I am trying to solve the exercise which is described and solved here
But I cannot understand the following claim:
In bullet (a) op says:
Since $$ \mu^{*}\left(A_{j}\right)=\inf\left\{ \sum_{j=1}^{\infty}\mu_{0}\left(B_{k}^{j}\right)\thinspace:\thinspace A_{j}\subseteq\bigcup_{k=1}^{\infty}B_{k}^{j}\thinspace\thinspace,\thinspace\left\{ B_{k}^{j}\right\} _{k=1}^{\infty}\subseteq\mathcal{A}\right\} $$
We have $$ \mu^{*}\left(A_{j}\right)\leq\mu_{0}\left(A_{j}\right)+2^{-j}\varepsilon $$
Why is this true? cannot see it.
Also, in bullet b, how come this is correct?
Shouldnt it be $\mu^{*}(B_n \cap E)$ instead $\mu^{*}(E)$ ? (and then the equality does not hold..
Thanks in advance
