@MISC{INTEGR22.MIZ,
  AUTHOR = {Narita, Keiko and Nakasho, Kazuhisa and Shidama, Yasunari},
  TITLE = {Riemann-Stieltjes Integral},
  ACKNOWLEDGEMENT = {This work was supported by JSPS KAKENHI 22300285.},
  SECTION1 = {Properties of Real Finite Sequences},
  SECTION2 = {The Definitions of Bounded Variation},
  SECTION3 = {The Definitions of Riemann-Stieltjes Integral},
  SECTION4 = {Linearity of Riemann-Stieltjes Integral},
  DAY = {30},
  MONTH = {June},
  YEAR = {2016},
  ADDRESS1 = {Hirosaki-city, Aomori, Japan},
  ADDRESS2 = {Akita Prefectural University\\Akita},
  ADDRESS3 = {Shinshu University\\Nagano},
  MSC2010 = {26A42 26A45},
  KEYWORDS = {Riemann-Stieltjes integral; bounded variation; linearity},
  SUMMARY = {In this article, the definitions and basic properties
  of Riemann-Stieltjes integral are formalized.
  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 \cite{INTEGR18.ABS}, 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 \cite{INTEGR18.ABS}.
  These formalizations are based on \cite{Stroock1999},
  \cite{Kestelman1960}, \cite{Gupta1986}, and \cite{Hille1974}.}}

@BOOK{Stroock1999,
  TITLE = {A Concise Introduction to the Theory of Integration}
  AUTHOR = {Daniel W. Stroock},
  PUBLISHER = {Springer Science \& Business Media}
  YEAR  = {1999},
}

@BOOK{Kestelman1960,
  TITLE={Modern theories of integration},
  AUTHOR={H. Kestelman},
  PUBLISHER={Dover Publications},
  YEAR={1960},
  EDITION={2nd}
}

@BOOK{Gupta1986,
  TITLE = {Fundamental Real Analysis},
  AUTHOR = {S.L. Gupta and Nisha Rani},
  PUBLISHER = {Vikas Pub},
  YEAR = {1986}
}

@BOOK{Hille1974,
  TITLE={Methods in classical and functional analysis},
  AUTHOR={Hille, Einar},
  PUBLISHER={Addison-Wesley Publishing Co., Halsted Press}
  YEAR={1974},
}
