Real Analysis Exchange

Integration by Parts and Other Theorems for R3S-Integrals

E. R. Love

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This paper is a continuation of [3], in which was introduced the Refinement-Ross-Riemann-Stieltjes $(R^3S)$ Integral, and in which some of its advantages were exhibited. After a brief summary of [3], this paper proves an integration by parts theorem which shows incidentally that if $f$ is $R^3S$-integrable with respect to $g$ then $g$ is $R^3S$-integrable with respect to $f$. Theorems on term-by-term integration of sequences analogous to the Helly-Bray Theorem are next proved, in a context of Wiener's functions of bounded generalized variation as developed by L. C. Young and me. In a similar context I prove also a theorem resembling the classical theorem of Riesz representing linear functionals by Stieltje.

Article information

Real Anal. Exchange, Volume 24, Number 1 (1998), 315-336.

First available in Project Euclid: 23 March 2011

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Mathematical Reviews number (MathSciNet)

Primary: 26A42: Integrals of Riemann, Stieltjes and Lebesgue type [See also 28-XX]

integration by parts


Love, E. R. Integration by Parts and Other Theorems for R 3 S -Integrals. Real Anal. Exchange 24 (1998), no. 1, 315--336.

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