Sugeno integral

In mathematics, the Sugeno integral, introduced by Michio Sugeno as a fuzzy integral in work on fuzzy measures at the Tokyo Institute of Technology, is a type of integral with respect to a fuzzy measure.^[1][2]

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

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

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

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

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

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

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