Real Analysis Exchange

Integration by Parts and Other Theorems for R3S-Integrals

E. R. Love

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Abstract

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

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

Dates
First available in Project Euclid: 23 March 2011

Permanent link to this document
https://projecteuclid.org/euclid.rae/1300906031

Mathematical Reviews number (MathSciNet)
MR1691754

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

Keywords
integration by parts

Citation

Love, E. R. Integration by Parts and Other Theorems for R 3 S -Integrals. Real Anal. Exchange 24 (1998), no. 1, 315--336. https://projecteuclid.org/euclid.rae/1300906031


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