When I learned about the Lebesgue integral, the definition I was taught of measurable set was the following: $A\subseteq\mathbb{R}^n$ is measurable if and only if $1_A$ is an almost everywhere limit of step functions.
However, these days I saw on the Internet another definition of measurability: $A$ is measurable if and only if for all $B\subseteq\mathbb{R}^n$ one has $m(A\cap B)+m(A^c\cap B)=m(B)$, where $m$ is the Lebesgue measure.
I would like to know the relation between these two definitions.