after seeing some proofs of the Vitali covering lemma for the Lebesgue outer measure, I asked myself why no one used compacity since this aroses naturally in this kind of problem.
If $\mathcal {V}$ is a Vitali covering (composed intervals, not necessarilly closed) for some set $E$ such that $m*(E) < +\infty$, then picking the intervals such that if itś not open, suppose $(a,b]$, then you add $(b-\epsilon, b+\epsilon)$ to it. After this you get a open covering for $E$, then picking an compact subset $K$ of $E$ such that they differs in outer measure by an tiny $\epsilon'$, then, by compacity, you can get a finite open covering for $K$ and since itś finite you can now exclude the $\epsilon$ 's added in the beggining, resulting on a finite subcovering of $\mathcal{V}$. Is there anything wrong in this proof(idea).
Thanks in advance.