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 $A_j=A\text{└}{S}_j$ 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 $\mathbb{N}$-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.