Let $X$ be some non-empty set and let $\mathcal{K}\subseteq\mathcal{P}(X)$ countably cover $X$ with $\emptyset\in\mathcal{K}$. Then, given $\nu\colon\mathcal{K}\to[0,\infty]$ with $\nu(\emptyset)=0$, $$ \mu^*(A):=\inf\left\{\sum_j\nu(K_j)\colon (K_j)\in\mathcal{K}^\mathbb{N}, A\subseteq\bigcup_jK_j\right\},\quad A\subseteq X$$ defines an outer measure $\mu^*$ on X.
Now, given a pre-measure $\mu'$ on a semi-algebra $\mathcal{A}$ on $X$ they would define an outer measure via the above mechanism which doesn't require $\mu'$ to be $\sigma$-additive and $\mathcal{A}$ to be a semi-ring of sets.
So, why do some texts use pre-measures and semi-alegbras to construct outer measures this way? Is this motivational, only?