@MISC{MESFUN9C.MIZ,
  AUTHOR = {Narita, Keiko and Endou, Noboru and Shidama, Yasunari},
  TITLE = {Lebesgue's Convergence Theorem of Complex-Valued Function},
  SECTION1 = {Partial Sums of Real-Valued Functional Sequences},
  SECTION2 = {Partial Sums of Complex-Valued Functional Sequences},
  SECTION3 = {Selected Properties of Complex-Valued Simple Functions},
  SECTION4 = {Lebesgue's Convergence Theorem of Complex-Valued Function},
  DAY = {17},
  MONTH = {March},
  YEAR = {2009},
  ADDRESS1 = {Hirosaki-city\\ Aomori, Japan},
  ADDRESS2 = {Gifu National College of Technology\\ Japan},
  ADDRESS3 = {Shinshu University\\ Nagano, Japan},
  SUMMARY = {In this article, we formalized Lebesgue's Convergence 
  theorem of complex-valued function. We proved Lebesgue's Convergence 
  Theorem of real-valued function using the theorem of extensional 
  real-valued function. Then applying the former theorem 
  to real part and imaginary part of complex-valued functional 
  sequences, we proved Lebesgue's Convergence Theorem of 
  complex-valued function. We also defined partial sums 
  of real-valued functional sequences and complex-valued 
  functional sequences and showed their properties. 
  In addition, we proved properties of complex-valued simple functions.}}

