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How to construct a Vitali set of outer measure 1. I couldn't understand the argument given here. Isn't there any easier way? I would also like if someone explains that to me.

Thank you in advance!

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  • $\begingroup$ @Abhishanka Saha there seem to be a lot of questions on this same topic like math.stackexchange.com/questions/14591/…. Hope this helps. $\endgroup$ – happymath Jul 29 '14 at 10:48
  • $\begingroup$ I have seen this question already, but I wanted to make a set of outer measure 1 using only the Vitali construction and not taking complement. $\endgroup$ – Abishanka Saha Jul 29 '14 at 11:08
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create a vitali set like $V$ in $\mathbb{R}$. It's exterior measure $m_*(V)$ is none-zero. now the set $\frac{1}{m_*(V)}V$ that is a scale of elements of $V$ is a set with exterior measure 1.

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  • $\begingroup$ What if the outer measure is $\infty$? How do I calculate the outer measure of any Vitali set? $\endgroup$ – Abishanka Saha Jul 29 '14 at 11:49
  • $\begingroup$ @AbishankaSaha its enough that you create the first vitali set a subset of $[0,1]$ and with this assumption the exterior measure is finite. $\endgroup$ – user120269 Jul 30 '14 at 12:35

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