1
$\begingroup$

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.

$\endgroup$
  • $\begingroup$ If I understood you correctly, you don't get a subcovering of $E$, only one for $K$. $\endgroup$ – Thomas Nov 29 '12 at 14:22
  • $\begingroup$ @Thomas yes and what's the problem?It does not change the final result. $\endgroup$ – user40276 Nov 29 '12 at 14:25
  • $\begingroup$ Sorry, that was not the problem, but see my answer below. $\endgroup$ – Thomas Nov 30 '12 at 8:58
1
$\begingroup$

You cannot guarantee that the subcovering is disjoint.

$\endgroup$
  • $\begingroup$ However you can obtain a disjoint subcovering by this one, you just have to cut the overlapping intervals, and since there are only a finite numbers of intervals you will just lose some $\epsilon$'s. Furthermore these ones will belong to $\mathcal{V}$ since it's a Vitali covering. $\endgroup$ – user40276 Nov 30 '12 at 16:27
  • $\begingroup$ I don't see how truncating the intervals gives you elements of $\mathcal{V}$. A Vitali covering only guarantees you the membership in the collection of some interval around the point smaller than a prescribed length. $\endgroup$ – Thomas Nov 30 '12 at 21:57

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.