Definition 2

A subset $A$ of $X$ is said to be measurable if its characteristic function $\varphi_A$ is measurable.

In view of Def. 1, it comes to the same to say that a measurable set $A$ is a set such that, for every compact set $K$, there exist a negligible set $N \subset K$ and a partition $(K_n)$ of $K - N$ formed by a sequence of compact sets each of which is contained either in $K \cap A$ or in $K \cap \complement A$.

This definition yields at once the following criterion: