• Title, Summary, Keyword: Lebesgue integral

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Development of the Integral Concept (from Riemann to Lebesgue) (적분개념의 발달 (리만적분에서 르베그적분으로의 이행을 중심으로))

  • Kim, Kyung-Hwa
    • Journal for History of Mathematics
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    • v.21 no.3
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    • pp.67-96
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    • 2008
  • In the 19th century Fourier and Dirichlet studied the expansion of "arbitrary" functions into the trigonometric series and this led to the development of the Riemann's definition of the integral. Riemann's integral was considered to be of the highest generality and was discussed intensively. As a result, some weak points were found but, at least at the beginning, these were not considered as the criticism of the Riemann's integral. But after Jordan introduced the theory of content and measure-theoretic approach to the concept of the integral, and after Borel developed the Jordan's theory of content to a theory of measure, Lebesgue joined these two concepts together and obtained a new theory of integral, now known as the "Lebesgue integral".

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THE LEBESGUE DELTA INTEGRAL

  • Park, Jae Myung;Lee, Deok Ho;Yoon, Ju Han;Lim, Jong Tae
    • Journal of the Chungcheong Mathematical Society
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    • v.27 no.3
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    • pp.489-494
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    • 2014
  • In this paper, we define the extension $f^*:[a,b]{\rightarrow}\mathbb{R}$ of a function $f:[a,b]_{\mathbb{T}}{\rightarrow}\mathbb{R}$ for a time scale $\mathbb{T}$ and investigate the properties of the Lebesgue delta integral of f on $[a,b]_{\mathbb{T}}$ by using the function $f^*$.

A note on distance measure and similarity measure defined by Choquet integral on interval-valued fuzzy sets (구간치 퍼지집합 상에서 쇼케이적분에 의해 정의된 거리측도와 유사측도에 관한 연구)

  • Jang, Lee-Chae
    • Journal of Korean Institute of Intelligent Systems
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    • v.17 no.4
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    • pp.455-459
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    • 2007
  • Interval-valued fuzzy sets were suggested for the first time by Gorzafczany(1983) and Turksen(1986). Based on this, Zeng and Li(2006) introduced concepts of similarity measure and entropy on interval-valued fuzzy sets which are different from Bustince and Burillo(1996). In this paper, by using Choquet integral with respect to a fuzzy measure, we introduce distance measure and similarity measure defined by Choquet integral on interval-valued fuzzy sets and discuss some properties of them. Choquet integral is a generalization concept of Lebesgue inetgral, because the two definitions of Choquet integral and Lebesgue integral are equal if a fuzzy measure is a classical measure.

The denjoy extension of the mcshane integral

  • Park, Jae-Myung;Lee, Deok-Ho
    • Bulletin of the Korean Mathematical Society
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    • v.33 no.3
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    • pp.411-417
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    • 1996
  • Some generalizations of the Riemann integral have been studied for real-valued functions. One of these generalizations leads to an integral, often called the McShane integral, that is equivalent to the Lebesgue integral.

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A New Approach to the Lebesgue-Radon-Nikodym Theorem. with respect to Weighted p-adic Invariant Integral on ℤp

  • Rim, Seog-Hoon;Jeong, Joo-Hee
    • Kyungpook Mathematical Journal
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    • v.52 no.3
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    • pp.299-306
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    • 2012
  • We will give a new proof of the Lebesgue-Radon-Nikodym theorem with respect to weighted p-adic q-measure on $Z_p$, using Mahler expansion of continuous functions, studied by the authors in 2012. In the special case, q = 1, we can derive the same result as in Kim, 2012, Kim et al, 2011.

DENJOY-TYPE INTEGRALS OF BANACH-VALUED FUNCTIONS

  • Cho, Sung-Jin;Lee, Byung-Soo;Lee, Gue-Myung
    • Communications of the Korean Mathematical Society
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    • v.13 no.2
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    • pp.307-316
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    • 1998
  • In this paper Denjoy*-Dunford, Denjoy*-Pettis, Denjoy*-McShane and Denjoy*-Bochner integrals of functions which map an interval [a,b] into a Banach space X are defined. And we give the relations among the integrals.

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Some characterizations of a mapping defined by interval-valued Choquet integrals

  • Jang, Lee-Chae;Kim, Hyun-Mee
    • International Journal of Fuzzy Logic and Intelligent Systems
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    • v.7 no.1
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    • pp.66-70
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    • 2007
  • Note that Choquet integral is a generalized concept of Lebesgue integral, because two definitions of Choquet integral and Lebesgue integral are equal if a fuzzy measure is a classical measure. In this paper, we consider interval-valued Choquet integrals with respect to fuzzy measures(see [4,5,6,7]). Using these Choquet integrals, we define a mappings on the classes of Choquet integrable functions and give an example of a mapping defined by interval-valued Choquet integrals. And we will investigate some relations between m-convex mappings ${\phi}$ on the class of Choquet integrable functions and m-convex mappings $T_{\phi}$, defined by the class of closed set-valued Choquet integrals with respect to fuzzy measures.

WEIGHTED LEBESGUE NORM INEQUALITIES FOR CERTAIN CLASSES OF OPERATORS

  • Song, Hi Ja
    • Korean Journal of Mathematics
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    • v.14 no.2
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    • pp.137-160
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    • 2006
  • We describe the weight functions for which Hardy's inequality of nonincreasing functions is satisfied. Further we characterize the pairs of weight functions $(w,v)$ for which the Laplace transform $\mathcal{L}f(x)={\int}^{\infty}_0e^{-xy}f(y)dy$, with monotone function $f$, is bounded from the weighted Lebesgue space $L^p(w)$ to $L^q(v)$.

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