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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 (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, pBV(a,b) such that for all xG L (a,b), F(x)=qx(a)+ 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 a b fdg for any fBV(a,b), gG(a,b) is proved; if fG(a,b), gBV(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
MSC:
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