Real Analysis Exchange

Topologizing the Denjoy space by measuring equiintegrability.

J. Alan Alewine and Eric Schechter

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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

Permanent link to this document
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

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

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


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References

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