A subset of a finite interval is Lebesgue measurable

Question

Can you help me out to prove this?

Let $[a,b]$ be a finite interval in $\mathbb{R}$ and $E \subset [a,b]$. Prove that if $m^*(E) + m^*([a,b]\setminus E) = b - a$, then $E$ is Lebesgue measurable.

I tried approximating $[a,b]$ through $G_{\delta}$ sets or $F_{\delta}$, but it did not work. Any ideas?

Answer 1

Let $F=[a,b]\setminus E$. There exist $G_{\delta}$ sets $G_E\supset E$ and $G_F\supset F$ s.t. $m(G_E)=m^{*}(E)$ and $m(G_F)=m^{*}(F)$. Then \begin{align} m(G_E\cup G_F)&\le m(G_E)+m(G_F)=m^{*}(E)+m^{*}(F) \\ &=m(E\cup F)\le m(G_E\cup G_F), \end{align} which implies that $m(G_E\cap G_F)=0$. Therefore, $E$ is measurable because $$ E=(([0,1])\setminus G_F)\cup(E\cap G_F) $$ and $E\cap G_F\subset G_E\cap G_F$ is a null set.