I've been working through an exercise which deals with showing that two definitions of a set being Lebesgue measurable are equivalent. The definition that I was thinking about was the following:
$E \subseteq \mathbb{R}^{d}$ is measurable if for every $\epsilon$ there is a closed set $F$ contained in $E$ with $m_{*}(E-F)<\epsilon$.
The $m_{*}$ is the outer measure obtained from taking the infimum of the total volume of rectangular covers.
I am having trouble showing that the compliment $E^{c}$ is also measurable when $E$ is measurable
My hunch is to show that $E^{c}$ is the intersection of such measurable sets. One observation I made is that for every $n \in \mathbb{N}$, we have a closed set $F_{n}$ such that
$m_{*}(E-F_{n}) < \epsilon$
If we set $O_{n}=F_{n}^{c}$ and $S=\bigcap_{n=1} O_{n}$, I think we should be might be able to show that $E^{c}=S$ or use $S$ in some way to show that $E^{c}$ is measurable.
I'm happy or accept to hear any alternative methods, but some help along the above lines would be very much appreciated.