Let $E\subset [a,b]$. Show that $E$ is Lebesgue measurable if and only if the Lebesgue outer measure of $E$ is equal to the Lebesgue inner measure of $E$.
I have seen the proof for this above statement for the Caratheodory definition of Lebesgue measurable, but I was wondering if someone could help me prove it for a different (but equivalent) definition of Lebesgue measurable set.
The definition my book is using:
$E\subset \mathbb{R}$ is said to be Lebesgue measurable if $E$ can be squeezed between an open set $G$ and a closed set $F$ where we have that $m^*(G\setminus F)<\varepsilon$
I need some help to start the problem please. Thanks!
Just for completeness I provide the definitions of inner and outer lebesgue measure below:
Lebesgue outer measure: For a subset $E$ of $\mathbb{R}$, we have that $m^*(E)=\inf\{\sum_{n=1}^{\infty}\ell(I_n):E\subset \cup_{n=1}^{\infty}I_n\}$
where $\ell(\cdot)$ denotes the length
Lebesgue inner measure: For a subset $E$ of a bounded interval $[a,b]$, we have that $m_*(E)=b-a-m^*([a,b]\setminus E)$