Proving the Vitali Set: A Real Line Challenge

[jem05]

Homework Statement

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:

Homework 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.

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 doesn't 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]

If S is a set of positive Lebesgue outer measure, not every coset of $\mathbb{Q}$ 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 $S \cap I$ has outer measure > 0. So replace S by $S \cap I$ and before constructing V.