# zbMATH — the first resource for mathematics

Extension of a class of decomposable measures using fuzzy pseudometrics. (English) Zbl 1284.28017
Summary: In this paper, we consider a topological approach to extension of t-conorm-based decomposable measures by introducing a fuzzy pseudometric structure on an algebra of sets. We prove that every non-strict continuous Archimedean t-conorm-based decomposable measure can be extended from an algebra to the completion of this algebra under the fuzzy pseudometric and then to the sigma-algebra generated by this algebra. The existence of such an extension follows very simply from the well-known Carathéodory result. However, our topological proof offers an intuitive interpretation of the extension of decomposable measures.

##### MSC:
 28E10 Fuzzy measure theory 54A40 Fuzzy topology 54H05 Descriptive set theory (topological aspects of Borel, analytic, projective, etc. sets)
Full Text:
##### References:
 [1] Ban, A. I., Intuitionistic fuzzy measurestheory and applications, (2006), Nova Science New York [2] Carathéodory, C., Vorlesungen über reelle funktionen, (1927), Leipzig Berlin · JFM 53.0225.16 [3] Cavallo, B.; D’Apuzzo, L.; Squillante, M., Independence and convergence in non-additive settings, Fuzzy Optim. Decis. Making, 8, 29-43, (2009) · Zbl 1177.28040 [4] Dubois, D.; Prade, M., A class of fuzzy measures based on triangular norms, Int. J. Gen. Syst., 8, 43-61, (1982) · Zbl 0473.94023 [5] Dubois, D.; Prade, H., Aggregation of decomposable measures with application to utility theory, Theory Decis., 41, 59-95, (1996) · Zbl 0863.90020 [6] Dubois, D.; Prade, H.; Sabbadin, R., Decision-theoretic foundations of possibility theory, Eur. J. Oper. Res., 128, 459-478, (2001) · Zbl 0982.90028 [7] Dunford, N.; Schwartz, J. T., Liner operators part 1general theory, (1988), Wiley Interscience New York [8] Fillmore, P. A., On topology induced by measure, Proc. Am. Math. Soc., 17, 854-857, (1966) · Zbl 0151.29701 [9] Frić, R., Extension of measuresa categorical approach, Math. Boh., 130, 397-407, (2005) · Zbl 1107.54014 [10] George, A.; Veeramani, P., On some results in fuzzy metric spaces, Fuzzy Sets Syst., 64, 395-399, (1994) · Zbl 0843.54014 [11] George, A.; Veeramani, P., On some results of analysis for fuzzy metric spaces, Fuzzy Sets Syst., 90, 365-368, (1997) · Zbl 0917.54010 [12] Gregori, V.; Romaguera, S., Characterizing completable fuzzy metric spaces, Fuzzy Sets Syst., 144, 411-420, (2004) · Zbl 1057.54010 [13] Gregori, V.; Morillas, S.; Sapena, A., On a class of completable fuzzy metric spaces, Fuzzy Sets Syst., 161, 2193-2205, (2010) · Zbl 1201.54011 [14] Gregori, V.; Morillas, S.; Sapena, A., Examples of fuzzy metrics and applications, Fuzzy Sets Syst., 170, 95-111, (2011) · Zbl 1210.94016 [15] Hadžić, O.; Pap, E., Probabilistic multi-valued contractions and decomposable measures, Int. J. Uncert. Fuzzy Knowl.-Based Syst., 10, Suppl., 59-74, (2002) · Zbl 1060.54017 [16] Halmos, P. R., Measure theory, (1974), Springer-Verlag New York · Zbl 0283.28001 [17] Ichihashi, H.; Tanaka, H., A fuzzy fault tree formulated by a class of fuzzy measures, Bull. Univ. Osaka Ser. A, 35, 181-193, (1986) · Zbl 0626.94020 [18] Johnson, R. A., Extending a measure from a ring to a sigma-ring, Proc. Am. Math. Soc., 79, 431-434, (1980) · Zbl 0424.28001 [19] Klement, E. P.; Mesiar, R.; Pap, E., Integration with respect to decomposable measures, based on a conditionally distributive semiring on the unit interval, Int. J. Uncert. Fuzzy Knowl.-Based Syst., 8, 701-717, (2000) · Zbl 0991.28014 [20] Klement, E. P.; Mesiar, R.; Pap, E., Triangular norms, (2000), Kluwer Academic Publishers Dordrecht · Zbl 0972.03002 [21] Kramosil, I.; Michalek, J., Fuzzy metric and statistical metric spaces, Kybernetika, 11, 326-334, (1975) [22] Mihailović, B.; Pap, E., Asymmetric general Choquet integrals, Acta Polytech. Hung., 6, 161-173, (2009) [23] Murofushi, T., Extensions of (weakly) null-additive, monotone set functions from rings of subsets to generated algebras, Fuzzy Sets Syst., 158, 2422-2428, (2007) · Zbl 1143.28007 [24] Pap, E., Lebesgue and Saks decompositions of $$\bot$$-decomposable measures, Fuzzy Sets Syst., 38, 345-353, (1990) · Zbl 0722.28015 [25] Pap, E., Extension of the continuous t-conorm decomposable measure, Univ. Nov. Sadu Zb. Rad. Prirod.-Mat. Fak. Ser. Mat., 20, 121-130, (1990) · Zbl 0754.28006 [26] Pap, E., Decompositions of supermodular functions and $$\square \operatorname{-} \operatorname{decomposable}$$ measures, Fuzzy Sets Syst., 65, 71-83, (1994) · Zbl 0859.28012 [27] Pap, E., Decomposable measures and nonlinear equations, Fuzzy Sets Syst., 92, 205-221, (1997) · Zbl 0934.28015 [28] Pap, E., Extension of null-additive set functions on algebra of subsets, Novi Sad J. Math., 31, 2-13, (2001) · Zbl 1003.28012 [29] Pap, E., Applications of decomposable measures on nonlinear differential equations, Novi Sad J. Math., 31, 89-98, (2001) · Zbl 1014.28002 [30] (Pap, E., Handbook of Measure Theory, (2002), Elsevier Science Amsterdam) · Zbl 0998.28001 [31] Qiao, Z., On the extension of possibility measures, Fuzzy Sets Syst., 32, 315-320, (1989) · Zbl 0677.94021 [32] Saminger, S.; Mesiar, R., A general approach to decomposable bi-capacities, Kybernetika, 39, 631-642, (2003) · Zbl 1249.28022 [33] Savchenko, A.; Zarichnyi, M., Fuzzy ultrametrics on the set of probability measures, Topology, 48, 130-136, (2009) · Zbl 1191.54008 [34] M. Sugeno, Theory of Fuzzy Integrals and Its Applications, Ph.D. Thesis, Tokyo Institute of Technology, Tokyo, 1974. [35] Varadarajan, V., Measures on topological spaces, Am. Math. Soc. Transl., 48, 161-228, (1965) [36] Wang, Z., Une class de mesures floues - LES quasi-mesures, BUSEFAL, 6, 28-37, (1981) [37] Wang, Z., Semi-lattice structure of all extensions of possibility measure and consonant belief function, (Feng, D.; Liu, X., Fuzzy Mathematics in Earthquake Research, (1985), Seismological Press Beijing), 332-336 [38] Wang, Z., Semi-lattice isomorphism of the extensions of possibility measure and the solutions of fuzzy relation equation, (Trappl, R., Cybernetics and Systems ’86, (1986), Kluwer Boston), 581-583 [39] Wang, Z., Some recent advances on the possibility measure theory, (Bouchon, B.; Yager, R. R., Uncertainty and Knowledge-Based Systems, (1987), Springer-Verlag New York), 173-175 [40] Wang, Z., Absolute continuity and extension of fuzzy measures, Fuzzy Sets Syst., 36, 395-399, (1990) · Zbl 0701.28012 [41] Wang, Z.; Klir, G. J., Generalized measure theory, (2009), Springer Press New York · Zbl 1184.28002 [42] Weber, S., $$\bot \operatorname{-} \operatorname{Decomposable}$$ measures and integrals for Archimedean t-conorm, J. Math. Anal. Appl., 101, 114-138, (1984) [43] Wu, C.; Sun, B., A note on the extension of null-additive set function on algebra of subsets, Appl. Math. Lett., 20, 770-772, (2007) · Zbl 1131.28002 [44] Yue, Y.; Shi, F. G., On fuzzy pseudo-metric spaces, Fuzzy Sets Syst., 161, 1105-1116, (2010) · Zbl 1194.54014
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.