实值函数Riemann—Stieltjes可积的另一个充要条件及某些有关问题

努尔买买提

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实值函数Riemann—Stieltjes可积的另一个充要条件及某些有关问题

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- 实值函数Riemann—Stieltjes可积的另一个充要条件及某些有关问题
- Journal of Xinjiang University (Natural Science Edition in Chinese and English) Issue 2, (1987)
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- Published：1987
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[1]努尔买买提.实值函数 Riemann—Stieltjes 可积的另一个充要条件及某些有关问题[J].新疆大学学报(自然科学版),1987(02):107-110.

努尔买买提. 实值函数Riemann—Stieltjes可积的另一个充要条件及某些有关问题[J]. Journal of Xinjiang University (Natural Science Edition in Chinese and English), 1987, (2).
[1]努尔买买提.实值函数 Riemann—Stieltjes 可积的另一个充要条件及某些有关问题[J].新疆大学学报(自然科学版),1987(02):107-110. DOI：

努尔买买提. 实值函数Riemann—Stieltjes可积的另一个充要条件及某些有关问题[J]. Journal of Xinjiang University (Natural Science Edition in Chinese and English), 1987, (2). DOI：

摘要

设[a

b]是有界闭区间

f是[a

b]上的有界实值函数

a是[a

b]上实值单调增函数。若f在[a

b]上关于a Riemann—Stieltjes可积(即积分■f(x)da(x)存在)

则简记为f∈R(a)。我们已知

在[a

b]上f∈R(a)的充要条件是

对任意ε>O

总存在划分p={a=x01n=b}

使U(p

a)-L(p

a) 关键词： Abstract： This paper proves following proposition:Suppose that[a

b]is a closed bounded interval

f and a real-valued functionon a

f being bounded and a being monotonically increasing on[a

b].Thenf∈R(a) on[a

b](that is

integral from n=a to b(f(x)da(a))exists)if and only if for any σ>0and ω>0

there is a partition p of[a

b] such that sum of the oscillations of a insubintervals in which the oscillations of f is greater than or equal to ω is lessthanσ. KeyWords：

Abstract

This paper proves following proposition:Suppose that[a

b]is a closed bounded interval

f and a real-valued functionon a

f being bounded and a being monotonically increasing on[a

b].Thenf∈R(a) on[a

b](that is

integral from n=a to b(f(x)da(a))exists)if and only if for any σ>0and ω>0

there is a partition p of[a

b] such that sum of the oscillations of a insubintervals in which the oscillations of f is greater than or equal to ω is lessthanσ.

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