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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?

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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.

It isn't determined from the usual description alone. – t.b. Aug 15 '12 at 15:41
@t.b. Many thanks, I guess this is a duplicate then. I will flag it. – Vital Aug 15 '12 at 15:43
No need to flag it. See also:, – t.b. Aug 15 '12 at 15:43
@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. – Vital Aug 15 '12 at 15:47
up vote 4 down vote accepted

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"

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