Abstract
The validity of the monotone convergence theorem, the Fatou and the reverse Fatou lemmas, and the dominated convergence theorem of the Choquet integral of measurable functions converging in measure are fully characterized by the conditional versions of the monotone autocontinuity and the autocontinuity. In those theorems the nonadditive measure may be infinite and the functions may be unbounded. The dual measure forms and the extension to symmetric and asymmetric Choquet integrals are also discussed.
This work was supported by JSPS KAKENHI Grant Number 17K05293.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsReferences
Choquet, G.: Theory of capacities. Ann. Inst. Fourier (Grenoble) 5, 131–295 (1953–1954)
Couso, I., Montes, S., Gil, P.: Stochastic convergence, uniform integrability and convergence in mean on fuzzy measure spaces. Fuzzy Sets Syst. 129, 95–104 (2002)
Denneberg, D.: Non-additive Measure and Integral, 2nd edn. Kluwer Academic Publishers, Dordrecht (1997)
Li, J., Mesiar, R., Pap, E., Klement, E.P.: Convergence theorems for monotone measures. Fuzzy Sets Syst. 281, 103–127 (2015)
Kawabe, J.: The bounded convergence in measure theorem for nonlinear integral functionals. Fuzzy Sets Syst. 271, 31–42 (2015)
Kawabe, J.: A unified approach to the monotone convergence theorem for nonlinear integrals. Fuzzy Sets Syst. 304, 1–19 (2016)
Kawabe, J.: The Vitali type theorem for the Choquet integral. Linear Nonlinear Anal. 3, 349–365 (2017)
Murofushi, T., Sugeno, M., Suzaki, M.: Autocontinuity, convergence in measure, and convergence in distribution. Fuzzy Sets Syst. 92, 197–203 (1997)
Pap, E.: Null-additive Set Functions. Kluwer Academic Publishers, Bratislava (1995)
Ralescu, D., Adams, G.: The fuzzy integral. J. Math. Anal. Appl. 75, 562–570 (1980)
Rébillé, Y.: Autocontinuity and convergence theorems for the Choquet integral. Fuzzy Sets Syst. 194, 52–65 (2012)
Šipoš, J.: Integral with respect to a pre-measure. Math. Slovaca 29, 141–155 (1979)
Schmeidler, D.: Integral representation without additivity. Proc. Am. Math. Soc. 97, 255–261 (1986)
Sugeno, M.: Theory of fuzzy integrals and its applications. Ph.D. dissertation, Tokyo Institute of Technology, Tokyo (1974)
Wang, Z.: Convergence theorems for sequences of Choquet integrals. Int. J. Gen. Syst. 26, 133–143 (1997)
Wang, Z., Klir, G.J.: Generalized Measure Theory. Springer, New York (2009)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2019 Springer Nature Switzerland AG
About this paper
Cite this paper
Kawabe, J. (2019). Convergence in Measure Theorems of the Choquet Integral Revisited. In: Torra, V., Narukawa, Y., Pasi, G., Viviani, M. (eds) Modeling Decisions for Artificial Intelligence. MDAI 2019. Lecture Notes in Computer Science(), vol 11676. Springer, Cham. https://doi.org/10.1007/978-3-030-26773-5_2
Download citation
DOI: https://doi.org/10.1007/978-3-030-26773-5_2
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-26772-8
Online ISBN: 978-3-030-26773-5
eBook Packages: Computer ScienceComputer Science (R0)