Choquet integral

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)