I am studying Measure theory form Stein and Shakarachi:Real Analysis.
I come across observation regarding the outer measure.
For any $E\in R^d$ $m_*(E)=\inf m_*(O)$ where $O $ is the open set containg E.
i.e $\forall \epsilon>0 \exists O\in R^d$ such that $m_*(O)<m_*(E)+\epsilon$ and $E\subset O$.
Also By Lebesgue measurable set definition
E is said to be Lebesgue measurable iff $\forall \epsilon>0,\exists O\in R^d$ such that $m_*(O\setminus E)<\epsilon$ and $E\subset O$
From this, I thought both definitions are the same. then every set in $R^d$ become Lebesgue measurable which is of course not true .where is my interpretation fails?
When it is possible that $m_*(O \setminus E)\neq m_*(O)-m_*(E)$
Please help me out to solve this misinterpretation.
Any help will be appreciated