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

Abstract

We introduce the notion of set-decomposition of a normal G -flat chain A in \mathbb{R}^{n} as a sequence A_{j}=A\,\text{\Large{$\llcorner$}}S_{j} associated to a Borel partition S_{j} of \mathbb{R}^{n} such that \mathbb{N}(A)=\sum \mathbb{N}(A_{j}) . 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.