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Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130100/c1301001.png" /> be a [[Measurable space|measurable space]]. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130100/c1301002.png" /> be a monotone set function (cf. also [[Set function|Set function]]) on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130100/c1301003.png" />, vanishing at the empty set, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130100/c1301004.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130100/c1301005.png" /> be a non-negative [[Measurable function|measurable function]] and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130100/c1301006.png" />. The Choquet integral of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130100/c1301007.png" /> on A with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130100/c1301008.png" /> is defined by
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the semi-automatic procedure described at https://encyclopediaofmath.org/wiki/User:Maximilian_Janisch/latexlist
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If the TeX and formula formatting is correct, please remove this message and the {{TEX|semi-auto}} category.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130100/c1301009.png" /></td> </tr></table>
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Out of 47 formulas, 47 were replaced by TEX code.-->
  
where the right-hand side is an improper integral and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130100/c13010010.png" /> is the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130100/c13010011.png" />-cut of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130100/c13010012.png" />, [[#References|[a1]]], [[#References|[a2]]], [[#References|[a6]]]. Specially, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130100/c13010013.png" /> be a simple measurable non-negative function on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130100/c13010014.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130100/c13010015.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130100/c13010016.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130100/c13010017.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130100/c13010018.png" /> whenever <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130100/c13010019.png" />. One can rewrite <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130100/c13010020.png" /> in the following form:
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Let $( X , \mathcal{A} )$ be a [[Measurable space|measurable space]]. Let $m : \mathcal{A} \rightarrow [ 0 , \infty ]$ be a monotone set function (cf. also [[Set function|Set function]]) on $\mathcal{A}$, vanishing at the empty set, $m ( \emptyset ) = 0$. Let $f$ be a non-negative [[Measurable function|measurable function]] and $A \in \mathcal{A}$. The Choquet integral of $f$ on A with respect to $m$ is defined by
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130100/c13010021.png" /></td> </tr></table>
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\begin{equation*} ( C ) \int _ { A } f d m = \int _ { 0 } ^ { + \infty } m ( A \bigcap F _ { \alpha } ) d \alpha, \end{equation*}
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where the right-hand side is an improper integral and $F _ { \alpha } = \{ x : f ( x ) \geq \alpha \}$ is the $\alpha$-cut of $f$, [[#References|[a1]]], [[#References|[a2]]], [[#References|[a6]]]. Specially, let $f$ be a simple measurable non-negative function on $( X , \mathcal{A} )$, $f = \sum _ { i = 1 } ^ { n } a _ { i } \chi _ {A_ i }$, $0 &lt; a _ { 1 } &lt; \ldots &lt; a _ { n }$, $\{ A ; \} _ { i = 1 } ^ { n } \subset \mathcal{A}$ and $A _ { i } \cap A _ { j } = \emptyset$ whenever $i \neq j$. One can rewrite $f$ in the following form:
 +
 
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\begin{equation*} f = \vee _ { i = 1 } ^ { n } a _ { i } \chi _ { B _ { i } } , \quad B _ { i } = \bigcup _ { j = i } ^ { n } A _ { i }. \end{equation*}
  
 
Then
 
Then
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130100/c13010022.png" /></td> </tr></table>
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\begin{equation*} ( C ) \int _ { X } f d m = \sum _ { i = 1 } ^ { n } ( a _ { i } - a _ { i - 1 } ) m ( B _ { i } ), \end{equation*}
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130100/c13010023.png" />. Note that for a [[Measure|measure]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130100/c13010024.png" /> (i.e., for a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130100/c13010026.png" />-additive measure) the [[Lebesgue integral|Lebesgue integral]] and the Choquet integral coincide.
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where $a_{ 0 } = 0$. Note that for a [[Measure|measure]] $m$ (i.e., for a $\sigma$-additive measure) the [[Lebesgue integral|Lebesgue integral]] and the Choquet integral coincide.
  
 
The Choquet integral has the following properties:
 
The Choquet integral has the following properties:
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130100/c13010027.png" />.
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$( C ) \int _ { A } f d m = ( C ) \int f . \chi _ { A } d m$.
  
For any constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130100/c13010028.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130100/c13010029.png" />.
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For any constant $a \in [ 0 , + \infty [$, $(C) \int a \cdot f d m = a \cdot ( C ) \int f d m$.
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130100/c13010030.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130100/c13010031.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130100/c13010032.png" />.
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If $f _ { 1 } \leq f _ { 2 }$ on $A$, then $(C)\int _ { A } f _ { 1 } d m \leq ( C ) \int _ { A } f_2 dm$.
  
For co-monotone functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130100/c13010033.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130100/c13010034.png" />, i.e., <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130100/c13010035.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130100/c13010036.png" />, one has
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For co-monotone functions $f _ { 1 }$ and $f _ { 2 }$, i.e., $( f _ { 1 } ( x ) - f _ { 1 } ( y ) ) . ( f _ { 2 } ( x ) - f _ { 2 } ( y ) ) \geq 0$ for all $x , y \in X$, one has
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130100/c13010037.png" /></td> </tr></table>
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\begin{equation*} (C) \int ( f _ { 1 } + f _ { 2 } ) d m = ( C ) \int f _ { 1 } d m + ( C ) \int f _ { 2 } d m. \end{equation*}
  
 
For other properties of the Choquet integral, see [[#References|[a2]]], [[#References|[a6]]], [[#References|[a7]]].
 
For other properties of the Choquet integral, see [[#References|[a2]]], [[#References|[a6]]], [[#References|[a7]]].
  
 
==Related integrals and generalizations.==
 
==Related integrals and generalizations.==
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130100/c13010038.png" /> be a non-negative extended real-valued measurable function on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130100/c13010039.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130100/c13010040.png" />. The Sugeno integral [[#References|[a8]]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130100/c13010041.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130100/c13010042.png" /> with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130100/c13010043.png" /> is defined by
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Let $f$ be a non-negative extended real-valued measurable function on $( X , \mathcal{A} , m )$ and $A \in \mathcal{A}$. The Sugeno integral [[#References|[a8]]] of $f$ on $A$ with respect to $m$ is defined by
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130100/c13010044.png" /></td> </tr></table>
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\begin{equation*} (S) \int _ { A } f d m = \operatorname { sup } _ { \alpha \in [ 0 , + \infty ] } [ \alpha \bigwedge m ( A \bigcap F _ { \alpha } ) ], \end{equation*}
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130100/c13010045.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130100/c13010046.png" />.
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where $F _ { \alpha } = \{ x : f ( x ) \geq \alpha \}$, $\alpha \in [ 0 , + \infty ]$.
  
The restrictions of Choquet-like integrals to the unit interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130100/c13010047.png" /> (both for functions and for fuzzy measures) are a special case of the more general <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130100/c13010049.png" />-conorm integrals defined in [[#References|[a3]]], [[#References|[a4]]], [[#References|[a5]]].
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The restrictions of Choquet-like integrals to the unit interval $[ 0,1 ]$ (both for functions and for fuzzy measures) are a special case of the more general $t$-conorm integrals defined in [[#References|[a3]]], [[#References|[a4]]], [[#References|[a5]]].
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  G. Choquet,  "Theory of capacities"  ''Ann. Inst. Fourier (Grenoble)'' , '''5'''  (1953)  pp. 131–295</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  D. Denneberg,  "Non-additive measure and integral" , Kluwer Acad. Publ.  (1994)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  M. Grabisch,  H.T. Nguyen,  E.A. Walker,  "Fundamentals of uncertainity calculi with application to fuzzy inference" , Kluwer Acad. Publ.  (1995)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  R. Mesiar,  "Choquet-like integrals"  ''J. Math. Anal. Appl.'' , '''194'''  (1995)  pp. 477–488</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  T. Murofushi,  M. Sugeno,  "A theory of fuzzy measures. Representation, the Choquet integral and null sets"  ''J. Math. Anal. Appl.'' , '''159'''  (1991)  pp. 532–549</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  E. Pap,  "Null-additive set functions" , Kluwer Acad. Publ. /Ister  (1995)</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  D. Schmeidler,  "Integral representation without additivity"  ''Proc. Amer. Math. Soc.'' , '''97'''  (1986)  pp. 253–261</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  M. Sugeno,  "Theory of fuzzy integrals and its applications"  ''PhD Thesis Tokyo Inst. Technol.''  (1974)</TD></TR></table>
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<table><tr><td valign="top">[a1]</td> <td valign="top">  G. Choquet,  "Theory of capacities"  ''Ann. Inst. Fourier (Grenoble)'' , '''5'''  (1953)  pp. 131–295</td></tr><tr><td valign="top">[a2]</td> <td valign="top">  D. Denneberg,  "Non-additive measure and integral" , Kluwer Acad. Publ.  (1994)</td></tr><tr><td valign="top">[a3]</td> <td valign="top">  M. Grabisch,  H.T. Nguyen,  E.A. Walker,  "Fundamentals of uncertainity calculi with application to fuzzy inference" , Kluwer Acad. Publ.  (1995)</td></tr><tr><td valign="top">[a4]</td> <td valign="top">  R. Mesiar,  "Choquet-like integrals"  ''J. Math. Anal. Appl.'' , '''194'''  (1995)  pp. 477–488</td></tr><tr><td valign="top">[a5]</td> <td valign="top">  T. Murofushi,  M. Sugeno,  "A theory of fuzzy measures. Representation, the Choquet integral and null sets"  ''J. Math. Anal. Appl.'' , '''159'''  (1991)  pp. 532–549</td></tr><tr><td valign="top">[a6]</td> <td valign="top">  E. Pap,  "Null-additive set functions" , Kluwer Acad. Publ. /Ister  (1995)</td></tr><tr><td valign="top">[a7]</td> <td valign="top">  D. Schmeidler,  "Integral representation without additivity"  ''Proc. Amer. Math. Soc.'' , '''97'''  (1986)  pp. 253–261</td></tr><tr><td valign="top">[a8]</td> <td valign="top">  M. Sugeno,  "Theory of fuzzy integrals and its applications"  ''PhD Thesis Tokyo Inst. Technol.''  (1974)</td></tr></table>

Revision as of 16:55, 1 July 2020

Let $( X , \mathcal{A} )$ be a measurable space. Let $m : \mathcal{A} \rightarrow [ 0 , \infty ]$ be a monotone set function (cf. also Set function) on $\mathcal{A}$, vanishing at the empty set, $m ( \emptyset ) = 0$. Let $f$ be a non-negative measurable function and $A \in \mathcal{A}$. The Choquet integral of $f$ on A with respect to $m$ is defined by

\begin{equation*} ( C ) \int _ { A } f d m = \int _ { 0 } ^ { + \infty } m ( A \bigcap F _ { \alpha } ) d \alpha, \end{equation*}

where the right-hand side is an improper integral and $F _ { \alpha } = \{ x : f ( x ) \geq \alpha \}$ is the $\alpha$-cut of $f$, [a1], [a2], [a6]. Specially, let $f$ be a simple measurable non-negative function on $( X , \mathcal{A} )$, $f = \sum _ { i = 1 } ^ { n } a _ { i } \chi _ {A_ i }$, $0 < a _ { 1 } < \ldots < a _ { n }$, $\{ A ; \} _ { i = 1 } ^ { n } \subset \mathcal{A}$ and $A _ { i } \cap A _ { j } = \emptyset$ whenever $i \neq j$. One can rewrite $f$ in the following form:

\begin{equation*} f = \vee _ { i = 1 } ^ { n } a _ { i } \chi _ { B _ { i } } , \quad B _ { i } = \bigcup _ { j = i } ^ { n } A _ { i }. \end{equation*}

Then

\begin{equation*} ( C ) \int _ { X } f d m = \sum _ { i = 1 } ^ { n } ( a _ { i } - a _ { i - 1 } ) m ( B _ { i } ), \end{equation*}

where $a_{ 0 } = 0$. Note that for a measure $m$ (i.e., for a $\sigma$-additive measure) the Lebesgue integral and the Choquet integral coincide.

The Choquet integral has the following properties:

$( C ) \int _ { A } f d m = ( C ) \int f . \chi _ { A } d m$.

For any constant $a \in [ 0 , + \infty [$, $(C) \int a \cdot f d m = a \cdot ( C ) \int f d m$.

If $f _ { 1 } \leq f _ { 2 }$ on $A$, then $(C)\int _ { A } f _ { 1 } d m \leq ( C ) \int _ { A } f_2 dm$.

For co-monotone functions $f _ { 1 }$ and $f _ { 2 }$, i.e., $( f _ { 1 } ( x ) - f _ { 1 } ( y ) ) . ( f _ { 2 } ( x ) - f _ { 2 } ( y ) ) \geq 0$ for all $x , y \in X$, one has

\begin{equation*} (C) \int ( f _ { 1 } + f _ { 2 } ) d m = ( C ) \int f _ { 1 } d m + ( C ) \int f _ { 2 } d m. \end{equation*}

For other properties of the Choquet integral, see [a2], [a6], [a7].

Related integrals and generalizations.

Let $f$ be a non-negative extended real-valued measurable function on $( X , \mathcal{A} , m )$ and $A \in \mathcal{A}$. The Sugeno integral [a8] of $f$ on $A$ with respect to $m$ is defined by

\begin{equation*} (S) \int _ { A } f d m = \operatorname { sup } _ { \alpha \in [ 0 , + \infty ] } [ \alpha \bigwedge m ( A \bigcap F _ { \alpha } ) ], \end{equation*}

where $F _ { \alpha } = \{ x : f ( x ) \geq \alpha \}$, $\alpha \in [ 0 , + \infty ]$.

The restrictions of Choquet-like integrals to the unit interval $[ 0,1 ]$ (both for functions and for fuzzy measures) are a special case of the more general $t$-conorm integrals defined in [a3], [a4], [a5].

References

[a1] G. Choquet, "Theory of capacities" Ann. Inst. Fourier (Grenoble) , 5 (1953) pp. 131–295
[a2] D. Denneberg, "Non-additive measure and integral" , Kluwer Acad. Publ. (1994)
[a3] M. Grabisch, H.T. Nguyen, E.A. Walker, "Fundamentals of uncertainity calculi with application to fuzzy inference" , Kluwer Acad. Publ. (1995)
[a4] R. Mesiar, "Choquet-like integrals" J. Math. Anal. Appl. , 194 (1995) pp. 477–488
[a5] T. Murofushi, M. Sugeno, "A theory of fuzzy measures. Representation, the Choquet integral and null sets" J. Math. Anal. Appl. , 159 (1991) pp. 532–549
[a6] E. Pap, "Null-additive set functions" , Kluwer Acad. Publ. /Ister (1995)
[a7] D. Schmeidler, "Integral representation without additivity" Proc. Amer. Math. Soc. , 97 (1986) pp. 253–261
[a8] M. Sugeno, "Theory of fuzzy integrals and its applications" PhD Thesis Tokyo Inst. Technol. (1974)
How to Cite This Entry:
Choquet integral. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Choquet_integral&oldid=18610
This article was adapted from an original article by E. Pap (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article