Vitali set

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1. The problem statement, all variables and given/known data
Let V a subset of the real line be called a vitali set if V contains precisely one point from each coset of the group of rational numbers. Prove:


2. Relevant equations
1) every lebesgue measurable subset of V is a nullset.
2) V is not lebesgue measurable
3) every set of positive Lebesgue outer measure contains a set that is not lebesgue measurable.



3. The attempt at a solution

i already did 1 and 2.
i am stuck at 3,
i think i need to consruct a vitali set in the given one which i could do if the set contains an interval,..
but it doesnt need to , all i could know about it is that it is not countable since if it were, then its outermeasure will be zero.

thanks a lot
any help is appreciated
 

jbunniii

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If S is a set of positive Lebesgue outer measure, not every coset of [itex]\mathbb{Q}[/itex] will necessarily intersect S, but uncountably many of them will. Try defining V as follows: from each coset that does intersect S, choose one point from the intersection. Then proceed as you did in 2.

[Edit]: My construction will assure that V is not too small, but you also need to make sure that it isn't too big. But S has outer measure > 0, so there must exist some interval I (say, of length 1) such that [itex]S \cap I[/itex] has outer measure > 0. So replace S by [itex]S \cap I[/itex] and before constructing V.
 
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