I am currently using Stein's book to self study measure theory and now I'm stuck on proving this property of Lebesgue measure below.
Property: Every open set in $\mathbb R^d$ is measurable
The book just says this immediately follows from the definition of Lebesgue measure (a subset $E$ of $\mathbb R^d$ is Lebesgue measurable, is for any $\varepsilon>0$ there exists an open set $O$ with $E\subset O$ and $m^*\left(O\smallsetminus E\right)<\varepsilon$), but I'm not sure how the property is derived from the definition.
Thanks in advance.
Best Answer
Modern usage is that $E\subset E.$ The author must be using this, and not requiring that $E\subsetneqq O.$ Because in the case $E=\Bbb R^n$ there is no $O$ such that $E\subsetneqq O\subset \Bbb R^n$, but if $E=\Bbb R^n$ then $E $ $ is $ measurable. So we can let $O=E,$ and it should not be hard to prove that $0=m^*(\phi)=m^*(O \backslash E).$