On Banach spaces of regulated functions of several variables. Analogue of the Riemann–Stieltjes integral

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.