Sugeno integral

In mathematics, the Sugeno integral, named after M. Sugeno, is a type of integral with respect to a fuzzy measure.

Let $(X,\Omega )$ be a measurable space and let $h:X\to [0,1]$ be an $\Omega$-measurable function.

The Sugeno integral over the crisp set $A\subseteq X$ of the function $h$ with respect to the fuzzy measure $g$ is defined by:

$\int _{A}h(x)\circ g={\sup _{E\subseteq X}}\left[\min \left(\min _{x\in E}h(x),g(A\cap E)\right)\right]={\sup _{\alpha \in [0,1]}}\left[\min \left(\alpha ,g(A\cap F_{\alpha })\right)\right]$

where $F_{\alpha }=\left\{x|h(x)\geq \alpha \right\}$.

The Sugeno integral over the fuzzy set ${\tilde {A}}$ of the function $h$ with respect to the fuzzy measure $g$ is defined by:

$\int _{A}h(x)\circ g=\int _{X}\left[h_{A}(x)\wedge h(x)\right]\circ g$

where $h_{A}(x)$ is the membership function of the fuzzy set ${\tilde {A}}$.

References

• Gunther Schmidt (2006) Relational measures and integration, Lecture Notes in Computer Science # 4136, pages 343−57, Springer books
• M. Sugeno & T. Murofushi (1987) "Pseudo-additive measures and integrals", Journal of Mathematical Analysis and Applications 122: 197−222 MR0874969