Definition 21.4.1 (Indicator Function).label Let $(X, \cm)$ be a measurable space and $E \in \cm$, then the function
\[\chi_{E} = \one_{E}: X \to \bracs{0, 1}\quad x \mapsto \begin{cases}1 &x \in E \\
0 &x \not\in E\end{cases}\]
is the characteristic/indicator function of $E$.