I'm trying to understand the section of the proof of Caratheodory's Extension Theorem which shows that the outer measure $\mu^*$ agrees with the defined set function $\mu$ on the algebra $\mathcal{A}$, i.e. $\mu^*(A) = \mu(A) \text{ }\forall A \in \mathcal{A}$ My notes state the following:
"Suppose $A \subseteq \cup_{n \in \mathbb{N}} A_n$ with $A_n \in \mathcal{A}$, given that the elements of $\mathcal{A}$ form an algebra we can assume that $A_n \subseteq A$ for all $n \in \mathbb{N}$ and that $A_n$ are pairwise disjoint."
My notes don't state it explicitly but my understanding of how to show $\mu^*(A) \geq \mu(A)$ is to argue that for any $\epsilon \geq 0$ there is a covering of $A$ where $A \subseteq \cup_n A_n$ and $\sum_n \mu(A_n) \leq \mu^*(A) + \epsilon$. We'd then use the countability property of $\mu$ on $\mathcal{A}$ to conclude that $\mu(A) = \sum_n \mu(A_n)$, but this is predicated on $\{A_n\}$ being a partition of $A$. This is the part I'm confused about, I don't understand why we can assume that the specific covering is a partition of $A$. More specifically, as stated above, why $A_n \subseteq A$ for all $n$.
Any clarification would be greatly appreciated, thanks.