TITLE: extensions of lebesgue measure
QUESTION [10 upvotes]: The Hahn-Banach theorem implies that Lebesgue measure can be extended give a "measure" on all subsets of [0,1], but this measure is only guaranteed to be finitely additive.  It might magically turn out that this measure is countably additive, but this can only happen if the continuum is a real-valued measurable cardinal, a strong set-theoretic assumption.  My question is: if it turns out that measure is countably additive on the measure zero sets, does this imply that the measure is countably additive everywhere?

REPLY [3 votes]: The answer is no. A proof can be found here.