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Russia Izhevsk
Year
2023
Volume
33
Issue
3
Pages
387-401
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Section Mathematics
Title On Banach spaces of regulated functions of several variables. An analogue of the Riemann integral
Author(-s) Baranov V.N.a, Rodionov V.I.a, Rodionova A.G.a
Affiliations Udmurt State Universitya
Abstract The paper introduces the concept of a regulated function of several variables $f\colon X\to\mathbb R$, where $X\subseteq \mathbb R^n$. The definition is based on the concept of a special partition of the set $X$ and the concept of oscillation of the function $f$ on the elements of the partition. It is shown that every function defined and continuous on the closure $X$ of the open bounded set $X_0\subseteq\mathbb R^n$, is regulated (belongs to the space $\langle{\rm G(}X),\|\cdot\ |\rangle$). The completeness of the space ${\rm G}(X)$ in the $\sup$-norm $\|\cdot\|$ is proved. This is the closure of the space of step functions. In the second part of the work, the space ${\rm G}^J(X)$ is defined and studied, which differs from the space ${\rm G}(X)$ in that its definition uses $J$-partitions instead of partitions, whose elements are Jordan measurable open sets. The properties of the space ${\rm G}(X)$ listed above carry over to the space ${\rm G}^J(X)$. In the final part of the paper, the notion of $J$-integrability of functions of several variables is defined. It is proved that if $X$ is a Jordan measurable closure of an open bounded set $X_0\subseteq\mathbb R^n$, and the function $f\colon X\to\mathbb R$ is Riemann integrable, then it is $J$-integrable. In this case, the values of the integrals coincide. All functions $f\in{\rm G}^J(X)$ are $J$-integrable.
Keywords step function, regulated function, Jordan measurability, Riemann integrability
UDC 517.982.22, 517.518.12
MSC 46B99, 26B15
DOI 10.35634/vm230301
Received 21 February 2023
Language Russian
Citation Baranov V.N., Rodionov V.I., Rodionova A.G. On Banach spaces of regulated functions of several variables. An analogue of the Riemann integral, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2023, vol. 33, issue 3, pp. 387-401.
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