Choquet integral

Let be a measurable space. Let be a monotone set function (cf. also Set function) on , vanishing at the empty set, . Let be a non-negative measurable function and . The Choquet integral of on A with respect to is defined by

where the right-hand side is an improper integral and is the -cut of , [a1], [a2], [a6]. Specially, let be a simple measurable non-negative function on , , , and whenever . One can rewrite in the following form:

Then

where . Note that for a measure (i.e., for a -additive measure) the Lebesgue integral and the Choquet integral coincide.

The Choquet integral has the following properties:

.

For any constant , .

If on , then .

For co-monotone functions and , i.e., for all , one has

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

Related integrals and generalizations.

Let be a non-negative extended real-valued measurable function on and . The Sugeno integral [a8] of on with respect to is defined by

where , .

The restrictions of Choquet-like integrals to the unit interval (both for functions and for fuzzy measures) are a special case of the more general -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)