# Rectifiability of flat chains

### Abstract

We prove (without using Federer’s structure theorem) that a finite-mass flat chain over any coefficient group is rectifiable if and only if almost all of its $0$-dimensional slices are rectifiable. This implies that every flat chain of finite mass and finite size is rectifiable. It also leads to a simple necessary and sufficient condition on the coefficient group in order for every finite-mass flat chain to be rectifiable.

Brian White