Real Analysis Exchange

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

Tuo-Yeong Lee

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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

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

First available in Project Euclid: 12 June 2006

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

Henstock-Kurzweil integral equi-integrability


Lee, Tuo-Yeong. The sharp Riesz-type definition for the Henstock-Kurzweil integral. Real Anal. Exchange 28 (2002), no. 1, 55--71.

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