Outer measure and g_delta set

Question

I think that my logic is not wrong, but I can't prove this problem completely. How can I correct my solution if my logic is wrong? If there is nothing wrong, how can I complete proof?

Oh, Korean word [풀이] was not edited... 풀이 means solution.

Answer 1

If $V$ is a family of open intervals let $f(V)= \sum_{I\in V}l(I).$ Then $f(V)\ge m^*(\cup V).$ It may be that $f(V)>m^*(\cup V),$ as the members of $V$ may fail to be disjoint.

Let $W$ be the set of all families of open intervals such that $V\in W\implies \cup V\supset A.$ Then $$m^*(A)=\inf_{V\in W}f(V)\ge \inf_{V\in W}m^*(\cup V).$$

You assume that the "$\ge$" in the line above can be replaced by "$=$". It can be, but you need to prove it, as follows: If $V\in W$ then $\cup V\supset A$ so $m^*(\cup V)\ge m^*(A),$ so $$\inf_{V\in W}m^*(\cup V)\ge m^*(A).$$