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Regulated functions and the Perron-Stieltjes integral. (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 ${G}_{L}\left(a,b\right)$ of left-continuous, on $\right]a,b\left[$, 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 ${G}_{L}\left(a,b\right)$ iff there exist $q\in ℝ$, $p\in BV\left(a,b\right)$ such that for all $x\in {G}_{L}\left(a,b\right)$, $F\left(x\right)=qx\left(a\right)+{\int }_{a}^{b}pdx$, 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 $f\in BV\left(a,b\right),$ $g\in G\left(a,b\right)$ is proved; if $f\in G\left(a,b\right)$, $g\in BV\left(a,b\right)$ 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).

Reviewer: P.S.Bullen
##### 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