Doubts on the Vitali Set - Marco's Story

[muzialis]

Hello all,

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

 

[HallsofIvy]

Hello all,

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.

No, there is no such set containing all irrationals. For example, e- \pi is not a rational number so e and \pi cannot be in the same equivalence class.

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

Your error, as I said, is in thinking that there was such a set containing all irrational numbers.
The set of all irrational numbers, between 0 and 1, is not, of course, "non-measurable"- it has measure 1.

 

[lavinia]

Hello all,


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

This is where you went wrong.

 

[muzialis]

Correct indeed, many thanks