@MISC{MESFUN16.MIZ,
  AUTHOR = {Endou, Noboru and Shidama, Yasunari},
  TITLE = {Integral of Continuous Functions of Two Variables},
  DAY = {18},
  MONTH = {December},
  YEAR = {2023},
  ADDRESS1 = {National Institute of Technology, Gifu College\\2236-2 Kamimakuwa, Motosu, Gifu, Japan},
  ADDRESS2 = {Karuizawa Hotch 244-1 Nagano, Japan},
  SECTION1 = {Preliminaries},
  SECTION2 = {Continuity of Two-variable Functions},
  SECTION3 = {Properties of Projective Functions},
  SECTION4 = {Integral of Continuous Functions of Two Variables},
  SUMMARY = {So far, the authors have proved many theorems on the integral 
  theory of one-variable functions for Riemann and Lebesgue integrals.
  As a result, we have shown that if a function bounded on a closed interval 
  (i.e., a continuous function) is Riemann integrable, then it is Lebesgue integrable, 
  and both integrals coincide \cite{MESFUN14.abs}.
  Furthermore, for the Lebesgue integral, there exist integral theorems on the product 
  measure spaces \cite{MESFUN13.abs}.
  From these results, this article shows that the Lebesgue integral of a continuous 
  function of two variables coincides with the Riemann iterated integral of a projective
  function \cite{Apostol:1969II}.
  In the first three sections of this article, we summarize the basic properties about
  the projection of functions of two variables.
  In the last section, we prove integrability and iterated integrals of continuous 
  functions of two variables.\par
  Note that the continuity of functions of multivariables not directly addressed in this 
  article, since there are many definitions (although they are essentially the same).}
  ACKNOWLEDGEMENT = {This work was supported by JSPS KAKENHI 23K11242.}}
  
@BOOK{Apostol:1967II,
 AUTHOR = {Apostol, Tom M.},
 TITLE = {Calculus},
 PUBLISHER = {Wiley},
 YEAR = {1969},
 VOLUME = {II},
 EDITION = {Second}}
  