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?



Is the intersection of two Caccioppoli (i.e. finite perimeter) sets Caccioppoli?
**That is true.** Caccioppoli sets are also known as sets of finite perimeter. 



> 
> **Theorem.** *Suppose $f\in L^1(\mathbb{R}^n)$ vanishes outside the unit cube $[0,1]^n$. For $i=1,2,\ldots,n$ consider the function
>  $V\_if(x\_1,\ldots,x\_{i-1},x\_{i+1},\ldots,x\_n)=
>  V\_0^1f(x\_1,\ldots,x\_{i-1},\cdot,x\_{i+1},\ldots,x\_n)$, i.e. the
>  (essential) variation of the one dimensional sections. Then $f\in
>  BV(\mathbb{R}^n)$ if and only if for every $i$, $V\_i\in
>  L^1([0,1]^{n-1}$).*
> 
> 
> 


This is Theorem 5.3.5 in: 


**W. P. Ziemer**, *Weakly differentiable functions. Sobolev spaces and functions of bounded variation.* Graduate Texts in Mathematics, 120. Springer-Verlag, New York, 1989. 


Basically it is a characterization of the functions of bounded variation by one dimensional slices. 


If $h\_1,h\_2$ are characteristic functions of sets in $[0,1]$, then $V\_0^1(h\_1h\_2)\leq V\_0^1h\_1+V\_0^1 h\_2$, see the proof of Theorem 2 in <http://mathonline.wikidot.com/multiples-and-products-of-functions-of-bounded-variation>. Now if $S$ and $T$ are Caccioppoli sets cotained in the unit cube, the characteristic functions $\chi\_S$ and $\chi\_T$ have bounded variation and the one dimensional result mentioned here shows that
$$
V\_i(\chi\_S\chi\_T)\leq V\_i(\chi\_S)+V\_i(\chi\_T)\in L^1([0,1]^{n-1}).
$$
That implies that $\chi\_S\chi\_T=\chi\_{S\cap T}$ has bounded variation so $S\cap T$ is a Caccioppoli set.


I assumed here that the sets are contained in the unit cube, but the argument applies to any bounded set.


Another answer is provided in a comment by Manfred Sauter (see above).


