Characteristic function of a set
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 .
|||P.R. Halmos, "Measure theory" , v. Nostrand (1950)|
The characteristic function of a set is also called the indicator function of that set. The symbols or are often used instead of .
Indicator function. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Indicator_function&oldid=35383