Let $\Omega$ be a bounded open subset of $\mathbb{R}^d$ with $d\ge 2$, and denote by $BV(\Omega)$ the space of functions of bounded variation. Then, is $BV(\Omega) \cap L^\infty(\Omega)$ dense in $BV(\Omega)$?
I know that $BV(\Omega) \cap C^\infty(\Omega)$ is not dense in $BV(\Omega)$ because the Sobolev space $W^{1,1}(\Omega)$ is a proper subset of $BV(\Omega)$, but $W^{1,1}(\Omega)$ is the $BV$-norm closure of $C^\infty(\Omega)$. However, an example of $f\in BV(\Omega)\setminus W^{1,1}(\Omega)$ in my mind is the indicator function of a set with a smooth boundary (cf. Giusti's textbook), which obviously belongs to $L^\infty(\Omega)\setminus C^\infty(\Omega)$. This is the reason why I asked this question.