Real Analysis Exchange

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

Tuo-Yeong Lee

Full-text: Open access

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

Permanent link to this document
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

Keywords
Henstock-Kurzweil integral equi-integrability

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


Export citation

References

  • B. Bongiorno, L. Di Piazza and V. Skvortsov, A new descriptive characterization of Denjoy-Perron integral, Real Anal. Exchange 21 (1995/96), 656–663.
  • R. A. Gordon, The integrals of Lebesgue, Denjoy, Perron, and Henstock, Graduate Studies in Mathematics, 4, American Mathematical Society, Providence, RI, 1994.
  • J. Kurzweil, Henstock-Kurzweil integration, its relation to topological vector spaces, Series in Real Analysis Volume, 7, World Scientific 2000.
  • J. Kurzweil and J. Jarník, Equiintegrability and Controlled Convergence of Perron-type integrable functions, Real Anal. Exchange, 17 (1991/92), 110–139.
  • Lee Peng Yee and Chew Tuan Seng, A Riesz-type definition of the Denjoy integral, Real Anal. Exchange, 11 (1985/86), 221–227.
  • Lee Peng Yee, Lanzhou Lectures on Henstock integration, Series in Real Analysis, 2, World Scientific 1989.
  • Lee Peng Yee and Rudolf Výborný, The integral, An Easy Approach after Kurzweil and Henstock, Australian Mathematical Society Lecture Series, 14, Cambridge University Press 2000.
  • Lee Tuo Yeong, A full descriptive definition of the Henstock-Kurzweil integral in the Euclidean space, Proc. London Math. Soc. to appear.
  • E. J. McShane, Integration, Princeton University Press, 1944.
  • S Nakanishi, A new definition of the Denjoy's special integral by the method of successive approximation, Math. Japonica 41, No. 1 (1995), 217–230.
  • W. F. Pfeffer, A note on the Generalized Riemann integral, Proc. Amer. Math. Soc., 103, No. 4 (1988), 1161–1166.
  • W. F. Pfeffer, The Lebesgue and Denjoy-Perron integrals from a descriptive point of view, Ricerche Mat. 48 (1999), no. 2, 211–223.
  • S. Saks, Theory of the integral, 2nd edn, New York, 1964.
  • B. S. Thomson, Derivates of Interval Functions, Mem. Amer. Math. Soc. 452, Providence, 1991.