Is Measure Countably Additive?

A measure must further be countably additive: if a 'large' subset can be decomposed into a finite (or countably infinite) number of 'smaller' disjoint subsets that are measurable, then the 'large' subset is measurable, and its measure is the sum (possibly infinite) of the measures of the "smaller" subsets.

Is Lebesgue measure countably additive?

Ultimately we want to show that Lebesgue measure is countably additive on any collection of disjoint measurable sets, so this is a step both towards showing that LRd is closed under complements and that Lebesgue measure is countably additive.

Is Outer measure countably additive?

(2) Outer measure is countably subadditive but is not countably additive, and indeed there are disjoint sets A and B such that m∗(A ∪ B) < m∗(A) + m∗(B).