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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5For one thing, it is not measureable! – 2012-08-17
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0I am aware that it is not measurable, but i'm in doubts of the construction of the vitali set itself... it is formed by the collection of representatives of partitions of all real numbers, but why can't the representatives form the interval [0,1] itself? – 2012-08-17
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2Because no two elements of the Vitali set differ by a rational number: that is precisely its characteristic property. The same is not true of the interval $[0,1]$. – 2012-08-17
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2For another thing, the Vitali set contains only one rational number. – 2012-08-17
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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