Rectifiable and flat \(G\) chains in a metric space (Q2879623)

The authors provide some new definitions and results in order to handle simultaneously some generalizations of rectifiable chains or currents considered in the works of B. White, L. Ambrosio and B. Kirchheim as well as U. Lang. The authors work in a general complete metric space \(X\) with chains having coefficients in any complete normed Abelian group. Sometimes it is assumed that the coefficient group \(G\) satisfy a no-Lipschitz condition. After some preliminaries concerning the metric space \(X\) and the coefficient group \(G\), the authors study the group \(\mathcal{R}(X;G)\) of \(m\)-dimensional rectifiable \(G\)-chains in \(X\). Then they study the rectifiable sets in a metric space, \(0\)-dimensional rectifiable \(G\)-chains, \(m\)-dimensional parametrized and rectifiable \(G\)-chains, restriction and sum, push-forward and some characterizations of rectifiable \(G\)-chains. The next subjects are the slicing, the group \(\mathcal{F}_0(X,G)\) of \(0\)-dimensional flat \(G\)-chains in \(X\), \(G\)-oriented Lipschitz curves, the boundary of a \(1\)-dimensional Lipschitz \(G\)-chain, the flat norm \(\mathcal{F}\) and flat completion, the boundary of an \(m+1\)-dimensional Lipschitz \(G\)-chain. The authors study the properties of the flat chains in finite dimensional spaces: a slice-null flat chain is zero, the slicing mass of a rectifiable chain and its comparability to \(\mathbb{M}\), the slicing mass of a flat chain, the Borel regular measure, the restriction of a finite mass \(T\) to a \({\mu}_T\) measurable set \(A\), the group of \(0\)-dimensional flat chains of finite mass, the \(G\)-valued Borel measure, rectifiability, the group of \(m\)-dimensional flat chains of finite mass, rectifiability and virtual flat chains.