In my text book Lebesgue measure is shown to have countable additivity on disjoint sets, i.e. if $A_k$ is a countable sequence of disjoint measurable sets, then $\mu(\bigcup A_k)=\sum\mu(A_k)$.
Thus Lebesgue measure is a measure with an additional property that the empty set has zero measure.
But for outer measure, my text book never says outer measure is a measure, and it does not say anything if outer measure has countable additivity on disjoint sets. So my question is does outer measure has such countable additivity? If not, is there a counter example?
Best Answer
The following seems to be a perfect match to this question. In the following, $|*|$ denotes a Lebesgue measure, and $|*|_e$ denotes an outer measure. The outer measure does not satisfy the countable additivity on disjoint sets because of the existence of non-measurable sets. As a result, outer measure is not a measure.