Weakening Carathéodory's criterion? 
Assume that $E$ has finite outer measure. Show that $E$ is measurable iff for each open, bounded interval $(a,b)$, $$b-a = m^*\big((a,b)\cap E\big) + m^*\big((a,b)/E\big)$$

This is an exercise on page 43, Real Analysis, H.L.Royden et al(4ed). The "only if" part is obvious, but I have absolutely no idea how to show the other direction.
 A: Let $A$ be an arbitrary subset of $\mathbb R$. It is enough to show that 
$$m^*(A\cap E)+m^*(A\setminus E)\leq m^*(A)\, . $$
Let $(I_k)_{k\in\mathbb N}$ be any sequence of bounded intervals covering $A$. By the assumption on $E$, we may write 
$$\sum_{k\in\mathbb N} \vert I_k\vert =\sum_{k\in\mathbb N} m^*(I_k\cap E)+\sum_{k\in\mathbb N} m^*(I_k\setminus E)$$
Moreover, since $m^*$ is an outer measure and $\bigcup_k I_k\supset A$, we also have
$$\sum_{k\in\mathbb N} m^*(I_k\cap E)\geq m^*\left(\bigcup_{k\in\mathbb N} I_k\cap E \right)\geq m^*(A\cap E)\, , $$
and likewise $\sum\limits_{k\in\mathbb N} m^*(I_k\setminus E)\geq m^*(A\setminus E) $. It follows that 
$$\sum_{k\in\mathbb N}\vert I_k\vert\geq m^*(A\cap E)+m^*(A\setminus E)$$
for any sequence of bounded intervals covering $A$, which gives the result.
A: IS the following enough? (If not: give me an idea)!!
Suppose $E$  is not a measurable set.
$∀ε>0$ If there is an open interval $(c,d)$  such that $E⊂(c,d)$ with $ m^*((c,d) )<m^*(E)+ε$.
Then $m^*((c,d)$~$E)= m^*((c,d) )- m^*(E)   $(given)$ < m^*(E) + ε - m^*(E) < ε$
Thus $m^*((c,d)$~$E) < ε$  and that equivalent to the measurability of $E$ $(contradiction)$
Therfore, $E$ is measurable. 
