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The Kolmogorov Ideas on the Integration Theory in Modern Research

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Abstract

Generalizations of construction of Kolmogorov integral to the case of Banach space-valued functions are considered. We demonstrate how the Kolmogorov ideas on integration theory, in particular, the notion of differential equivalence, have been developed in the theory of the Henstock–Kurzweil integral. In this connection, a variational version of a Henstock type integral with respect to a rather general derivation basis is studied. An example of application of this integral to harmonic analysis is given. Some results related to the Kolmogorov \(A\)-integral are also considered.

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Funding

The work of A.P. Solodov is supported by the Foundation for Development of Theoretical Physics and Mathematics Basis.

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Authors and Affiliations

  1. Chair of Mathematical Analysis, Faculty of Mechanics and Mathematics, Lomonosov Moscow State University, Moscow, Russia

    T. P. Lukashenko & A. P. Solodov

  2. Moscow Center of Fundamental and Applied Mathematics, Moscow, Russia

    T. P. Lukashenko, V. A. Skvortsov & A. P. Solodov

  3. Chair of Theory of Functions and Functional Analysis, Faculty of Mechanics and Mathematics, Lomonosov Moscow State University, Moscow, Russia

    V. A. Skvortsov

Authors
  1. T. P. Lukashenko
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  2. V. A. Skvortsov
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  3. A. P. Solodov
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Correspondence to T. P. Lukashenko, V. A. Skvortsov or A. P. Solodov.

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The authors of this work declare that they have no conflicts of interest.

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Translated by E. Oborin

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Lukashenko, T.P., Skvortsov, V.A. & Solodov, A.P. The Kolmogorov Ideas on the Integration Theory in Modern Research. Moscow Univ. Math. Bull. 79, 22–33 (2024). https://doi.org/10.3103/S0027132224700037

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