## Vitali sets versus real-valued measurable cardinals

If there exists a real-valued measurable cardinal, then there is a countably additive extension of Lebesgue measure to all sets of real numbers. This would include then the Vitali sets, which are an example of sets that are not Lebesgue measurable -- or at least, apparently for weaker assumptions than the existence of a real-valued measureable cardinal. However, after going over the proof that a Vitali set is not measurable, for example in Vitali set - Wikipedia, the free encyclopedia, I do not see where the proof would fail under the assumption of a real-valued measureable, i.e., assuming that there exists a cardinal κ so that there is an atomless κ-additive measure on the power set of κ. I presume I am missing something breathtakingly obvious. Could someone point this out to me?