한국수학교육학회지시리즈B:순수및응용수학 (The Pure and Applied Mathematics)

  • 제31권4호
  • /
  • Pages.453-476
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  • 2024
  • /
  • 1226-0657(pISSN)
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  • 2287-6081(eISSN)
한국수학교육학회 (Korean Society of Mathematical Education)
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RIEMANN-STIELTJES INTEGRALS AND THEIR REPRESENTING MEASURES

  • Joong Kwoen Lee (Department of Mathematics Education, Dongguk University) ;
  • Han Ju Lee (Department of Mathematics Education, Dongguk University)
  • 투고 : 2024.08.07
  • 심사 : 2024.10.05
  • 발행 : 2024.11.30
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초록

The Riemann-Stieltjes integrals of continuous functions with respect to a function of bounded variation can be represented by a regular, Borel, complex measure. In this paper, we study the link between the Riemann-Stieltjes integral and measure theory using this representation. Specifically, we investigate the Riemann-Stieltjes integrability and its measurability. Furthermore, we derive a criterion for Riemann-Stieltjes integrability through a method different from known proofs. In particular, we calculate the upper and lower Riemann-Stieltjes integrals with respect to a monotone increasing function.

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과제정보

The second named author was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology [NRF-2020R1A2C1A01010377].

참고문헌

  1. T. M. Apostol: Mathematical Analysis. Second Edition. Pearson, 1974.
  2. G. B. Folland: Real analysis : modern techniques and their applications. 2nd ed. A Wiley-Interscience publication, 1999.
  3. H. J. Ter Horst: Riemann-Stieltjes and Lebesgue-Stieltjes Integrability. Amer. Math. Monthly 91 (1984), 551-559. https://doi.org/10.2307/2323739
  4. M. Safonov: Appendix B. Criterion of Riemann-Stieltjes Integrability, Lecture note. https://www-users.cse.umn.edu/~safon002/Archive/Math5616H/NOTES/B.pdf
  5. T. J. Stieltjes: Recherches sur les fractions continues. Ann. Fac. Sci. Toulouse VIII (1894), 1-122. https://afst.centre-mersenne.org/item/AFST_1995_6_4_1_J1_0/