Recall that we say that a bounded measurable set $S\subset\mathbb R^n$ is said to be *Caccioppoli* if the indicator function $1_S$ is BV, and we set
$$
\operatorname{perim}(S)=\| \nabla 1_S\|_{TV}
$$
where $\|\cdot\|_{TV}$ denotes the total variation. So, if $S$ and $T$ are Caccioppoli sets, is it known whether $S\cap T$ is Caccioppoli?