# Find the sup of a non-measurable set

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 $N$.

• By "the supremum of the outer measure of $N$", are you taking the supremum over all $N$ that we can construct this way? – user125932 Mar 31 '17 at 5:36
• yeah, all the N. The difference just lies in how to choose element in each classes. – Ivon Mar 31 '17 at 7:15

EDIT: The answer is that the supremum is $1$, but my proof for this fact was incorrect (so I am deleting it). Correct answers to the question are already available on this site, so I am linking to them instead.

• Sorry, could you explain why $(a, b) \subset E \cup X$? – Ivon Apr 2 '17 at 14:45
• I had missed this when writing the proof, but $[a, b]$ cannot be in $E \cup X$. However, the original question has already been answered on this site, so I am editing my answer to link there instead. – Nicolas Pelletier Apr 3 '17 at 10:08