A Riesz-type definition of the Denjoy integral
type: Item
instance of: scholarly article
title: A Riesz-type definition of the Denjoy integral
zbMATH Open document ID: 0604.26007
author: Tuan Seng Chew
author: Peng-Yee Lee
published in: Real Analysis Exchange
publication date: 1986-01-01T00:00:00Z
review text: A sequence \((f_ n)\) is said to be control-convergent to f on [a,b] iff \((i)\quad f_ n(x)\to f(x)\) a.e. in [a,b] as \(n\to \infty\) and each \(f_ n\) is Denjoy integrable on [a,b]; (ii) the primitives \(F_ n\) of \(f_ n\) are ACG\({}_*\) uniformly in n; (iii) \(F_ n(x)\) converges uniformly on [a,b], and \(n\to \infty.\) If \((f_ n)\) is control-convergent to f on [a,b], then f is Denjoy integrable on [a,b] and \[ \int^{b}_{a}f(x)dx=\lim_{n\to \infty}\int^{b}_{a}f_ n(x)dx. \] A function f is said to be RD integrable on [a,b] iff it is an a.e. limit of a control-convergent sequence of step functions \(\phi_ n\) and \[ \int^{b}_{a}f(x)dx=\lim_{n\to \infty}\int^{b}_{a}\phi_ n(x)dx. \] A function f is RD integrable on [a,b] if and only if it is Denjoy integrable on [a,b].
Mathematics Subject Classification ID: 26A39
zbMATH DE Number: 3975262
zbMATH Keywords: Denjoy integral
zbMATH Keywords: Riesz-type definition
zbMATH Keywords: \(ACG_ *\)
zbMATH Keywords: control-convergent sequence
reviewed by: Andrzej Smajdor
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 3975262