I am trying to get my head around understanding the Lebesgue measure and I have come across two equivalent definitions. I am struggling to see how these definitions are equivalent, and was hoping someone would be able to shed some light on these.
Let $A \subseteq \mathbb{R}$ be a set with $|A|<\infty $. $A$ is Lebesgue measurable if and only if for every $\epsilon>0$ there is a set $G$ which is a union of finitely many disjoint bounded open intervals with $|A \setminus G|+|G \setminus A|<\epsilon$.
$A \subseteq \mathbb{R}$ is Lebesgue measurable if and only if for every $ a$ and $b$ we have $|[a, b] \cap A|+|[a, b] \setminus A|=b-a$
I understand that these come from manipulating the properties of the Lebesgue measure but I don't seem to be getting anywhere.