# Indecomposable sets of finite perimeter in doubling metric measure spaces

## TL;DR

This paper investigates the structure of finite perimeter sets in doubling metric measure spaces, providing a decomposition into indecomposable parts and characterizing extreme points of BV functions under isotropicity conditions.

## Contribution

It introduces a measure-theoretic connectedness concept and proves a decomposition theorem and a characterization of extreme points in BV spaces in this setting.

## Key findings

- Decomposition of finite perimeter sets into indecomposable components.
- Characterization of extreme points in BV functions.
- Requires isotropicity assumption related to perimeter measure.

## Abstract

We study a measure-theoretic notion of connectedness for sets of finite perimeter in the setting of doubling metric measure spaces supporting a weak $(1,1)$-Poincar\'{e} 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.