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 $$\tilde{A}$$.

Usage and Relationships
Sugeno integral is related to h-index.