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Home » Issues » Archive of Issues » 2024 - 2 issue » pp. 182-203

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Russia Izhevsk
Year
2024
Volume
34
Issue
2
Pages
182-203
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Section Mathematics
Title On Banach spaces of regulated functions of several variables. Analogue of the Riemann–Stieltjes integral
Author(-s) Baranov V.N.a, Rodionov V.I.a, Rodionova A.G.a
Affiliations Udmurt State Universitya
Abstract In the previous work of the authors, the concept of a regulated function of several variables $f\colon X\to\mathbb R$ was introduced, where $X\subseteq \mathbb R^n.$ The definition is based on the concept of a special partition of the set $X$ and the concept oscillation of the function $f$ on the elements of the partition. The space ${\rm G}(X)$ of such functions is Banach in the $\sup$-norm and is the closure of the space of step functions. In this paper, the space ${\rm G}^F(X)$ is defined and studied, which differs from ${\rm G}(X)$ in that here, in defining regulated functions of several variables, instead of special partitions, $F$-partitions are used: their elements are non-empty open sets measurable by the generalized Jordan measure (by the measure $m_{_{\!F}}$). (Symbol $F$ denotes the function generating the measure $m_{_{\!F}}.$) In the second part of the work, the concept of $F$-integrability of functions of several variables is defined. It is proved that if $X$ is the closure of a non-empty open bounded set $X_0\subseteq {\mathbb R}^n,$ measurable with respect to measure $m_{_{\!F}},$ and the function $f\colon X\to {\mathbb R}$ is integrable in the Riemann–Stieltjes sense with respect to the measure $m_{_{\!F}}$, then it is $F$-integrable. In this case, the values of the multiple integrals coincide. All functions from the space ${\rm G}^F(X)$ are $F$-integrable. The main properties of the Riemann–Stieltjes $F$-integral are proved.
Keywords step function, regulated function, generalized Jordan measure, Riemann–Stieltjes integral
UDC 517.982.22, 517.518.12
MSC 46B99, 26A42
DOI 10.35634/vm240202
Received 2 November 2023
Language Russian
Citation Baranov V.N., Rodionov V.I., Rodionova A.G. On Banach spaces of regulated functions of several variables. Analogue of the Riemann–Stieltjes integral, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2024, vol. 34, issue 2, pp. 182-203.
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