Real Analysis Exchange

Topologizing the Denjoy space by measuring equiintegrability.

J. Alan Alewine and Eric Schechter

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Basic limit theorems for the KH integral involve equiintegrable sets. We construct a family of Banach spaces $X_\Delta$ whose bounded sets are precisely the subsets of ${\cal KH}[0,1]$ that are equiintegrable and pointwise bounded. The resulting inductive limit topology on $\bigcup_\Delta X_\Delta = {\cal KH}[0,1]$ is barreled, bornological, and stronger than both pointwise convergence and the topology given by the Alexiewicz seminorm, but it lacks the countability and compatibility conditions that are often associated with inductive limits.

Article information

Real Anal. Exchange, Volume 31, Number 1 (2005), 23-44.

First available in Project Euclid: 5 June 2006

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

barreled convergence rate Denjoy space equiintegrable Henstock integral inductive limit KH integral Kurzweil integral norm


Alewine, J. Alan; Schechter, Eric. Topologizing the Denjoy space by measuring equiintegrability. Real Anal. Exchange 31 (2005), no. 1, 23--44.

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  • J. Alan Alewine, An inductive limit topology on the Denjoy space, Ph.D. thesis, Vanderbilt University, 2003.
  • J. Alan Alewine and Eric Schechter, Rates of uniform convergence for Riemann integrals, in preparation.
  • A. Alexiewicz, Linear functionals on Denjoy-integrable functions, Colloquium Math., 1 (1948), 289–293.
  • Robert G. Bartle, A modern theory of integration, Graduate Studies in Mathematics 32, American Mathematical Society, Providence, RI, 2001.
  • Russell A. Gordon, The integrals of Lebesgue, Denjoy, Perron, and Henstock, Graduate Studies in Mathematics, 4, American Mathematical Society, Providence, RI, 1994.
  • Russell A. Gordon, An iterated limits theorem applied to the Henstock integral, Real Anal. Exchange, 21(2) (1995/96), 774–781.
  • Henri Hogbe-Nlend, Bornologies and functional analysis, North-Holland Publishing Co., Amsterdam, 1977.
  • Jaroslav Kurzweil, Henstock-Kurzweil integration: its relation to topological vector spaces, Series in Real Analysis, 7, World Scientific Publishing Co. Inc., River Edge, NJ, 2000.
  • Peng Yee Lee, Lanzhou lectures on Henstock integration, Series in Real Analysis, 2 World Scientific Publishing Co. Inc., Teaneck, NJ, 1989.
  • Peng Yee Lee and Rudolf Výborný, Integral: an easy approach after Kurzweil and Henstock, Australian Mathematical Society Lecture Series, 14, Cambridge University Press, Cambridge, 2000.
  • Balmohan Vishnu Limaye, Functional analysis, John Wiley & Sons Inc., New York, 1981, A Halsted Press Book.
  • Eric Schechter, Handbook of analysis and its foundations, Academic Press Inc., San Diego, CA, 1997.
  • Štefan Schwabik and Ivo Vrkoč, On Kurzweil-Henstock equiintegrable sequences, Mathematica Bohemica, 121(2) (1996), 189–207.
  • Charles Swartz, Norm convergence and uniform integrability for the Henstock-Kurzweil integral, Real Anal. Exchange, 24(1) (1998/99), 423–426.
  • Brian S. Thomson, The space of Denjoy-Perron integrable functions, Real Anal. Exchange, 25(2) (1999/00), 711–726.