MATHEMATICA BOHEMICA, Vol. 141, No. 2, pp. 153-168, 2016

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

Abraham Racca, Emmanuel Cabral

Abraham Racca, Adventist University of the Philippines, Santa Rosa-Tagaytay Road, Puting Kahoy, Silang, 4118 Cavite, Republic of the Philippines, e-mail: abraham.racca@yahoo.com; Emmanuel Cabral, Ateneo de Manila University, Katipunan Avenue, Loyola Heights, Quezon City, 1108 Metro Manila, Republic of the Philippines, e-mail: ecabral@ateneo.edu

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.

Keywords: Kurzweil-Henstock integral; $g$-integral; double Lusin condition; uniform double Lusin condition

Classification (MSC 2010): 26A39

DOI: 10.21136/MB.2016.13


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