Characteristic function of a set

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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 .


[1] 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 .

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Indicator function. Encyclopedia of Mathematics. URL: