Lattice-valued fuzzy measure and lattice-valued fuzzy integral.

*(English)*Zbl 0824.28015Summary: In this paper, (1) the concepts of lattice-valued fuzzy measure (with no valuation property) and lower (resp. upper) lattice-valued fuzzy integral are proposed, which give the unified description to the fuzzy measures and fuzzy integrals studied by Delgado and Moral, Qiao, Ralescu, Adams, Sugeno, Wang, and Zhang; (2) some asymptotic structural characteristics of lattice-valued fuzzy measures are introduced, and some relations between them are given; (3) some concepts of convergences for lattice- valued functions are defined, and Riesz’ theorem, Egoroff’s theorem and Lebesgue’s theorem for lattice-valued measurable functions are proved; (4) the monotone increasing (resp. decreasing) convergence theorem and almost (resp. pseudo almost) everywhere convergence theorem for lower (resp. upper) lattice-valued fuzzy integral are shown under some weak conditions.

##### MSC:

28E10 | Fuzzy measure theory |

##### Keywords:

lattice-valued fuzzy measure; lattice-valued fuzzy integral; Riesz’ theorem; Egoroff’s theorem; Lebesgue’s theorem; convergence
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\textit{X. Liu} and \textit{G. Zhang}, Fuzzy Sets Syst. 62, No. 3, 319--332 (1994; Zbl 0824.28015)

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##### References:

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