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