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

In the previous work of the authors, the concept of a regulated function of several variables f X→ R was introduced, where X⊆ Rⁿ. 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 G(X) of such functions is Banach in the -norm and is the closure of the space of step functions. In this paper, the space GF(X) is defined and studied, which differs from 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 X₀⊆ Rⁿ, measurable with respect to measure m_\!F, and the function f X→ 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 GF(X) are F-integrable. The main properties of the Riemann–Stieltjes F-integral are proved.