Real Analysis Exchange

The Wide Denjoy Integral as the Limit of a Sequence of Stepfunctions in a Suitable Convergence

Vasile Ene

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Abstract

In this paper we shall prove that a function \(f:[a,b] \to \overline{{\mathbb R}}\) that is \({\mathcal D}\)--integrable on \([a,b]\) can be defined as the limit of a \({\mathcal D}\)-controlled convergent sequence of stepfunctions (see the second part of Theorem 2). In the last section we show that Ridder’s \(\alpha\)- and \(\beta\)-integrals can also be defined as the limit of some controlled convergent sequences of stepfunctions (see Theorem 4).

Article information

Source
Real Anal. Exchange, Volume 23, Number 2 (1999), 719-734.

Dates
First available in Project Euclid: 14 May 2012

Permanent link to this document
https://projecteuclid.org/euclid.rae/1337001377

Mathematical Reviews number (MathSciNet)
MR1639945

Zentralblatt MATH identifier
0943.26020

Subjects
Primary: 26A39: Denjoy and Perron integrals, other special integrals 26A46: Absolutely continuous functions

Keywords
{the wide Denjoy integral} {\(AC\)} {\(UACG\)} {controlled convergence}

Citation

Ene, Vasile. The Wide Denjoy Integral as the Limit of a Sequence of Stepfunctions in a Suitable Convergence. Real Anal. Exchange 23 (1999), no. 2, 719--734. https://projecteuclid.org/euclid.rae/1337001377


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