From "An Introduction to Lebesgue Integration and Fourier Series" by Howard J. Wilcox and David L. Myers:
For the remainder of this chapter, we will denote the unit interval $[0,1]$ by $E$.
7.2 Theorem: Every non-empty open set $G \subset \mathbb{R}$ can be expressed uniquely as a finite or countably infinite union of pairwise disjoint open intervals.
7.3 Definition: The outer measure $m^{*}(G)$ of an open set $G \subset E$ is defined as the real number $\sum_{i} (b_{i} - a_{i})$, where $G = \bigcup_{i} (a_{i}, b_{i})$ as in Theorem 7.2.
7.4 Definition: The outer measure $m^{*}(A)$ of any set $A \subset E$ is defined to be the real number $\text{glb } \{ m^{*}(G) \mid A \subset G \text{ and } G \text{ open in } E \}$.
9.15 Prove that $m^{*}$ is countably additive on the class of open subsets of $E$.
9.18 If $A \subset [0, 1/2]$ and $B \subset (1/2, 1]$, show that $m^{*}(A \cup B) = m^{*}(A) + m^{*}(B)$.
I am trying to prove 9.18. The furthest I have been able to get is:
If $A$ and $B$ are open sets, then $A$ and $B$ are disjoint, and hence $m^{*}(A \cup B) = m^{*}(A) + m^{*}(B)$ by 9.15.
and
$m^{*}(A) + m^{*}(B) = \\ \text{glb } \{ m^{*}(G) \mid A \subset G \text{ and } G \text{ open in } E \} + \text{glb } \{ m^{*}(G) \mid B \subset G \text{ and } G \text{ open in } E \} = \\ \text{glb } \{ m^{*}(G_{1}) + m^{*}(G_{2}) \mid A \subset G_{1} \text{ and } G_{1} \text{ open in } E \text{ and } B \subset G_{2} \text{ and } G_{2} \text{ open in } E \}$
Any help would be appreciated.