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 of left-continuous, on , 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 iff there exist , such that for all , , 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 for any is proved; if , 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).