Let's consider a non-empty open set $A \subset \mathbb{R}^{n}$. How to prove that there exist a finite or a countable family of open pairwise disjoint balls $$\bigcup_{n \in I} {A_{n}}$$ $$ \forall n \in I, A_{n} \subset A$$ so that $$m(A/ \bigcup{A_{n}}) = 0$$ where $m$ is Lebesgue measure.
Despite the fact that the by the light of the nature this fact seems to be correct, i failed in proceeding a rigorous proof. How to obtain it?
The fact is extremely useful, for example, while proving that the Lebesgue outer measure is invariant under the rotations, since we can approximate the instant set not only by the rectanges, but also by open balls, which are clearly invariant under the transformation.
Any sort of help would be much appreciated.
I think you mean to say $A_n \subset A$ rather than $A_n \in A$.
For each $x \in A$ let $r(x) = \sup\{r > 0 : B(x,r) \subset A\}$. Then $r(x) > 0$ for all $x \in A$, and if $0 < r < r(x)$ then $B(x,r) \subset A$. Define $$ {\cal V} = \{\overline{B(x,r)} \mid x \in A,\ 0 < r < r(x)\}.$$
${\cal V}$ is a Vitali cover of $A$. Now apply Vitali's covering theorem to extract an an most countable disjoint subcollection of closed balls in ${\cal V}$ whose union covers almost all of $A$. Their interiors are the sets you need.