Putting $A=E$ in $\mu^*(A)+\epsilon=\sum_{n=1}^{\infty}[\lambda(F_n^{'} \cap E) + \lambda(F_n^{'} \cap E^{c})]$
We get $\mu^*(E)+\epsilon=\sum_{n=1}^{\infty}[\lambda(F_n^{'} \cap E) + \lambda(F_n^{'} \cap E^{c})]\geq \sum_{n=1}^{\infty}[\lambda(F_n^{'} \cap E)]$
I am not getting $\mu^*(E) \geq \lambda(E)$ or How $\sum_{n=1}^{\infty}[\lambda(F_n^{'} \cap E)]=\lambda(E)$?
Thanks in advance!