On Banach spaces of regulated functions of several variables. An analogue of the Riemann integral

The paper introduces the concept of a regulated function of several variables f : X -> 8, where X subset of 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 subset of R-n, is regulated (belongs to the space (G(X), & Vert;<middle dot> |>). The completeness of the space G(X) in the sup-norm & Vert;<middle dot>& Vert; is proved. This is the closure of the space of step functions. In the second part of the work, the space G(J) (X) is defined and studied, which differs from the space G(X) in that its definition uses J-partitions instead of partitions, whose elements are Jordan measurable open sets. The properties of the space G(X) listed above carry over to the space 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 subset of R-n, and the function f :X -> 8 is Riemann integrable, then it is J-integrable. In this case, the values of the integrals coincide. All functions f is an element of G(J) (X) are J-integrable.