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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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    To answer your last question, see [this](http://math.stackexchange.com/q/14591/8271)2012-12-29

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