if outer measure of $E$ is $\sup \{ m^*(K) : K \subset E \text{ with K compact} \}$ then $E$ is measurable

Question

Let $E \subset \mathbb{R}$ be a bounded set such that $m^*(E) = \sup \{ m^*(K) : K \subset E \text{, with K compact} \}$. Show $E$ is measurable.

$m^*$ is Lebesgue outer measure.

Answer 1

An outline:

1. Build an increasing sequence of compact sets $K_n\subset K_{n+1}$ so that $m^*(K_n)\to m^*(E)$.

2. Recall that, since the $K_n$ are measurable, we have $m^*(E)=m^*(E\setminus \cup K_n)+m^*(\cup K_n)$ (this is Caratheodory's criterion).

3. Lebesgue measure is complete (meaning zero outer measure sets are measurable).

Obviously step 2 might fail to give enough info if $m^*(E)=\infty$, as was noted in the comments.