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Title: On the double Lusin condition and convergence theorem for Kurzweil-Henstock type integrals (English)
Author: Racca, Abraham
Author: Cabral, Emmanuel
Language: English
Journal: Mathematica Bohemica
ISSN: 0862-7959 (print)
ISSN: 2464-7136 (online)
Volume: 141
Issue: 2
Year: 2016
Pages: 153-168
Summary lang: English
Category: math
Summary: Equiintegrability in a compact interval $E$ may be defined as a uniform integrability property that involves both the integrand $f_n$ and the corresponding primitive $F_n$. The pointwise convergence of the integrands $f_n$ to some $f$ and the equiintegrability of the functions $f_n$ together imply that $f$ is also integrable with primitive $F$ and that the primitives $F_n$ converge uniformly to $F$. In this paper, another uniform integrability property called uniform double Lusin condition introduced in the papers E. Cabral and P. Y. Lee (2001/2002) is revisited. Under the assumption of pointwise convergence of the integrands $f_n$, the three uniform integrability properties, namely equiintegrability and the two versions of the uniform double Lusin condition, are all equivalent. The first version of the double Lusin condition and its corresponding uniform double Lusin convergence theorem are also extended into the division space. (English)
Keyword: Kurzweil-Henstock integral
Keyword: $g$-integral
Keyword: double Lusin condition
Keyword: uniform double Lusin condition
MSC: 26A39
idZBL: Zbl 06587860
idMR: MR3499782
DOI: 10.21136/MB.2016.13
Date available: 2016-05-19T09:03:45Z
Last updated: 2020-07-01
Stable URL:
Reference: [1] Aye, K. K., Lee, P. Y.: The dual of the space of functions of bounded variation.Math. Bohem. 131 (2006), 1-9. Zbl 1112.26008, MR 2210998
Reference: [2] Cabral, E., Lee, P.-Y.: The primitive of a Kurzweil-Henstock integrable function in multidimensional space.Real Anal. Exch. 27 (2002), 627-634. Zbl 1069.26013, MR 1922673, 10.14321/realanalexch.27.2.0627
Reference: [3] Cabral, E., Lee, P.-Y.: A fundamental theorem of calculus for the Kurzweil-Henstock integral in {$\Bbb R^m$}.Real Anal. Exch. 26 (2001), 867-876. MR 1844400
Reference: [4] Lee, P. Y.: The integral à la Henstock.Sci. Math. Jpn. 67 (2008), 13-21. Zbl 1162.26004, MR 2384584
Reference: [5] Lee, P. Y.: Lanzhou Lectures on Henstock Integration.Series in Real Analysis 2 World Scientific, London (1989). Zbl 0699.26004, MR 1050957


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