Indicator function

An indicator function of a subset $A\subseteq X$ is the function $\mathbf 1_A:X\to\{0,1\}$ defined by $\mathbf 1_A(x)=1$ for $x\in A$ and $\mathbf 1_A(x)=0$ for $x\notin A$.

Indicator functions translate set operations into algebraic ones and are the basic building blocks of simple functions . In a measurable space $(X,\Sigma)$, $\mathbf 1_A$ is measurable exactly when $A$ is a measurable set (with $\{0,1\}$ carrying its power-set sigma-algebra).

Examples:

• On $(\mathbb R,\mathcal B(\mathbb R))$, the indicator function $\mathbf 1_{(0,1)}$ of the open interval $(0,1)$ is measurable.
• If $N$ is a null set in a measure space, then $\mathbf 1_N$ equals $0$ almost everywhere .