I have a question regarding sets of finite perimeter. I feel that it should be true, but I didn't manage to prove or find a reference about it. Suppose $D$ is an open, bounded subset of $\Bbb{R}^n$, and define the perimeter of a measurable set $A \subset D$ as

$$P_D(A)=\sup \left \lbrace\\ \int_D \chi_A {\rm div} \varphi \ dx \ : \ \varphi \in C_c^1(U;\Bbb{R}^N),\ |\varphi|\leq 1\\ \right\rbrace $$

If $|A|=V$ with $V\in (0,|\Omega|)$ and $P_D(A)<\infty$ is it true that we can modify $A$ up to a set of measure zero in order to find a small ball included in $A$?

If yes, is there any reference, or easy proof for this?

Thank you.