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Lattice-valued fuzzy measure and lattice-valued fuzzy integral. (English) Zbl 0824.28015
Summary: 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.

28E10Fuzzy measure theory
Full Text: DOI
[1] Birkhoff, G.: Lattice theory. Amer. math. Soc. coll. Publ. (1979) · Zbl 0505.06001
[2] Delgado, M.; Moral, S.: On the concept of possibility-probability consistency. Fuzzy sets and systems 21, 311-318 (1987) · Zbl 0618.60003
[3] Greco, G. H.: On L-fuzzy integrals of measurable functions. J. math. Anal. appl. 128, 581-585 (1987) · Zbl 0624.28015
[4] Greco, G. H.: Fuzzy measures and fuzzy integrals with their values in complete lattices. J. math. Anal. appl. 126, 594-603 (1987) · Zbl 0625.28015
[5] Goguen, J. A.: L-fuzzy sets. J. math. Anal. appl. 18, 145-174 (1967) · Zbl 0145.24404
[6] Halmos, P. R.: Measure theory. (1967)
[7] Kelley, J. A.: General topology. (1955) · Zbl 0066.16604
[8] Klement, E. P.; Weber, S.: General fuzzy measure. Fuzzy sets and systems 40, 375-394 (1991)
[9] Liu, X.: On Levi-like theorem, Borel-cantelli lemma and Lusin theorem to fuzzy measure and fuzzy integral. Cybernetics and systems’92, 391-398 (1992)
[10] Liu, X.: Further discussion on the convergence theorems for seminormed fuzzy integrals and semiconormed fuzzy integrals. Fuzzy sets and systems 55, 219-226 (1993) · Zbl 0782.28014
[11] Qiao, Z.: On fuzzy measure and fuzzy integral on fuzzy sets. Fuzzy sets and systems 37, 77-92 (1990) · Zbl 0701.28011
[12] Qiao, Z.: Fuzzy integrals on L-fuzzy sets. Fuzzy sets and systems 38, 61-79 (1990) · Zbl 0708.28010
[13] Relescu, D.; Adams, G.: The fuzzy integral. J. math. Anal. appl. 75, 469-474 (1980)
[14] Squillante, M.; Ventre, A. G. S.: Fuzzy measure and convergence. Fuzzy sets and systems 25, 251-257 (1987) · Zbl 0641.28013
[15] Sugeno, M.: Theory of fuzzy integrals and its applications. Thesis (1974)
[16] Sun, Q.; Wang, Z.: On the autocontinuity of fuzzy measures. Cybernetics and systems’88, 717-721 (1988)
[17] Wang, Z.: The autocontinuity of set function and the fuzzy integral. J. math. Anal. appl. 99, 195-218 (1984) · Zbl 0581.28003
[18] Wang, Z.: Asymptotic structural characteristics of fuzzy measures and their applications. Fuzzy sets and systems 16, 277-290 (1985) · Zbl 0593.28007
[19] Zhang, G.: Fuzzy continuous function and its properties. Fuzzy sets and systems 43, 159-171 (1991) · Zbl 0735.26013
[20] Zhang, G.: Some theorems on fuzzy number-valued fuzzy measure and fuzzy number-valued fuzzy measure space. Proc. of the 5th conference of fuzzy mathematics and systems society of China, 78-81 (1990)
[21] G. Zhang, Fuzzy number-valued fuzzy measure and fuzzy number-valued integral, Fuzzy Sets and Systems (to appear). · Zbl 0782.28017
[22] Zhang, G.: Fuzzy number-value fuzzy measure and fuzzy number-valued fuzzy integral on the L-fuzzy set. Quarterly of mathematics 2, 74-95 (1991)