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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 $G\sb L(a,b)$ of left-continuous, on $]a,b[$, regulated functions can be represented by means of a Dushnik-Stieltjes integral [{\it 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 $G\sb L(a,b)$ iff there exist $q\in {\bbfR}$, $p\in BV(a,b)$ such that for all $x\in G\sb L(a,b)$, $F(x)=qx(a)+\int\sp{b}\sb{a}p dx$, 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 $\int\sp{b}\sb{a}f dg$ 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).
Reviewer: P.S.Bullen

26A39Special integrals of functions of one real variable
26A45Functions of bounded variation (one real variable)
28A25Integration with respect to measures and other set functions
46E99Linear function spaces and their duals
Full Text: EuDML