@MISC{MEASUR13.MIZ,
  AUTHOR = {Endou, Noboru and Shidama, Yasunari},
  TITLE = {Multidimensional Measure Space and Integration},
  DAY = {21},
  MONTH = {November},
  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 = {Construction of $m$-dimensional Measure Space},
  SECTION3 = {Integrability of Functions on $(n+1)$-dimensional Space},
  SUMMARY = {In this paper, we introduce multidimensional measure spaces and the integration    \cite{Rao2004} and \cite{bogachev2007measure} of functions on these spaces.
  In constructing measures on multidimensional spaces, we constructed a finite sequence of   Cartesian product spaces of sets in Section 1.
  In Section 2, using the Fubini's Theorem \cite{MEASUR11.abs}, we have constructed measures on   general multidimensional spaces by introducing measures one by one into the finite sequence of   direct product spaces obtained in Section 1.
  In Section 3, integrals on the m-dimensional Cartesian product measure space obtained in   Section 2 are presented, and the concept of sequentially integrable, which is useful in   considering integrability \cite{MESFUN13.abs} for functions on multidimensional spaces, is   introduced and its effectiveness is shown.},
  ACKNOWLEDGEMENT = {This work was supported by JSPS KAKENHI 23K11242.}}
  
@BOOK{Apostol:1967,
 AUTHOR = {Apostol, Tom M.},
 TITLE = {Calculus},
 PUBLISHER = {John Wiley \& Sons},
 YEAR = {1967},
 VOLUME = {I},
 EDITION = {Second}}

@BOOK{Courant:1988,
 AUTHOR = {Courant, R and McShane, E.J.},
 TITLE = {Differential and Integral Calculus},
 PUBLISHER = {John Wiley \& Sons},
 YEAR = {1988}}
 