Question: I understand that Lebesgue outer measure on $\mathbb R$ is not countably additive. But if there are two disjoint sets, does the outer measure of their union equal the sum of their outer measure? Can someone give me a counterexample?

Answer: 
Take $\Omega = \{a, b\}$. Define
$$
 \mu^\*(A)
 =
 \left\{
 \begin{array}{ll}
 0, & A = \emptyset
 \\
 1, & A \neq \emptyset
 \end{array}
 \right.
$$
This is an outer measure.

In this case,
$$
 \mu^\*(\{a,b\}) = 1
 \neq
 2 = \mu^\*(\{a\}) + \mu^\*(\{b\}).
$$
Countably additive holds only If the sets $\{ E\_n \} $ are measurable and pairwise disjoint: That is. If $\{ E\_n \}\_{n\geq 1} $ are measurable, then 

$$ \mu^{\*} ( \bigcup E\_n ) = \sum \mu^\*(E\_n) $$