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# Can a Vitali set be Lebesgue measurable? (ZF)

Here is the definition of Lebesgue measure.

The standard proof that Vitali sets are not Lebesgue measurable uses countable additivity of Lebesgue measure, which is not a theorem of ZF. (In particular, it is consistent that the real line is a countable union of countable sets, and thus a countable union of measure zero sets.) Since ZF does prove that Lebesgue measure is super-additive, that proof can be easily adapted to show in ZF that if a Vitali set is measurable, then its measure is zero. By the Caratheodory construction, this is equivalent to having outer measure zero.

Does ZF prove that all Vitali sets have positive outer measure?

If no, does ZF prove "if there exists a Vitali set, then there exists a Vitali set with positive outer measure"?