l’étude des grands théorèmes d’intégration que sont le Théorème Fondamental de l’Analyse, le Théorème de la Divergence et le Théorème de Stokes. Dans cette thèse, ces outils sont transposés à l’intégration sur des domaines singuliers, vus comme des courants entiers au sens de Federer et Fleming. On obtient un critère d’effaçabilité pour les singularités des courants considérés : les courants ayant un ensemble singulier de contenu de Minkowski relatif fini satisfont un Théorème de Stokes général, c’est le cas notamment des courants définissables dans une structure o-minimale quelconque, c’est aussi le cas de courants minimiseurs de masse sans singularité au bord. A contrario, on construit un courant de dimension 2 dans ℝ3 ayant un ensemble singulier réduit à un point, qui ne vérifie pas ce Théorème de Stokes général.Cette thèse contient aussi les définitions de méthodes d’intégration non absolument convergentes sur tout courant entier de dimension 1, ainsi que sur les courants entiers de dimension quelconque dans un espace euclidien dont les singularités sont effaçables. / Methods of gauge integration, like those developped by W. F. Pfeffer on bounded sets of finite perimeter, are well suited to the study of integration theorems, such as the Fundamental Theorem of Calculus, The Divergence Theorem and Stokes’ Theorem. In this thesis, Pfeffer Integration is transposed to the context of integral currents in the sense of Federer and Fleming. Not all integral currents are adapted to this type of gauge integration and a criterion on the singular set of the current is obtained. Well behaved currents include all 1-dimensional integral currents, integral currents definable in an o-minimal structure and mass minimizing integral currents whenever the boundary singularities are controlled. All those currents are shown to satisfy a general Stokes’ Theorem. On the other hand, an example is given of an integral current of dimension 2 in ℝ3 with only one singular point, which does not satisfy such a general Stokes-Cartan Theorem. This thesis also contains the definitions of non-absolutely convergent integrations methods on 1-dimensionalintegral currents as well as on integral currents of any dimension in Euclidean space, whenever their singular set has controlled relative Minkowski content.