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Keywords:
Kurzweil-Henstock integral; $g$-integral; double Lusin condition; uniform double Lusin condition
Kurzweil-Henstock integral; $g$-integral; double Lusin condition; uniform double Lusin condition
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.
References:
[1] Aye, K. K., Lee, P. Y.: The dual of the space of functions of bounded variation. Math. Bohem. 131 (2006), 1-9. MR 2210998 | Zbl 1112.26008
[2] Cabral, E., Lee, P.-Y.: The primitive of a Kurzweil-Henstock integrable function in multidimensional space. Real Anal. Exch. 27 (2002), 627-634. DOI 10.14321/realanalexch.27.2.0627 | MR 1922673 | Zbl 1069.26013
[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
[4] Lee, P. Y.: The integral à la Henstock. Sci. Math. Jpn. 67 (2008), 13-21. MR 2384584 | Zbl 1162.26004
[5] Lee, P. Y.: Lanzhou Lectures on Henstock Integration. Series in Real Analysis 2 World Scientific, London (1989). MR 1050957 | Zbl 0699.26004
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