It seems to me that there at least two different ways to "construct" a measure. The first way is to first define an outer measure using a set function and then "restrict" the outer measure to the set that satisfy the Catheodory's Criterion would create a measure. On the other hand, we can "extend" the pre-measure to a measure using the Catheodory's extension theorem. However, the later construction seems to require more structure. i.e. That of an algebra and the concept of a pre-measure to construct a measure. What is the motivation for this second way of constructing a measure?
From what I have seen so far, the motivation mainly comes from the construction of "desirable" measures, such as the product measure. That is, we would normally start off with an elementary desirable notion of pre-measure defined on an algebra, and then extend this measure uniquely to a $\sigma$-algebra. Whereas in the first way of constructing the measure, there seems to be less of a control what kind of measure that we would end up with. Is this correct?
Definitions:
Alegebra: Let $X$ be a set. We define an algebra $\mathcal{A}$ on $X$ to be $\mathcal{A} \subseteq P(X)$ that satisfy the following conditions: 1. $\emptyset, X \in \mathcal{A}$. 2. If $A \in \mathcal{A}$, then $A^c \in \mathcal{A}$. 3. If the finite sequence $\{ A_n \}_{n = 1} ^N \subseteq \mathcal{A}$, then the finite union $\bigcup_{n = 1} ^N A_n \in \mathcal{A}$.
Pre-Measure: Let $X$ be a set and $\mathcal{A} \subseteq P(X)$ be an algebra defined above. Then $\mu_0: \mathcal{A} \to [0, +\infty]$ is a pre-measure if it satisfy the following conditions: 1. $\mu_0(\emptyset) = 0$. 2. If the disjoint sequence $\{ A_n \}_{n = 1} ^\infty \subseteq P(X)$ is in fact such that $\{ A_n \}_{n = 1} ^\infty \subseteq \mathcal{A}$, then we have $\mu_0(\bigcup_{n = 1} ^\infty A_n) = \sum_{n = 1} ^\infty \mu_0(A_n)$.
Note that even when starting with a premeasure $\mu_0$, we still construct an outer measure $\mu^*$ from it, and restrict that to a $\sigma$-algebra contained in the $\mu^*$-measurable sets. So, Caratheodory's extension theorem is a more refined statement of Caratheodory's "outer-measure theorem" regarding what happens when we impose additional hypotheses. So these aren't really two different methods for constructing measures. One just gives us more refined conclusions than the other.
The benefit of the premeasure approach is that first, the algebra we start with will be $\mu^*$-measurable and second, that $\mu^*$ agrees with $\mu_0$ on the sets of the algebra (hence it's called Caratheodory's extension theorem). We do this not just for product measures, but also for Lebesgue measure on $\Bbb{R}^n$.