In mathematics, the Sugeno integral, named after M. Sugeno,[1] is a type of integral with respect to a fuzzy measure.
Let 
be a measurable space and let ![{\displaystyle h:X\to [0,1]}](//wikimedia.org/api/rest_v1/media/math/render/svg/cb92ae376f146c67ce48e0580a38f17fa2454ff5)
be an 
-measurable function.
The Sugeno integral over the crisp set 
of the function 
with respect to the fuzzy measure 
is defined by:
![{\displaystyle \int _{A}h(x)\circ g={\sup _{E\subseteq X}}\left[\min \left(\min _{x\in E}h(x),g(A\cap E)\right)\right]={\sup _{\alpha \in [0,1]}}\left[\min \left(\alpha ,g(A\cap F_{\alpha })\right)\right]}](//wikimedia.org/api/rest_v1/media/math/render/svg/97ed855d6d1645da9706763f4fd0620d86221a85)
where 
.
The Sugeno integral over the fuzzy set 
of the function with respect to the fuzzy measure is defined by:
![{\displaystyle \int _{A}h(x)\circ g=\int _{X}\left[h_{A}(x)\wedge h(x)\right]\circ g}](//wikimedia.org/api/rest_v1/media/math/render/svg/6c02653de5840607a3c4936aafdebed331c5be00)
where 
is the membership function of the fuzzy set .
![{\displaystyle h:X\to [0,1]}](http://wikimedia.org/api/rest_v1/media/math/render/svg/cb92ae376f146c67ce48e0580a38f17fa2454ff5)
![{\displaystyle \int _{A}h(x)\circ g={\sup _{E\subseteq X}}\left[\min \left(\min _{x\in E}h(x),g(A\cap E)\right)\right]={\sup _{\alpha \in [0,1]}}\left[\min \left(\alpha ,g(A\cap F_{\alpha })\right)\right]}](http://wikimedia.org/api/rest_v1/media/math/render/svg/97ed855d6d1645da9706763f4fd0620d86221a85)
![{\displaystyle \int _{A}h(x)\circ g=\int _{X}\left[h_{A}(x)\wedge h(x)\right]\circ g}](http://wikimedia.org/api/rest_v1/media/math/render/svg/6c02653de5840607a3c4936aafdebed331c5be00)
