Possible Duplicate:
Vitali-type set with given outer measure

Given that the construction of Vitali set is based on the axiom of choice. How can the outer measure of this set be calculated?


marked as duplicate by t.b., Asaf Karagila, Martin Sleziak, Jonas Teuwen, Jack Schmidt Aug 15 '12 at 16:10

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

  • $\begingroup$ It isn't determined from the usual description alone. $\endgroup$ – t.b. Aug 15 '12 at 15:41
  • $\begingroup$ @t.b. Many thanks, I guess this is a duplicate then. I will flag it. $\endgroup$ – Vital Aug 15 '12 at 15:43
  • $\begingroup$ No need to flag it. See also: math.stackexchange.com/questions/157532, math.stackexchange.com/q/32214 $\endgroup$ – t.b. Aug 15 '12 at 15:43
  • $\begingroup$ @t.b. If you post your comment as an answer with the references, I will accept it. This is what I was looking for. Thanks. $\endgroup$ – Vital Aug 15 '12 at 15:47

On Vital's request:

There isn't just one Vitali set: each choice of representatives of the equivalence relation on $\mathbb{R}$ given by $x \sim y$ if and only if $y - x \in \mathbb{Q}$ yields what one calls a Vitali set. You can arrange them to have any given positive outer measure you want.

There are many threads on this site where Vitali sets were discussed, among which:

You can find a few more by Googling for "Vitali set" site:math.stackexchange.com


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