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1996/1997 Riesz type theorems for general integrals
Vasile Ene
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Real Anal. Exchange 22(2): 714-733 (1996/1997).


The author gives a general descriptive definition for integration, denoted by \({\mathcal P}\), which has as special cases the Lebesgue integral for bounded measurable functions, the Lebesgue integral, the Denjoy-Perron integral \({\mathcal D}^*\), the wide Denjoy integral \({\mathcal D}\), the Foran integral, the Iseki integral and the \(S{\mathcal F}\)-integral (\cite{Ene1}). This \({\mathcal P}\)-integral will admit Riesz type representation theorems (introducing an Alexiewicz norm, and identifying \(f\) with \(g\) whenever \(f = g\) \(a.e.\) on \([a,b]\)). The classical Riesz representation theorem for the linear and continuous functionals on \((C([a,b]),\|\cdot\|_\infty)\) is a consequence of Theorem 2. In addition it is shown that the space of \({\mathcal P}\)-integrable functions is of the first category in itself (see Section 5). Also a characterization of weak convergence on this space is given.


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Vasile Ene. "Riesz type theorems for general integrals." Real Anal. Exchange 22 (2) 714 - 733, 1996/1997.


Published: 1996/1997
First available in Project Euclid: 22 May 2012

zbMATH: 0942.26016
MathSciNet: MR1460983

Primary: 26A39 , 26A45 , 26A46 , 46A50 , 54E52

Keywords: Alexiewicz norm , bounded slope variation , completion , essentially bounded variation , weak convergence

Rights: Copyright © 1996 Michigan State University Press

Vol.22 • No. 2 • 1996/1997
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