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Some algebraic-topological properties of invariant measures and their applications (FR-18-6190)


Funded by

SRNSFGShota Rustaveli National Science Foundation of Georgia

Start Date: 2019-02-22       End Date: 2021-02-21

Investigation of various problem s connected with the fundamental concepts of measurability of sets and functions is necessary for further development of m any branches of modern mathematics, such as real and com plex analysis, abstract harmonic analysis, analysis in infinite-dimensional topological vector spaces, ergodic theory, probability theory, the theory of stochastic processes, gam e theory, general topology, etc. In this respect, the notion of measurability m ay be treated or construed in different ways, accordingly to the specific features of problem s under consideration.
The m ain goal of this project is an exam ination various approaches to the notion of m easurability and to the closely related notion of sm allness of sets in infinite-dim ensional topological vector spaces and their introduction in such areas of m athem atics as are the theory of linear functional, the theory of generalized integrals, m athem atical statistics, etc. For exam ple, we had used a notion of the Haar null set introduced by C hristensen(1973) for infinite-dim ensional Polish topological vector spaces for a classification of infinite-sam ple well-founded estim ates of a usefull signal into subjective and objective estimates in the case of the one dimensional stochastic model, which allowed us to get a strong m athematical analysis of conjectures of Jum Nunnally(1960) and Jacob Cohen(1993) arise under criticism of the theory of statistical decisions and asserted that the Null
Hypothesis is rejected for ''almost every" infinite sample by some Hypothesis Testing of “maximal reliability”.
So, one can anticipate that any progress in measure theory developed for infinite-dimensional Polish topological vector spaces will initiate the corresponding progress of the above-mentioned fields of mathematics. Moreover, the area of applications of measure theory developed for infinite-dimensional Polish topological vector spaces became substantially wider during several last decades.

Project members:

Publications

  • Alexander Kharazishvili, On orbits without the Baire property, Georgian Math. J., 2019 / 26, no. 4, 625-628, De Gruyter, 2019.
  • Alexander Kharazishvili, On finite sums of periodic functions, Georgian Math. J., 2020 / 27, no. 2, 265-269, De Gruyter, 2020.
  • Alexander Kharazishvili, Some remarks on Sierpiński-Zygmund functions in the strong sense, Georgian Math. J., 2020 / 27, no. 4, 569-575, De Gruyter, 2020.
  • Alexander Kharazishvili, A measure zero set in the plane with absolutely nonmeasurable linear sections, Trans. A. Razmadze Math. Inst., 2020 / 174, no. 3, 359-362, TSU, 2020.
  • Alexander Kharazishvili, On the generalized nonmeasurability of Vitali sets and Bernstein sets, Georgian Math. J., 2021 / 28, no. 4, 575–579, De Gruyter, 2020.
  • Alexander Kharazishvili, On oscillations of real-valued functions, Trans. A. Razmadze Math. Inst., 2021 /  175, no. 1, 63-67, TSU, 2021.
  • Alexander Kharazishvili, On the generalized nonmeasurability of some classical point sets, Trans. A. Razmadze Math. Inst., 2021 / 175, no. 1, 151-153, TSU, 2021.
  • Alexander Kharazishvili, On $T_2$-negligible $S_2$-absolutely nonmeasurable sets in the Euclidean plane, Bull. TICMI,  2021 /25, no. 1, 77-81, VIAM, 2021.