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''<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021660/c0216601.png" /> in a space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021660/c0216602.png" />''
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The function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021660/c0216603.png" /> that is equal to 1 when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021660/c0216604.png" /> and equal to 0 when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021660/c0216605.png" /> (where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021660/c0216606.png" /> is the complement to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021660/c0216607.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021660/c0216608.png" />). Every function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021660/c0216609.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021660/c02166010.png" /> with values in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021660/c02166011.png" /> is the characteristic function of some set, namely, the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021660/c02166012.png" />. Properties of characteristic functions are:
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1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021660/c02166013.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021660/c02166014.png" />;
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'' $  E $
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in a space  $  X $''
  
2) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021660/c02166015.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021660/c02166016.png" />;
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The function  $  \chi = \chi _ {E} $
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that is equal to 1 when  $  x \in E $
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and equal to 0 when  $  x \in CE $(
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where  $  CE $
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is the complement to  $  E $
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in  $  X $).  
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Every function  $  \chi $
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on  $  X $
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with values in  $  \{ 0, 1 \} $
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is the characteristic function of some set, namely, the set  $  E = \{ {x } : {\chi ( x) = 1 } \} $.
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Properties of characteristic functions are:
  
3) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021660/c02166017.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021660/c02166018.png" />;
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1) $  \chi _ {CE} = 1 - \chi _ {E} $,  
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$  \chi _ {E \setminus  F }  = \chi _ {E} ( 1 - \chi _ {F} ) $;
  
4) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021660/c02166019.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021660/c02166020.png" />;
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2) if $  F \subset  E $,  
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then $  \chi _ {E \setminus  F }  = \chi _ {E} - \chi _ {F} $;
  
5) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021660/c02166021.png" /> are pairwise disjoint, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021660/c02166022.png" />;
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3) if $  E = \cup _  \alpha  E _  \alpha  $,  
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then $  \chi _ {E} = \sup _  \alpha  \{ \chi _ {E _  \alpha  } \} $;
  
6) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021660/c02166023.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021660/c02166024.png" />.
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4) if $  E = \cap _  \alpha  E _  \alpha  $,
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then  $  \chi _ {E} = \inf _  \alpha  \{ \chi _ {E _  \alpha  } \} $;
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5) if  $  E _ {1} , E _ {2} \dots $
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are pairwise disjoint, then $  \chi _ {\cup E _ {K}  } = \sum _ {1}  ^  \infty  \chi _ {E _ {K}  } $;
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6) if  $  E = \cap _ {K} E _ {K} $,
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then  $  \chi _ {E} = \prod _ {1}  ^  \infty  \chi _ {E _ {K}  } $.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  P.R. Halmos,  "Measure theory" , v. Nostrand  (1950)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  P.R. Halmos,  "Measure theory" , v. Nostrand  (1950)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
The characteristic function of a set is also called the indicator function of that set. The symbols <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021660/c02166025.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021660/c02166026.png" /> are often used instead of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021660/c02166027.png" />.
+
The characteristic function of a set is also called the indicator function of that set. The symbols $  1 _ {E} $
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or $  \xi _ {E} $
 +
are often used instead of $  \chi _ {E} $.

Revision as of 16:43, 4 June 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 CE $( where $ CE $ 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 _ {CE} = 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 $ 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