The property of countable subadditivity of outer measures is
$$\mu^{*} \big( \bigcup_{k=1}^\infty A_k \big) \le \sum_{k=1}^\infty \mu^* (A_k) $$
where the definition of outer measures indicated is
$$\mu^* (A)= \inf \{ \sum_m \iota(C_m) \mid \{C_m\}_m\subseteq\mathcal{C}, A\subseteq\bigcup_m C_m \} $$ for non-empty $A\subseteq X$ and $\mu^*(\emptyset)=0$. In this case, $\mathcal {C}\subseteq \mathcal{P}(X)$ is a semi-algebra of the set $X$ and $\iota \colon \mathcal C\to[0,\infty]$ is a set function satisfying $\iota(\emptyset)=0$, finite additivity over disjoint unions and countable subadditivity.
The text that I am referring to is "A Course in Real Analysis" by MacDonald and Weiss. The proof of countable subadditivity utilises $\epsilon$-room technique. Is there any alternate proof of countable subadditivity of outer measures which does not utilise $\epsilon$-room technique?