View source for Choquet integral

<!--This article has been texified automatically. Since there was no Nroff source code for this article, 
the semi-automatic procedure described at https://encyclopediaofmath.org/wiki/User:Maximilian_Janisch/latexlist
was used.
If the TeX and formula formatting is correct, please remove this message and the {{TEX|semi-auto}} category.

Out of 47 formulas, 47 were replaced by TEX code.-->

{{TEX|semi-auto}}{{TEX|done}}
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

\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 $f$, [[#References|[a1]]], [[#References|[a2]]], [[#References|[a6]]]. Specially, let $f$ be a simple measurable non-negative function on $( X , \mathcal{A} )$, $f = \sum _ { i = 1 } ^ { n } a _ { i } \chi _ {A_ i }$, $0 < a _ { 1 } < \ldots < a _ { n }$, $\{ A ; \} _ { i = 1 } ^ { n } \subset \mathcal{A}$ 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|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:

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

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 [[#References|[a2]]], [[#References|[a6]]], [[#References|[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 [[#References|[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 [[#References|[a3]]], [[#References|[a4]]], [[#References|[a5]]].

====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>