Sugeno integral

In mathematics, the Sugeno integral, named after M. Sugeno,Sugeno, M. (1974) Theory of fuzzy integrals and its applications, Doctoral. Thesis, Tokyo Institute of Technology is a type of integral with respect to a fuzzy measure.

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

The Sugeno integral over the crisp set $A\; \backslash subseteq\; X$ 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 $F\_\backslash alpha\; =\; \backslash left\backslash \{x\; |\; h(x)\; \backslash geq\; \backslash alpha\; \backslash right\backslash \}$.

The Sugeno integral over the fuzzy set $\backslash tilde\{A\}$ 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 $\backslash tilde\{A\}$.