关于向量值函数Riemann积分的若干研究
On Riemann integration of vector-valued functions
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摘要: 讨论向量值函数Riemann可积与连续性之间的关系,以及空间的Lebesgue性质(即取值于该空间的所有Riemann可积向量值函数必几乎处处连续). 通过反例进一步说明
l^p\left(1 < p < \infty \right) 及l^\infty 均不具有Lebesgue性质. 通过细化改进现有文献中的证明思路,得到l^1 具有Lebesgue性质的另一证明.Abstract: We study the relationship between the Riemann integration and the continuity almost everywhere of vector-valued functions and the property of Lebesgue (that is, every Riemann integrable vector-valued function is continuous almost everywhere). We prove thatl^p(1 < p < \infty ) andl^\infty do not possess the property of Lebesgue by constructing two counterexamples, Further, based on the existing literature, we show thatl^1 have the property of Lebesgue.