The sharp Riesz-type definition for the Henstock-Kurzweil integral.

Abstract

In this paper, we prove that if $f$ is Henstock-Kurzweil integrable on a compact subinterval $[a,b]$ of the real line, then the following conditions are satisfied: (i) there exists an increasing sequence $\{X_n\}$ of closed sets whose union is $[a,b]$; (ii) $\{f{\chi_{ _{X_n}}}\}$ is a sequence of Lebesgue integrable functions on $[a,b]$; (iii) the sequence $\{f{\chi_{ _{X_n}}}\}$ is Henstock-Kurzweil equi-integrable on $[a,b]$. Subsequently, we deduce that the gauge function in the definition of the Henstock-Kurzweil integral can be chosen to be measurable, and an indefinite Henstock-Kurzweil integral generates a sequence of uniformly absolutely continuous finite variational measures.