Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Formalized Mathematics

(a computer assisted approach)

Editor-in-Chief: Matuszewski, Roman

4 Issues per year

SCImago Journal Rank (SJR) 2016: 0.207
Source Normalized Impact per Paper (SNIP) 2016: 0.315

Open Access
See all formats and pricing
More options …
Volume 24, Issue 3


Riemann-Stieltjes Integral

Keiko Narita / Kazuhisa Nakasho / Yasunari Shidama
Published Online: 2017-02-21 | DOI: https://doi.org/10.1515/forma-2016-0016


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


  • [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:CrossrefGoogle Scholar

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

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

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

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

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

  • [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:CrossrefGoogle Scholar

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

About the article

Received: 2016-06-30

Published Online: 2017-02-21

Published in Print: 2016-09-01

Citation Information: Formalized Mathematics, Volume 24, Issue 3, Pages 199–204, 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 BY-SA 3.0 LEGALCODE

Citing Articles

Here you can find all Crossref-listed publications in which this article is cited. If you would like to receive automatic email messages as soon as this article is cited in other publications, simply activate the “Citation Alert” on the top of this page.

Roland Coghetto
Formalized Mathematics, 2017, Volume 25, Number 3

Comments (0)

Please log in or register to comment.
Log in