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what is the exterior measure of $\mathcal{N}$, which is defined by picking a representation of the coset $\mathbb{R}$/$\mathbb{Q}$ in the interval [0,1]. Is it zero? How to prove?

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    I do not believe it's $0$. Were it, for each $n$ we could find an open set $U_n$ containing $\mathcal{N}$ satisfying $m^*(U_n) < \frac{1}{n}$. Then $\mathcal{N}$ is contained in the intersection of all the $U_n$, and this intersection is a $G_{\delta}$ set of measure $0$. By completeness of Lebesgue measure, $\mathcal{N}$ is measurable.2012-12-28
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    http://math.stackexchange.com/questions/182870/what-is-the-outer-measure-of-vitali-set Answer here2012-12-28
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    then what is the exterior measure of $\mathcal{N}$,is it undetermined?2012-12-28
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    but if for every given n, we can find a representation that is in (0,$\frac{1}{n}$),then this $\mathcal{N}$is contained in (0,$\frac{1}{n}$). Is there a logical mistake here?2012-12-28
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    No, why should there be? For every $n$ you have a different $\mathcal{N}_n$ contained in $(0,\frac1n)$.2012-12-28
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    If so, then the outer measure of $\mathcal{N}$ can be any value. But how to construct such a set with a given value $a$ as its outer measure?2012-12-29
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    To answer your last question, see [this](http://math.stackexchange.com/q/14591/8271)2012-12-29

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