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We introduce the notion of set-decomposition of a normal G-flat chain. Weshow that any normal rectifiable $G$-flat chain admits a decomposition inset-indecomposable sub-chains. This generalizes the decomposition of sets offinite perimeter in their ``measure theoretic'' connected components due toAmbrosio, Caselles, Masnou and Morel. It can also be seen as a variant of thedecomposition of integral currents in indecomposable components by Federer.Asopposed to previous results, we do not assume that G is boundedly compact.Therefore we cannot rely on the compactness of sequences of chains withuniformly bounded N-norms. We deduce instead the result from a new abstractdecomposition principle. As in earlier proofs a central ingredient is thevalidity of an isoperimetric inequality. We obtain it here using the finitenessof some h-mass to replace integrality.

Set-decomposition of normal rectifiable G-chains via an abstract decomposition principle