Spaces of non-additive measures generated by triangular norms
Abstract
We consider non-additive measures on the compact Hausdorff spaces, which are generalizations of the idempotent measures and max-min measures. These measures are related to the continuous triangular norms and they are defined as functionals on the spaces of continuous functions from a compact Hausdorff space into the unit segment.
The obtained space of measures (called ∗-measures, where ∗ is a triangular norm) are endowed with the weak* topology. This construction determines a functor in the category of compact Hausdorff spaces. It is proved, in particular, that the ∗-measures of finite support are dense in the spaces of ∗-measures. One of the main results of the paper provides an alternative description of ∗-measures on a compact Hausdorff space X, namely as hyperspaces of certain subsets in X × [0, 1]. This is an analog of a theorem for max-min measures proved by Brydun and Zarichnyi.
References
S. Ageev, E. D. Tymchatyn, On exact atomless Milutin maps, Topology and its Applications, 153 (2005), No2-3, 227–238. https://doi.org/10.1016/j.topol.2003.07.021
W. Briec, C. Horvath, B-convexity, Optimization, 53 (2004), No2, 103–127. https://doi.org/10.1080/02331930410001695283
V. Brydun, A. Savchenko, M. Zarichnyi, Fuzzy metrization of the spaces of idempotent measures, European Journal of Mathematics, 6 (2020), No1, 98–109. https://doi.org/10.1007/s40879-019-00341-8
V. Brydun, M. Zarichnyi, Spaces of max-min measures on compact Hausdorff spaces, Fuzzy Sets and Systems, 396 (2020), 138–151. https://doi.org/10.1016/j.fss.2019.06.012
G. Cohen, S. Gaubert, J. Quadrat, I. Singer, Max-plus convex sets and functions. In: Litvinov, G.L., Maslov, V.P. (eds.): Idempotent Mathematics and Mathematical Physics. Contemporary Mathematics. American Mathematical Society, 2005, 105–129.
E. P. Klement, R. Mesiar, E. Pap, Triangular Norms. Dordrecht: Kluwer. 2000. ISBN 0-7923-6416-3.
V. Kolokoltsov, V. Maslov, Idempotent Analysis and Applications, Kluwer Acad. Publ., 1997.
G .L. Litvinov, The Maslov dequantization, idempotent and tropical mathematics: a brief introduction, J. Math. Sc. (New York), 140 (2007), No3, 426–444. https://doi.org/10.1007/s10958-007-0450-5
Sam B. Nadler, Hyperspaces of sets. Monographs and Textbooks on Pure and Applied Mathematics, No49, Marcel Dekker, Inc., New York, 1978, xvi + 707 pp.
T. Radul, Equilibrium under uncertainty with fuzzy payoff, Topol. Methods Nonlinear Anal., 59 (2002), No2B, 1029–1045. https://doi.org/10.12775/TMNA.2021.049
E.V. Shchepin, Functors and uncountable powers of compacta, Uspekhi Mat. Nauk, 31 (1981), 3–62.
T. Świrszcz, Monadic functors and convexity, Bull. Acad. Pol. Sci. Ser. Sci. Math. Astronom. Phys. 22 (1974), No1, 39–42.
M. Zarichnyi, Spaces and maps of idempotent measures, Izvestia: Mathematics, 74 (2010), No3, 481–499.
Copyright (c) 2023 Kh. Sukhorukova
This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.
Matematychni Studii is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International (CC BY-NC-ND 4.0) license.