What is the outer measure of Vitali set? [duplicate]

measure-theory

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?


Solution 1:

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:

  • Vitali-type set with given outer measure
  • Are Vitali sets dense in [0,1)?
  • Vitali set of outer-measure exactly $1$.

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

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