# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
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.

##### MSC:
 2.8e+11 Fuzzy measure theory
Full Text:
##### References:
 [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)