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In the following conservation (link below) https://www.researchgate.net/post/Bounds_on_the_outer_measure_of_Vitali_Set_in_0_1

There is a person who said that outer measure of the Vitali set on $[0,1]$ is greater than 0. Anyone can tells me why?

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  1. $m^{\star}(A) \ge 0$

  2. $m^{\star}(A) =0$ implies $A$ is measurable.

Let $\mathcal{V}$ denote the Vitali set.

If $m^{\star} (\mathcal{V}) \not >0$ , then $m^{\star}(\mathcal{V}) =0$ implies $\mathcal{V}$ is measurable.

But $\mathcal{V}$ is non measurable.

Infact $\mathcal{V}$ has full outer measure.

( See here)

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