Wikipedia states that
the Lebesgue measure \lambda is an extension of "the" Borel measure which possesses the crucial property that it is a complete measure (unlike the Borel measure).
However I have read that for every Lebesgue-measurable set a subset can be found, which is not measurable (some kind of Vitali set inside a measurable set, if I'm not mistaken). So I could take a null set, e.g. the Cantor set, and with help of the axiom of choice I can create some sort of Vitali set, which is a subset of the Cantor set. This, however would contradict Wikipedia's claim that the Lebesgue measure is a complete measure.
Pretty new to the whole measure theory stuff, so I guess my mistake is a pretty obvious one. Any help is greatly appreciated :)