I don't seem to get the special properties of Vitali sets which makes them different from the intervals, e.g. [0,1].
Why is the Vitali set not necessarily equal to the interval e.g. [0,1]?
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real-analysis
measure-theory
elementary-set-theory
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0See [here](http://math.stackexchange.com/q/137949/8271) for detailed explanations of the construction of Vitali set(s). – 2012-08-18
1 Answers
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The Vitali set can not be an interval. As Mariano Suarez-Alvarez mentioned, a well-known result is that the Vitali set is not measurable, but every interval is measureable.
However, even more elementary than that: A Vitali set contains exactly one element from each coset of $\mathbb{R} / \mathbb{Q}$ (as additive groups). Alternatively, you can define the equivalence relation on $\mathbb{R}$ by $a \sim b$ if and only if $a - b \in \mathbb{Q}$. The Vitali set is then a set containing exactly one element from each equivalence class. $\mathbb{Q}$ is the equivalence class containing $0$. Hence, Vitali set only contain one rational number. The Vitali set can not be a interval since intervals contain more than a single rational number.
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1@Matt: your accounts have been merged. Please register to avoid such difficulties in the future. – 2012-08-17