Let $\Omega\subseteq\mathbb{R}^n$ be an open set and $E$ be a set of finite perimeter in $\Omega$ (i.e., the indicator function $\chi_E\in BV(\Omega)$), $\|\partial E \|(\Omega)$ the perimeter of $E$ in $\Omega$. Let $B_k$ be the open ball with center in $0$ and radius $k$. Let $E_k:=E\cap B_k$.
Is it true that
$$ \|\partial (E_k) \|(\Omega) \xrightarrow{k\to\infty} \|\partial (E) \|(\Omega) \quad ? $$
No, take $\Omega=\mathbb R$ and $E=\mathbb R\setminus [-1,1]$. $|\partial E|=2$. But $|\partial E_k|=4$ for any $k>1$.