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).

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).

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.

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).