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I have a doubt on the Vitali set.

In its construction we define equivalence classes, two numbers belonging to one if their difference is rational.

Then we "pick" a member per class, forming a set. It is then shown that the interval [0,1] is a disjoint union of such set (after a rational translation), hence after some reasonsing this set is non-measurable.

It seems to me that one such set is formed by a rational number plus all irrationals.

Now, the interval [0,1] is also the disjoint union of the rationals and the irrationals.

Hence, as the rationals are a set of zero lebesgue meausure, and as the irrationals are not measurable (differing by only one member from the the set built during the Vitali construction), one could conclude the interval [0,1] is the disjoiunt union of a zero measure set and a non - measurable set, which seems a contradiction.

Can anybody point out my mistake as I can not?

Thanks

Marco

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# Vitali set

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