## Real Analysis Exchange

### Topologizing the Denjoy space by measuring equiintegrability.

#### Abstract

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

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

Dates
First available in Project Euclid: 5 June 2006

https://projecteuclid.org/euclid.rae/1149516795

Mathematical Reviews number (MathSciNet)
MR2218186

Zentralblatt MATH identifier
1129.26003

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

#### Citation

Alewine, J. Alan; Schechter, Eric. Topologizing the Denjoy space by measuring equiintegrability. Real Anal. Exchange 31 (2005), no. 1, 23--44. https://projecteuclid.org/euclid.rae/1149516795

#### References

• 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.