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 


$$P\_D(A)=\sup \left \lbrace \int\_D \chi\_A {\rm div} \varphi \ dx \ : \ \varphi \in C\_c^1(D;\Bbb{R}^N),\ |\varphi|\leq 1 \right\rbrace $$



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


Thank you. 



A property of sets of finite perimeter
I'm sorry that I answer my own question, but I found out the answer this morning from my teacher. There are examples of sets of finite perimeter with positive measure, which do not contain any open ball.


For example, take $D=B(0,1)$, the unit ball in $\Bbb{R}^2$ and denote $S=D \cap \Bbb{Q}^2=(x\_n)\_{n \geq 0}$. Then, we can find a sequence of positive real numbers $(r\_n)$ such that


* $\sum\_{n=1}^\infty 2\pi r\_n < \infty$
* $\sum\_{n=1}^\infty \pi r\_n^2 < \pi$


Take $B=\bigcup\_{n=1}^\infty B(x\_n,r\_n) $. Then $C=D\setminus B$ has the desired property. Indeed, $|C|=|D|-|B|>0$, and $Per\_D(C)=Per\_D(B)<\infty$. 


If $C$ would contain an open ball then that ball would intersect $S \subset B$, which is not possible.


