Henstock积分

Henstock积分

摘要

本文参照Riemann积分的基本性质,证明了Henstock积分的1些基本性质;在讨论微积分基本定理在Henstock可积函数中完全成立的基础上,文章给出了1个推广的结论的证明; 另外，在小测度集上用小Riemann和刻划了Henstock积分的实质,揭示了Henstock积分与Lebesgue积分的关系,文章给出了Henstock积分性质的另1种证明方法,还给出了不是Henstock可积的1个例子; 最后,证明了Henstock积分中的 满足1定条件时就恢复为Riemann积分.
关键词：Riemann积分；Lebesgue积分；Henstock积分； 精细分法.

Henstock integral

Abstract

In this paper, we proved the basic properties of Henstock integral in the light of the basic properties of Riemann integral. Basic Theorem (i.e., N-L formula) doesn’t hold for all Riemann integral functions, but it holds for all Henstock integral functions, and we gave a proof of the extended result.  By using small Riemannian sums on the small set, Henstock integrals are described, and it’s properties is given in another way, in addition, we gave an example that is not Henstock integral. At last, we proved that Henstock integral is just Riemann integral when  satisfies some conditions.
Keywords: Riemannian integral; Henstock integral; Lebesgue integral; -fine division.
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