I cannot prove this formula: $E$ is measurable and $A$ is any subset of $E$ show that $m(E)=m_*(A) +m^*(E-A)$.
Define inner measure of $A$ by $m_∗(A)=\sup(m(F))$, where the supremum is taken over all closed subsets $F$ of $E$.
$m(E)$ means $E$ is measurable and for outer measure of $E$, cover $E$ by countable collection $S$ of intervals $I_k$. i.e. $m^\ast(E)=\inf \sum \nu(I_k)$
Thanks and regards.