In mathematics, the Sugeno integral, named after M. Sugeno,[1] 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 A ~ {\displaystyle {\tilde {A}}} {\tilde {A}} of the function h {\displaystyle h} h with respect to the fuzzy measure g {\displaystyle g} 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
Sugeno, M. (1974) Theory of fuzzy integrals and its applications, Doctoral. Thesis, Tokyo Institute of Technology
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
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
Hellenica World - Scientific Library
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