• Title, Summary, Keyword: Lebesgue convergence theorem

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A Note on Set-Valued Choquet Integrals

  • Hong, Dug-Hun;Kim, Kyung-Tae
    • Journal of the Korean Data and Information Science Society
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    • v.16 no.4
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    • pp.1041-1044
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    • 2005
  • Recently, Zhang et al.(Fuzzy Sets and Systems 147(2004) 475-485) proved Fatou's lemma and Lebesgue dominated convergence theorem under some conditions of fuzzy measure. In this note, we show that these conditions of fuzzy measure is essential to prove Fatou's lemma and Lebesgue dominated convergence theorem by examples

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On Lebesgue-type theorems for interval-valued Choquet integrals with respect to a monotone set function. (단조집합함수에 의해 정의된 구간치 쇼케이적분에 대한 르베그형태 정리에 관한 연구)

  • Jang, Lee-Chae;Kim, Tae-Kyun
    • Proceedings of the Korean Institute of Intelligent Systems Conference
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    • pp.195-198
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    • 2007
  • In this paper, we consider Lebesgue-type theorems in non-additive measure theory and then investigate interval-valued Choquet integrals and interval-valued fuzzy integral with respect to a additive monotone set function. Furthermore, we discuss the equivalence among the Lebesgue's theorems, the monotone convergence theorems of interval-valued fuzzy integrals with respect to a monotone set function and find some sufficient condition that the monotone convergence theorem of interval-valued Choquet integrals with respect to a monotone set function holds.

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On Lebesgue-type theorems for interval-valued Choquet integrals with respect to a monotone set function (단조집합함수에 의해 정의된 구간치 쇼케이적분에 대한 르베그형태 정리에 관한 연구)

  • Jang, Lee-Chae;Kim, Tae-Kyun
    • Journal of the Korean Institute of Intelligent Systems
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    • v.17 no.6
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    • pp.749-753
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    • 2007
  • In this paper, we consider Lebesgue-type theorems in non-additive measure theory and then investigate interval valued Choquet integrals and interval-valued fuzzy integral with respect to a additive monotone set function. Furthermore, we discuss the equivalence among the Lebesgue's theorems, the monotone convergence theorems of interval-valued fuzzy integrals with respect to a monotone set function and find some sufficient condition that the monotone convergence theorem of interval-valued Choquet integrals with respect to a monotone set function holds.

A PROOF OF STIRLING'S FORMULA

  • Park, June-Sang
    • Communications of the Korean Mathematical Society
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    • v.9 no.4
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    • pp.853-855
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    • 1994
  • The object of present note is to give a very short proof of Stirling's formula which uses only a formula for the generalized zeta function. There are several proofs for this formula. For example, Dr. E. J. Routh gave an elementary proof using Wallis' theorem in lectures at Cambridge ([5, pp.66-68]). We can find another proof which used the Maclaurin summation formula ([5, pp.116-120]). In [1], they used the Central Limit Theorem or the inversion theorem for characteristic functions. In [2], pp. Diaconis and D. Freeman provided another proof similarly as in [1]. J. M. Patin [7] used the Lebesgue dominated convergence theorem.

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T-FUZZY INTEGRALS OF SET-VALUED MAPPINGS

  • CHO, SUNG JIN
    • Journal of the Korean Society for Industrial and Applied Mathematics
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    • v.4 no.1
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    • pp.39-48
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    • 2000
  • In this paper we define T-fuzzy integrals of set-valued mappings, which are extensions of fuzzy integrals of the single-valued functions defined by Sugeno. And we discuss their properties.

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SOME GENERALIZATIONS OF SUGENOS FUZZY INTEGRAL TO SET-VALUED MAPPINGS

  • Cho, Sung-Jin;Lee, Byung-Soo;Lee, Gue-Myung;Kim, Do-Sang
    • Proceedings of the Korean Institute of Intelligent Systems Conference
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    • pp.380-386
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    • 1998
  • In this paper we introduce the concept of fuzzy integrals for set-valued mappings, which is an extension of fuzzy integrals for single-valued functions defined by Sugeno. And we give some properties including convergence theorems on fuzzy integrals for set-valued mappings.

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ON THE LEBESGUE SPACE OF VECTOR MEASURES

  • Choi, Chang-Sun;Lee, Keun-Young
    • Bulletin of the Korean Mathematical Society
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    • v.48 no.4
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    • pp.779-789
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    • 2011
  • In this paper we study the Banach space $L^1$(G) of real valued measurable functions which are integrable with respect to a vector measure G in the sense of D. R. Lewis. First, we investigate conditions for a scalarly integrable function f which guarantee $f{\in}L^1$(G). Next, we give a sufficient condition for a sequence to converge in $L^1$(G). Moreover, for two vector measures F and G with values in the same Banach space, when F can be written as the integral of a function $f{\in}L^1$(G), we show that certain properties of G are inherited to F; for instance, relative compactness or convexity of the range of vector measure. Finally, we give some examples of $L^1$(G) related to the approximation property.