@MISC{MESFUN14.MIZ,
  AUTHOR = {Endou, Noboru},
  TITLE = {Relationship between the {R}iemann and {L}ebesgue Integrals},
  DAY = {30},
  MONTH = {September},
  YEAR = {2021},
  ADDRESS1 = {National Institute of Technology, Gifu College\\2236-2 Kamimakuwa, Motosu, Gifu, Japan},
  SECTION1 = {Preliminaries},
  SECTION2 = {Properties of Complete Measure Space},
  SECTION3 = {Relation Between Riemann and Lebesgue Integrals},
  SUMMARY = {The goal of this article is to clarify the relationship between Riemann
  and Lebesgue integrals. In previous article \cite{MEASUR12.abs}, we constructed a one-dimensional Lebesgue measure.
  The one-dimensional Lebesgue measure provides a measure of any intervals,
  which can be used to prove the well-known relationship \cite{FOLLAND} between the Riemann and
  Lebesgue integrals. We also proved the relationship between the integral of a given measure and that of its complete
  measure.
  As the result of this work, the Lebesgue integral of a
  bounded real valued function in the Mizar system can be
  calculated by the Riemann integral.}}
