Some answers to a question of P. Bullen
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.
Kurzweil-Henstock generalized Riemann type integrals
scientific article; zbMATH DE number 4081874