B.I. Ananyev. On some complements to Liu’s theory ... P. 5-20

In the framework of Baoding Liu’s uncertainty theory, some new concepts are introduced and their properties are considered. In particular, regular functions of uncertainty are introduced on an uncountable product of spaces. An analog of the Lomnitskii–Ulam theorem from traditional probability theory is obtained. Necessary and sufficient conditions are specified under which a function defined on a Banach space of bounded functions is a distribution function for some uncertain mapping. Some notions of Liu’s theory are generalized for uncountably many objects. Examples showing the similarity and the difference between Liu’s theory and probability theory are analyzed. An application of Liu’s theory to estimation theory is considered on examples.

Keywords: functions of uncertainty, uncertain mappings, distribution functions, theory of estimation

Received July 15, 2023

Revised October 20, 2023

Accepted October 23, 2023

Funding Agency: The work was performed as part of research conducted in the Ural Mathematical Center with the financial support of the Ministry of Science and Higher Education of the Russian Federation (Agreement number 075-02-2023-913).

Boris Ivanovich Ananyev, Dr. Phys.-Math. Sci., Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620108 Russia, e-mail: abi@imm.uran.ru

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Cite this article as: B.I. Ananyev. On some complements to Liu’s theory. Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2024, vol. 30, no. 1, pp. 5–20. Proceedings of the Steklov Institute of Mathematics, 2024, Vol. 325, Suppl. 1, pp. S1–S16.