Some answers to a question of P. Bullen
type: Item
instance of: scholarly article
title: Some answers to a question of P. Bullen
zbMATH Open document ID: 0662.26004
author: James Foran
author: Sandra Meinershagen
published in: Real Analysis Exchange
publication date: 1988-01-01T00:00:00Z
review text: Peter Bullen has raised the question whether it is possible to restrict the gauge function used in Kurzweil-Henstock generalized Riemann type integrals without restricting the class of integrable functions. It is shown in this paper that the gauge function for a Kurzweil-Henstock integrable function f can be chosen to be measurable and, if Z is a \(G_{\delta}\) subset of measure zero containing the set \(Z_ f\) of x where the restricted Denjoy integral F(x) does not exist or has not f for derivative, then the gauge restricted to the complement of Z can be chosen of Baire class 2. Moreover, if \(| f|\) is dominated by a Baire function, a Baire gauge will suffice and if f is bounded measurable, it suffices to take a gauge which is in the second class of Baire. Some results are also given for Stieltjes type integrals.
Mathematics Subject Classification ID: 26A39
Mathematics Subject Classification ID: 26A42
Mathematics Subject Classification ID: 28A20
zbMATH DE Number: 4081874
zbMATH Keywords: measurable functions
zbMATH Keywords: Kurzweil-Henstock generalized Riemann type integrals
zbMATH Keywords: restricted Denjoy integral
zbMATH Keywords: Baire class 2
zbMATH Keywords: Baire function
zbMATH Keywords: Baire gauge
zbMATH Keywords: Stieltjes type integrals
reviewed by: Jean Mawhin
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 4081874