Example of an outer measure that I don't understand So I understand everything in this except verifying (04) that it is sub-additive I don't understand the part of the existence of A covering {${B_{nk}}$} and afterwards. Could someone explain this part that would be great.


 A: It is my first answer in Mathematics Stack Exchange, thus please do not hesitate to point out my faults in mathematics or editing.
I assume that you have problem in understanding the following statements:
$\forall A_n$and $\forall \epsilon>0, \exists \{B_{n,k}\}_{k=1}^{\infty}$which satisfies $A_n \subset\bigcup_{k=1}^{\infty}B_{n,k}$ and $\sum_{k=1}^{\infty}\mu\left(B_{n,k}\right)\leq\mu^\ast\left(A_n\right)+\dfrac{\epsilon}{2^n}$.
This comes from the difinition of outer measurement $\mu^\ast\left(A_n\right)$ and the difinition of infimum.
Suppose $p$ is the infimum of the ordered set $A$, then $\forall \varepsilon>0$, there exists $a\in A, a\leq p+\varepsilon$.
Since $\mu^\ast\left(A_n\right) = \inf\{\sum_{k=1}^{\infty}\mu\left(B_{n,k}\right): A_n \subset\bigcup_{k=1}^{\infty}B_{n,k}\}$, thus by the difinition of infimum: $\forall \dfrac{\epsilon}{2^n}>0$, there exists some $\{B_{n,k}\}_{k=1}^{\infty}$ which satisfies $\sum_{k=1}^{\infty}\mu\left(B_{n,k}\right)\leq \mu^\ast(A_n)+\dfrac{\epsilon}{2^n}$.
