Zero volume boundary for extension domains from Sobolev to BV

In this note, we prove that the boundary of a \((W^{1, p}, BV)\) $(W^{1,p},BV)$ -extension domain is of volume zero under the assumption that the domain \({\Omega }\) $\mathrm{\Omega }$ is 1-fat at almost every \(x\in \partial {\Omega }\) $x\in \partial \mathrm{\Omega }$ . Especially, the boundary of any planar \((W^{1, p}, BV)\) $(W^{1,p},BV)$ -extension domain is of volume zero.