Intuition about outer measures. Folland
I have a question regarding this proof. First of all in the statement does he mean to use $\rho$ in the brackets right after summation, or did he mean $\mu$. I looked at the errata and still am not sure.
Furthermore, in the proof he claims that for a sequence of $A_j$ in the power set, we can always approximate them by elements of our elementary family with accuracy of $\epsilon/2^j$. This sounds very bizarre and i have a difficulty accepting this. What is the intuition behind this, and are we assuming that the elementary family contains all kinds of sets that cover?

He definitely means $\sum \rho(E_j)$.
For the second point, his specific claim is that we can find some sets $E_j$ with $\sum \rho(E_j) \le \mu^*(A) + \varepsilon2^{-j}$. Why should this be true? Because $\mu^*$ is the infimum of all such sums. If we cannot find such a sum within $\varepsilon 2^{-j}$, it can't have been the infimum.
