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} $.