On the double Lusin condition and convergence theorem for Kurzweil-Henstock type integrals

Abraham Racca; Emmanuel Cabral

Mathematica Bohemica (2016)

  • Volume: 141, Issue: 2, page 153-168
  • ISSN: 0862-7959

Abstract

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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.

How to cite

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Racca, Abraham, and Cabral, Emmanuel. "On the double Lusin condition and convergence theorem for Kurzweil-Henstock type integrals." Mathematica Bohemica 141.2 (2016): 153-168. <http://eudml.org/doc/276991>.

@article{Racca2016,
abstract = {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.},
author = {Racca, Abraham, Cabral, Emmanuel},
journal = {Mathematica Bohemica},
keywords = {Kurzweil-Henstock integral; $g$-integral; double Lusin condition; uniform double Lusin condition},
language = {eng},
number = {2},
pages = {153-168},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On the double Lusin condition and convergence theorem for Kurzweil-Henstock type integrals},
url = {http://eudml.org/doc/276991},
volume = {141},
year = {2016},
}

TY - JOUR
AU - Racca, Abraham
AU - Cabral, Emmanuel
TI - On the double Lusin condition and convergence theorem for Kurzweil-Henstock type integrals
JO - Mathematica Bohemica
PY - 2016
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 141
IS - 2
SP - 153
EP - 168
AB - 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.
LA - eng
KW - Kurzweil-Henstock integral; $g$-integral; double Lusin condition; uniform double Lusin condition
UR - http://eudml.org/doc/276991
ER -

References

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  1. Aye, K. K., Lee, P. Y., The dual of the space of functions of bounded variation, Math. Bohem. 131 (2006), 1-9. (2006) Zbl1112.26008MR2210998
  2. Cabral, E., Lee, P.-Y., 10.14321/realanalexch.27.2.0627, Real Anal. Exch. 27 (2002), 627-634. (2002) Zbl1069.26013MR1922673DOI10.14321/realanalexch.27.2.0627
  3. Cabral, E., Lee, P.-Y., A fundamental theorem of calculus for the Kurzweil-Henstock integral in ℝ m , Real Anal. Exch. 26 (2001), 867-876. (2001) MR1844400
  4. Lee, P. Y., The integral à la Henstock, Sci. Math. Jpn. 67 (2008), 13-21. (2008) Zbl1162.26004MR2384584
  5. Lee, P. Y., Lanzhou Lectures on Henstock Integration, Series in Real Analysis 2 World Scientific, London (1989). (1989) Zbl0699.26004MR1050957

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