## Real Analysis Exchange

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

Tuo-Yeong Lee

#### 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.

#### Article information

Source
Real Anal. Exchange, Volume 28, Number 1 (2002), 55-71.

Dates
First available in Project Euclid: 12 June 2006

https://projecteuclid.org/euclid.rae/1150118735

Mathematical Reviews number (MathSciNet)
MR1973968

Zentralblatt MATH identifier
1044.26007

Subjects
Primary: 26A39: Denjoy and Perron integrals, other special integrals

#### Citation

Lee, Tuo-Yeong. The sharp Riesz-type definition for the Henstock-Kurzweil integral. Real Anal. Exchange 28 (2002), no. 1, 55--71. https://projecteuclid.org/euclid.rae/1150118735

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