View source for Choquet integral

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

<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>

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:

<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>

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>

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.

The Choquet integral has the following properties:

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130100/c13010027.png" />.

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" />.

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" />.

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

<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>

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

==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

<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>

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" />.

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]]].

====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>