On the general form of a linear continuous functional in the space of regulated functions

Abstract

The author's research continues on the theory of regulated functions (functions having finite one-sided limits at each point) and $\sigma$-continuous functions (bounded functions having no more than a countable set of discontinuity points), as well as on the theory of the *-integral. The representability of a regulated function in the form of a sum of a right-continuous function and a left-continuous function is proved ($rl$-representability of the proper function). It is shown that the general form of a linear continuous functional in the space of regulated functions ($\sigma$-continuous functions) is the *-integral of a regulated ($\sigma$-continuous) function over a function of bounded variation.