Characteristic Function Measurable iff Set Measurable

Theorem
Let $\left({X, \Sigma}\right)$ be a measurable space.

Let $E \subseteq X$.

Then the following are equivalent:


 * $E \in \Sigma$; i.e., $E$ is a $\Sigma$-measurable set
 * $\chi_E: X \to \left\{{0, 1}\right\}$, the characteristic function of $E$, is $\Sigma$-measurable