I am currently studying E.Stein and R.Sharkarchi's Real Analysis. I have got a problem about the non-measurable set constructed in text named $N$. I have proved that the outer measure of $[0, 1]-N$ is 1. And the outer measure of $N$ is dependent upon the choosing of element in each equivalence class, because you can choose one in each class to make the outer measure however small as you like.
My question is : what is the supremum of the outer measure of $N$? Is it 1?
In case you need to revise the definition of $N$: Construct an equivalence among all the real numbers in $[0,1]$, by: If $a-b$ is rational, then a~b. Otherwise a and b are in different classes. Now we get a classification of real numbers between 0 and 1. Choose one element from each equivalent class, then these elements form a non-measurable subset of $[0,1]$. Note it as $N$.