I need help with this problem, i couldn't find a proof in any book related to measure theory.

Show that every Lebesgue measurable subset of a Vitali set V is a Lebesgue nullset.

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- Oct 24th 2010, 10:24 AMraa91Vitali set
I need help with this problem, i couldn't find a proof in any book related to measure theory.

Show that every Lebesgue measurable subset of a Vitali set V is a Lebesgue nullset. - Oct 24th 2010, 04:00 PMBruno J.
Translates of the Vitali set are disjoint from it.

Suppose it had a measurable subset of nonzero measure. By translating this subset by all rationals in the unit interval, you obtain a countable collection of disjoint sets of real numbers in the interval [0,2], who all have the same nonzero measure (by translation invariance of measure); their countable union is measurable, and since it's a subset of [0,2], it must have finite measure. See what you can make of all this. - Oct 25th 2010, 04:14 AMraa91
Thanks, i did not know these facts from before..:D