@MISC{MESFUNC6.MIZ,
  AUTHOR = {Shidama, Yasunari and Endou, Noboru},
  TITLE = {Integral of Real-Valued Measurable Function},
  NOTE = {This work has been partially supported by the MEXT grant Grant-in-Aid for Young Scientists (B)16700156.},
  SECTION1 = {The Measurability of Real-Valued Functions},
  SECTION2 = {The Measurability of $f+g$ and $f-g$ for Real-Valued Functions $f,g$},
  SECTION3 = {Basic Properties of Real-Valued Functions, $\max_{+}f$ and $\max_{-}f$},
  SECTION4 = {The Measurability of $\max_{+}f, \max_{-}f$ and $|f|$},
  SECTION5 = {The Definition and the Measurability of a Real-Valued Simple Function},
  SECTION6 = {Lemmas for a Real-Valued Measurable Function and a Simple Function},
  SECTION7 = {The Integral of a Real-Valued Function},
  DAY = {27},
  MONTH = {October},
  YEAR = {2006},
  ADDRESS1 = {Shinshu University\\Nagano, Japan},
  ADDRESS2 = {Gifu National College of Technology\\Gifu, Japan},
  SUMMARY = {Based on \cite{Halmos74}, authors formalized the integral
   of an extended real valued measurable function in \cite{MESFUNC5.ABS}
   before. However, the integral argued in \cite{MESFUNC5.ABS}
   cannot be applied to real-valued functions unconditionally.
   Therefore, in this article we have formalized the integral of
   a real-value function.}}

