Inserted: 25 jul 2019
Last Updated: 25 jul 2019
We study a measure-theoretic notion of connectedness for sets of finite perimeter in the setting of doubling metric measure spaces supporting a weak $(1,1)$-Poincaré inequality. The two main results we obtain are a decomposition theorem into indecomposable sets and a characterisation of extreme points in the space of BV functions. In both cases, the proof we propose requires an additional assumption on the space, which is called isotropicity and concerns the Hausdorff-type representation of the perimeter measure.
Keywords: Set of finite perimeter, PI space, Indecomposable set