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A Vitali set is a subset $V$ of $[0,1]$ such that for every $r\in \mathbb R$ there exists one and only one $v\in V$ for which $v-r \in \mathbb Q$. Equivalently, $V$ contains a single representative of every element of $\mathbb R / \mathbb Q$.

For any $\epsilon>0$, is it possible to construct a Vitali set $H$ so that the outer measure $\lambda^*(H)<\epsilon$?

Hermi
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  • Yes; you can generalize the construction to a small interval and outer measure still respects monotonicity. – kodiak Mar 01 '24 at 02:52
  • @kodiak Can you please say more details about this construction? – Hermi Mar 01 '24 at 03:00
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    The idea is that nothing goes wrong if you change $[0, 1]$ to $[0, \epsilon]$ in the usual construction of a Vitali set. When you do that, you get a set with outer measure at most $\epsilon$ by monotonicity. Discussed here as well. – Izaak van Dongen Mar 01 '24 at 03:47

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