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Difference between revisions of "Characteristic function of a set"

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m (ce)
 
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The function  $  \chi = \chi _ {E} $
 
The function  $  \chi = \chi _ {E} $
 
that is equal to 1 when  $  x \in E $
 
that is equal to 1 when  $  x \in E $
and equal to 0 when  $  x \in CE $(
+
and equal to 0 when  $  x \in \complement E $(
where  $ CE $
+
where  $\complement E $
 
is the complement to  $  E $
 
is the complement to  $  E $
 
in  $  X $).  
 
in  $  X $).  
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Properties of characteristic functions are:
 
Properties of characteristic functions are:
  
1)  $  \chi _ {CE} = 1 - \chi _ {E} $,  
+
1)  $  \chi _ {\complement E} = 1 - \chi _ {E} $,  
 
$  \chi _ {E \setminus  F }  = \chi _ {E} ( 1 - \chi _ {F} ) $;
 
$  \chi _ {E \setminus  F }  = \chi _ {E} ( 1 - \chi _ {F} ) $;
  
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====Comments====
 
====Comments====
The characteristic function of a set is also called the indicator function of that set. The symbols  $  1 _ {E} $
+
The characteristic function of a set is also called the indicator function of that set. The symbols  $  \mathbf{1} _ {E} $
 
or  $  \xi _ {E} $
 
or  $  \xi _ {E} $
 
are often used instead of  $  \chi _ {E} $.
 
are often used instead of  $  \chi _ {E} $.

Latest revision as of 18:24, 24 December 2020


$ 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} $.

How to Cite This Entry:
Characteristic function of a set. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Characteristic_function_of_a_set&oldid=46320
This article was adapted from an original article by A.A. Konyushkov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article