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.
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机译:摘要:紧凑区间$ E $中的等积性可以定义为一个统一的可积性属性,它涉及被积数$ f_n $和相应的原始$ F_n $。积分元$ f_n $到点$ f $的逐点收敛性以及函数$ f_n $的等积分性意味着$ f $也可以与原始$ F $积分,并且原始元$ F_n $统一收敛到$ F $ 。在本文中,再次探讨了E. Cabral和P. Y. Lee(2001/2002)论文中引入的另一种称为均匀双重Lusin条件的均匀可积性。在被积元$ f_n $逐点收敛的假设下,三个一致的可积性,即等积性和一致双Lusin条件的两个版本都相等。双重Lusin条件的第一个版本及其对应的一致双重Lusin收敛定理也扩展到了划分空间中。
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