共查询到16条相似文献,搜索用时 62 毫秒
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用可数分法代替有限分法,可去掉函数有界和定义域测度有限的限制,一次性地作出函数Lebesgue可积的定义并得出可积的充要条件. 相似文献
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约定1〈p〈∞,定义空间Cp[a,b],证明Cp[a,b]是Lp[a,b]的子空间.利用Lebesgue积分和Riemann积分在Lp[a,b]和Cp[a,b]上分别定义线性泛函L和R,证明二者有界且有相等范数.利用Taylor定理和已得结论证明L是R从Cp[a,b]到Lp[a,b]的唯一保范延拓. 相似文献
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总结Riemann积分的发展历史、基本思想及其各种推广,介绍Riemann积分的局限性和缺陷,以及由此推动产生Lebesgue积分的过程. 相似文献
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瑕积分的收敛性与Lebesgue可积性之间具有密不可分的关系.首先详细证明了瑕积分收敛定理,然后说明无穷积分的收敛定理与瑕积分的收敛定理是等价的,最后利用两条定理的等价性给出一个应用. 相似文献
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Lebesgue积分的产生及其影响(迎接ICM2002特约文章) 总被引:3,自引:0,他引:3
这是一篇由法国科学院院士 Kahane为纪念Lebesgue积分 100周年而写的文章,原文发表在《巴黎科学院通报》上,文章讲述了Lebesgue积分创立的历史背景及其影响,经作者同意,由武汉大学范爱华教授翻译成中文在本刊发表. 相似文献
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证明了非负有界函数的Lebesgue上积分等于函数下方图形的Lebesgue外测度,其Lebesgue下积分等于函数下方图形的Lebesgue内测度,从而将积分的几何意义从可测情形推广到一般情形. 相似文献
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D. Azagra J.B. Seoane-Sepúlveda 《Journal of Mathematical Analysis and Applications》2009,354(1):229-233
If f is continuous on the interval [a,b], g is Riemann integrable (resp. Lebesgue measurable) on the interval [α,β] and g([α,β])⊂[a,b], then f○g is Riemann integrable (resp. measurable) on [α,β]. A well-known fact, on the other hand, states that f○g might not be Riemann integrable (resp. measurable) when f is Riemann integrable (resp. measurable) and g is continuous. If c stands for the continuum, in this paper we construct a c2-dimensional space V and a c-dimensional space W of, respectively, Riemann integrable functions and continuous functions such that, for every f∈V?{0} and g∈W?{0}, f○g is not Riemann integrable, showing that nice properties (such as continuity or Riemann integrability) can be lost, in a linear fashion, via the composite function. Similarly we construct a c-dimensional space W of continuous functions such that for every g∈W?{0} there exists a c-dimensional space V of measurable functions such that f○g is not measurable for all f∈V?{0}. 相似文献
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U. B. Darji and M. J. Evans [1] showed previously that it is possible to obtain the integral of a Lebesgue integrable function
on the interval [0,1] via a Riemann type process, where one chooses the selected point in each partition interval using a
first-return algorithm based on a sequence {x
n} which is dense in [0,1]. Here we show that if the same is true for every rearrangement of {x
n}, then the function must be equal almost everywhere to a Riemann integrable function.
This revised version was published online in June 2006 with corrections to the Cover Date. 相似文献
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In this paper, we calculate the exact asymptotics with remainder of integrals of Laplace type in an arbitrary space, without imposing any constraints on the smoothness of the functions. As a special case, we derive the classical Laplace formula. Some examples are given. 相似文献
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Jack Grahl 《Journal of Mathematical Analysis and Applications》2008,340(1):358-365
We construct an integral of a measurable real function using randomly chosen Riemann sums and show that it converges in probability to the Lebesgue integral where this exists. We then prove some conditions for the almost sure convergence of this integral. 相似文献
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A complement to A. M. Olevskii's fundamental inequality on the logarithmic growth of Lebesgue functions of an arbitrary uniformly bounded orthonormal system on a set of positive measure is made. Namely, the index where the Lebesgue functions have growth slightly weaker than logarithmic can be chosen independently of the variable. The theorem proved in this paper improves a result established earlier by the author. 相似文献