*(English)*Zbl 0671.26006

Let G(a,b) be the Banach space of regulated functions on the compact interval [a,b]; the norm is the sup norm. Then it is known that the bounded linear functionals on its subspace ${G}_{L}(a,b)$ of left-continuous, on $]a,b[$, regulated functions can be represented by means of a Dushnik-Stieltjes integral [*H. S. Kaltenborn*, Bull. Am. Math. Soc. 40, 702-708 (1934; Zbl 0010.16905)]. Hence, using known relationships, F is a bounded linear functional on ${G}_{L}(a,b)$ iff there exist $q\in \mathbb{R}$, $p\in BV(a,b)$ such that for all $x\in {G}_{L}(a,b)$, $F\left(x\right)=qx\left(a\right)+{\int}_{a}^{b}pdx$, where the integral is a Perron-Stieltjes integral.

The object of this paper is to give a direct proof of this based on Kurzweil’s theory of the Perron-Stieltjes integral. The existence of the Perron-Stieltjes integral ${\int}_{a}^{b}fdg$ for any $f\in BV(a,b),$ $g\in G(a,b)$ is proved; if $f\in G(a,b)$, $g\in BV(a,b)$ the existence is known from Kurzweil’s theory. As well extensions of the integration by parts theorem and substitution theorem are proved; these will be useful in dealing with generalized differential equations and Volterra-Stieltjes integral equations in G(a,b).

##### MSC:

26A39 | Special integrals of functions of one real variable |

26A45 | Functions of bounded variation (one real variable) |

28A25 | Integration with respect to measures and other set functions |

46E99 | Linear function spaces and their duals |