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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 $N$.

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

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  • $\begingroup$ Sorry, could you explain why $(a, b) \subset E \cup X$? $\endgroup$ – Ivon Apr 2 '17 at 14:45
  • $\begingroup$ 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. $\endgroup$ – Nicolas Pelletier Apr 3 '17 at 10:08

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