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2002/2003 On the \(T\)-integration of Karták and Mařík.
Vasile Ene
Author Affiliations +
Real Anal. Exchange 28(2): 515-542 (2002/2003).


General notions of integration have been introduced by Saks \cite[p. 254]{S3}, Kart\'ak \cite[p. 482]{K14}, Kubota \cite[p. 389]{K8} and Sarkhel \cite[p. 299]{S19}. Kart\'ak's \(T\)-integration was further studied by Kart\'ak and Ma\v{r}\`{\i} in \cite{K15}, and by Kubota in \cite{K17}. \par In this paper, starting from Kartak and Ma\v{r}\`{\i}'s definition, we introduce another general integration (see Definition \ref{D3}), that allows a very general theorem of dominated convergence (see Theorem \ref{T2}). Then we present a general definition for primitives, and this definition contains many of the known nonabsolutely convergent integrals: the Denjoy\(^*\)-integral, the \(\alpha\)-Ridder integral, the wide Denjoy integral, the \(\beta\)-Ridder integral, the Foran integral, the AF integral, the Gordon integral. Using this integration and Theorem \ref{T2}, we obtain a generalization of a result on differential equations, of Bullen and Vyborny \cite{B17}. \par We further give a Banach-Steinhaus type theorem, a categoricity theorem, Riesz type theorems (as a particular case we obtain the Alexiewicz Theorem \cite{A2}), and study the weak convergence for the \(T\)-integration.


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Vasile Ene. "On the \(T\)-integration of Karták and Mařík.." Real Anal. Exchange 28 (2) 515 - 542, 2002/2003.


Published: 2002/2003
First available in Project Euclid: 20 July 2007

zbMATH: 1052.26007
MathSciNet: MR2010334

Primary: 26A39 , 26A45

Keywords: $VB$ , AC , essentially bounded variation , Kart\'ak and Ma\v{r}\`{\i}k's $T$-integration , nonabsolutely convergent integrals , normed space.

Rights: Copyright © 2002 Michigan State University Press

Vol.28 • No. 2 • 2002/2003
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