Sugeno integral

From Wikipedia, the free encyclopedia
Jump to: navigation, search

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}.