Is outer measure a measure?

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?

• An outer measure is usually defined on all subsets of the underlying space. In order to obtain a measure, one typically restricts to measurable subsets that are defined using Caratheodory's condition: $E$ is measurable if and only if $\mu(A) = \mu(A \cap E) + \mu(A \cap E^c)$ for all $A \subset X$. Jul 17, 2015 at 13:28

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.

• would you please write down the book name containing this page!? Nov 18, 2017 at 17:56
• @Sherry - Would you pls share the name/details of the resource you've quoted here. Dec 12, 2021 at 5:24

Lebesgue measure is shown to have countable additivity on disjoint sets, i.e. if $$A_k$$ is a countable sequence of disjoint measurable sets...;