Show that a set $A \subset \mathbb{R}^d$ is measurable if and only if $$m_*(E)=m_*(E\cap A)+m_*(E\cap A^c)$$ for every set $E\subset \mathbb{R}^d$.
Thank you guys for telling me this is a definition of Lebesgue measurable. In my textbook, a subset $E$ of $\mathbb{R}^d$ is Lebesgue measurable, or simply measurable, if for any $\epsilon >0$ there exists an open set $O$ with $E\subset O$ and $m_*(O-E)\leq \epsilon$. So the question is: Show that these two definitions are equivalent / Show that these two conditions imply each other.
My progress:
$\because E=(E\cap A)\cup (E\cap A^c)$
$\therefore m_*(E)\leq m_*(E\cap A)+m_*(E\cap A^c)$
Any help is much appreciated!