Vitali set of outer measure 1

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!

• @Abhishanka Saha there seem to be a lot of questions on this same topic like math.stackexchange.com/questions/14591/…. Hope this helps. – happymath Jul 29 '14 at 10:48
• 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. – Abishanka Saha Jul 29 '14 at 11:08

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