A set $E$ is Jordan-measurable if its outer Jordan measure and its inner Jordan measure coincide.
I am trying to prove that the following is an equivalent definition
A set $E$ is Jordan measurable if for every $\epsilon>0$, there exists an elementary set $A$ such that $m^*(A \triangle E) < \epsilon$, where $m^*$ denotes the Jordan outer measure.
I got stuck here because although $A$ is elementary, $A \cap E$ is not necessary an elementary set. So I do not see how to find a proper way to lower bound the inner Jordan measure of $E$.