I'm following Terry Tao's An introduction to measure theory. Here he has defined lebesgue measurability as:
Definition 1 (Lebesgue measurability): A set $E$ is said to be lebesgue measurable if for every $\epsilon > 0$, $\exists$ an open set $U$, containing $E$, such that $m^*(U \setminus E) \leq \epsilon$.
where $m^*$ denotes the outer lebesgue measure.
Then he has stated the Caratheodory Criterion:
Definition 2 (Lebesgue measurability): A set $E$ is said to be lebesgue measurable if for every elementary set $A$, we have $$m(A)=m^*(A \cap E)+m^*(A \setminus E)$$
where an elementary set is a finite union of boxes.
The problem is to show the equivalence of the two definitions. I have shown that Def $(1)$ $\implies$ Def $(2)$, but having difficulty in showing the reverse, i.e. Def $(2)$ $\implies$ Def $(1)$. Any help (full/brief solution or even some hints) would be greatly appreciated! Thanks in advance.