# Characteristic function of a set

From Encyclopedia of Mathematics

* in a space *

The function that is equal to 1 when and equal to 0 when (where is the complement to in ). Every function on with values in is the characteristic function of some set, namely, the set . Properties of characteristic functions are:

1) , ;

2) if , then ;

3) if , then ;

4) if , then ;

5) if are pairwise disjoint, then ;

6) if , then .

#### 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 or are often used instead of .

**How to Cite This Entry:**

Characteristic function of a set. A.A. Konyushkov (originator),

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Characteristic_function_of_a_set&oldid=12927

This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098