The Proof of Quantum and Fuzzy Measures as Generalization of Measure That Does Not Generalize Each Other

Miftahul Fikri; Zuherman Rustam; Kurniawan Atmadja; Nurhadi Hadi

doi:10.3889/oamjms.2022.8516

Authors

Miftahul Fikri Electrical Technology Study Program, PLN Institute of Technology, Jakarta, Indonesia
Zuherman Rustam Departement of Mathematics, University of Indonesia, Depok, Indonesia
Kurniawan Atmadja Mathematics Study Program, National Institute of Science and Technology, Jakarta, Indonesia
Nurhadi Hadi Islamic Family Law Study Program, Muhammadiyah University of Jakarta, Jakarta, Indonesia

DOI:

https://doi.org/10.3889/oamjms.2022.8516

Keywords:

Measure, Quantum measure, Fuzzy measure

Abstract

The studies on quantum and fuzzy theories by Planck and Zadeh, respectively, still continue presently. Based on the mathematical side, these two theories that directly related and become the basis for various studies, both theoretical and applied, are quantum and fuzzy measures. Although in the literature, these are measure generalizations but not substantiated by definition; therefore, the substance does not appear directly. Furthermore, there is also no discussion of the relationship between quantum and fuzzy measures on Boolean σ– algebra. This study accomplishes a proof based on the definition that both the quantum and the fuzzy measures are measure generalizations or do not reciprocally generalize; hence, the measure is the intersection of the two.

Downloads

Download data is not yet available.

Metrics

Metrics Loading ...

Plum Analytics Artifact Widget Block

References

Wattimena RA. Philosophy and Science. Indonesia: Grasindo; 2008.

Marsh GE. An Introduction to the Standard Model of Particle Physics for the Non-Specialist. Singapore: World Scientific; 2018. DOI: https://doi.org/10.1142/10779

Ter Haar D. On the Theory of the Energy Distribution Law of the Normal Spectrum. United Kingdom: Pergamon Press; 1967. p. 82. DOI: https://doi.org/10.1016/B978-0-08-012102-4.50013-9

Gudder S. Generalized measure theory. Found Phys. 1973;3(3):399-411. DOI: https://doi.org/10.1007/BF00708681

Sorkin RD. Quantum mechanics as quantum measure theory. Mod Phys Lett A. 1994;9(33):3119-27. DOI: https://doi.org/10.1142/S021773239400294X

Gudder S. Quantum measure and integration theory. J Math Phys. 2009;50:59. DOI: https://doi.org/10.1063/1.3267867

Gudder S. Quantum measure theory. Math Slovaca. 2010;60(5):681-700. DOI: https://doi.org/10.2478/s12175-010-0040-8

Zadeh LA. Fuzzy sets. Inf Control. 1965;8:338-53. DOI: https://doi.org/10.1016/S0019-9958(65)90241-X

Zadeh LA. Probability measures of fuzzy events. J Math Anal Appl. 1968;23:421-7. DOI: https://doi.org/10.1016/0022-247X(68)90078-4

Sugeno M. Theory of Fuzzy Integrals and its Applications. Japan: Tokyo Institute of Technology; 1974. 11. Wang Z. The autocontinuity of set function and the fuzzy integral. J Math Anal Appl. 1984;99:195-218. DOI: https://doi.org/10.1016/0022-247X(84)90243-9

Murofushi T, Sugeno M. An interpretation of fuzzy measures and the choquet integral as an integral with respect to a fuzzy measure. Fuzzy Sets Syst. 1989;29:201-27. DOI: https://doi.org/10.1016/0165-0114(89)90194-2

Chen L, Duan G, Wang S, Ma J. A choquet integral based fuzzy logic approach to solve uncertain multi-criteria decision making problem. Expert Syst Appl. 2020;149:1-12. DOI: https://doi.org/10.1016/j.eswa.2020.113303

Bezdek JC. FCM : The fuzzy c-means clustering algorithm. Comput Geosci. 1984;10(2):191-203. DOI: https://doi.org/10.1016/0098-3004(84)90020-7

Chung F, Lee T. Fuzzy Learning Vector Quantization. Nagoya, Japan: Proceedings of 1993 International Joint Confrence on Neral Networks; 1993. p. 2739-42.

Ibrahim OA, Member S, Keller JM, Fellow L, Bezdek JC, Fellow L. Evaluating evolving structure in streaming data with modified Dunn’s indices. IEEE Trans Emerg Top Comput Intell. 2019;5:1-12.

Dvurecenskij A, Chovanec F. Fuzzy quantum spaces and compatibility. Int J Theor Phys. 1988;27(9):1069-82. DOI: https://doi.org/10.1007/BF00674352

Piasecki K. Probability of fuzzy events defined as denumerable additivity measure. Fuzzy Sets Syst. 1985;17:271-84. DOI: https://doi.org/10.1016/0165-0114(85)90093-4

Duris V, Bartkova R, Tirpakova A. Several limit theorems on fuzzy quantum space. Mathematics. 2021;9:1-14. DOI: https://doi.org/10.3390/math9040438

Birkhoff G. Lattice Theory. New York: American Mathematical Society; 1948.

Gratzer G. General Lattice Theory. Berlin: Birkhause Verlag; 2007.

Jech T. Set Theory. Berlin, Germany: Springer; 2006.

Bogachev VI. Measure Theory. Vol. 1. Berlin, Germany: Springer Berlin Heidelberg; 2007. DOI: https://doi.org/10.1007/978-3-540-34514-5

Royden HL, Fitzpatrick PM. Real Analysis. United States: Prentice Hall; 2010.

Sugeno M. A way to Choquet calculus. IEEE Trans Fuzzy Syst. 2014;23(5):1-22. DOI: https://doi.org/10.1109/TFUZZ.2014.2362148