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Formalized Mathematics

(a computer assisted approach)

Editor-in-Chief: Matuszewski, Roman

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In This Section
Volume 24, Issue 3 (Sep 2016)

Issues

Riemann-Stieltjes Integral

Keiko Narita
  • Corresponding author
  • Hirosaki-city Aomori, Japan
/ Kazuhisa Nakasho
  • Akita Prefectural University Akita, Japan
/ Yasunari Shidama
  • Shinshu University Nagano, Japan
Published Online: 2017-02-21 | DOI: https://doi.org/10.1515/forma-2016-0016

Abstract

In this article, the definitions and basic properties of Riemann-Stieltjes integral are formalized in Mizar [1]. In the first section, we showed the preliminary definition. We proved also some properties of finite sequences of real numbers. In Sec. 2, we defined variation. Using the definition, we also defined bounded variation and total variation, and proved theorems about related properties.

In Sec. 3, we defined Riemann-Stieltjes integral. Referring to the way of the article [7], we described the definitions. In the last section, we proved theorems about linearity of Riemann-Stieltjes integral. Because there are two types of linearity in Riemann-Stieltjes integral, we proved linearity in two ways. We showed the proof of theorems based on the description of the article [7]. These formalizations are based on [8], [5], [3], and [4].

MSC: 26A42; 26A45; 03B35

Keywords: Riemann-Stieltjes integral; bounded variation; linearity

MML: identifier: INTEGR22; version: 8.1.05 5.37.1275

References

  • [1] Grzegorz Bancerek, Czesław Bylinski, Adam Grabowski, Artur Korniłowicz, Roman Matuszewski, Adam Naumowicz, Karol Pak, and Josef Urban. Mizar: State-of-the-art and beyond. In Manfred Kerber, Jacques Carette, Cezary Kaliszyk, Florian Rabe, and Volker Sorge, editors, Intelligent Computer Mathematics, volume 9150 of Lecture Notes in Computer Science, pages 261-279. Springer International Publishing, 2015. ISBN 978-3-319-20614-1. doi: [Crossref] [Crossref]

  • [2] Czesław Bylinski. The complex numbers. Formalized Mathematics, 1(3):507-513, 1990.

  • [3] S.L. Gupta and Nisha Rani. Fundamental Real Analysis. Vikas Pub., 1986.

  • [4] Einar Hille. Methods in classical and functional analysis. Addison-Wesley Publishing Co., Halsted Press, 1974.

  • [5] H. Kestelman. Modern theories of integration. Dover Publications, 2nd edition, 1960.

  • [6] Jarosław Kotowicz. Convergent real sequences. Upper and lower bound of sets of real numbers. Formalized Mathematics, 1(3):477-481, 1990.

  • [7] Keiichi Miyajima, Takahiro Kato, and Yasunari Shidama. Riemann integral of functions from ℝ into real normed space. Formalized Mathematics, 19(1):17-22, 2011. doi: [Crossref] [Crossref]

  • [8] Daniel W. Stroock. A Concise Introduction to the Theory of Integration. Springer Science & Business Media, 1999.

About the article

Received: 2016-06-30

Published Online: 2017-02-21

Published in Print: 2016-09-01



Citation Information: Formalized Mathematics, ISSN (Online) 1898-9934, DOI: https://doi.org/10.1515/forma-2016-0016. Export Citation

© by Keiko Narita. This work is licensed under version 3.0 of the Creative Commons Attribution–ShareAlike License (CC BY-SA 3.0 LEGALCODE)

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