A small question about Vitali covering lemma proof.

Question

The Lemma and the first part of its proof are given below:

My question is:

I think the first line in the proof by the following part (i) in the following theorem .... am I correct?

if so, I have this question, why $E$ being any set of real numbers leads to that $m^{*}(E)$ is finite, can not $E$ be an infinite set of real numbers?

Answer 1

I don't think so: in the theorem as stated there is no assumption that $E$ be measurable, and one would need that to apply theorem 11.

But that's not necessary. As $E$ has finite outer measure, there is a sequence of open intervals $(I_n)$ covering $E$ of finite total length. Take $\mathcal{O}=\bigcup_n I_n$.