Characteristic function of a set

$E$ in a space $X$

The function $\chi = \chi _ {E}$ that is equal to 1 when $x \in E$ and equal to 0 when $x \in \complement E$( where $\complement E$ is the complement to $E$ in $X$). Every function $\chi$ on $X$ with values in $\{ 0, 1 \}$ is the characteristic function of some set, namely, the set $E = \{ {x } : {\chi ( x) = 1 } \}$. Properties of characteristic functions are:

1) $\chi _ {\complement E} = 1 - \chi _ {E}$, $\chi _ {E \setminus F } = \chi _ {E} ( 1 - \chi _ {F} )$;

2) if $F \subset E$, then $\chi _ {E \setminus F } = \chi _ {E} - \chi _ {F}$;

3) if $E = \cup _ \alpha E _ \alpha$, then $\chi _ {E} = \sup _ \alpha \{ \chi _ {E _ \alpha } \}$;

4) if $E = \cap _ \alpha E _ \alpha$, then $\chi _ {E} = \inf _ \alpha \{ \chi _ {E _ \alpha } \}$;

5) if $E _ {1} , E _ {2} \dots$ are pairwise disjoint, then $\chi _ {\cup E _ {K} } = \sum _ {1} ^ \infty \chi _ {E _ {K} }$;

6) if $E = \cap _ {K} E _ {K}$, then $\chi _ {E} = \prod _ {1} ^ \infty \chi _ {E _ {K} }$.

References

[1]	P.R. Halmos, "Measure theory" , v. Nostrand (1950)

Comments

The characteristic function of a set is also called the indicator function of that set. The symbols $\mathbf{1} _ {E}$ or $\xi _ {E}$ are often used instead of $\chi _ {E}$.