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?



extensions of lebesgue measure
The answer is no. A proof can be found [here](http://www.math.wisc.edu/~akumar/INDUCED_IDEALS.pdf).


