I’m currently working my way through Real Analysis by Folland and, in his proof of theorem 1.14 he does something I don’t quite understand:
Take some algebra $\mathcal{A}$ on some set $X$ which generates some $\sigma$-algebra $\mathcal{M}$. If we have $E \in \mathcal{M}$ then we write $E \subset A$, where $A := \cup_{1}^{\infty} A_j$ for some $A_j \in \mathcal{A}$. Now, I believe this is possible because $X \in \mathcal{A}$ and $E \subset X$. However, what I don’t understand is that he then goes on to say that, given some $\epsilon > 0$ we can choose the $A_j$’s in such a way that $\mu(A) \leq \mu(E) + \epsilon$ for a measure $\mu$ on $(X, \mathcal{M})$. How can we make such a choice?