@MISC{LPSPACE1.MIZ,
  AUTHOR = {Watase, Yasushige and Endou, Noboru and Shidama, Yasunari},
  TITLE = {On $L^1$ Space Formed by Real-valued Partial Functions},
  SECTION1 = {Preliminaries of Real Linear Space},
  SECTION2 = {Quasi-Real Linear Space of Partial Functions},
  SECTION3 = {Quasi-Real Linear Space of Integrable Functions},
  SECTION4 = {Quotient Space of Quasi-Real Linear Space
    of Integrable Functions},
  SECTION5 = {Real Normed Space of Integrable Functions},
  DAY = {26},
  MONTH = {August},
  YEAR = {2008},
  ADDRESS1 = {Graduate School of Science and Technology\\Shinshu University\\4-17-1 Wakasato, Nagano, Japan},
  ADDRESS2 = {Gifu National College of Technology\\Gifu, Japan},
  ADDRESS3 = {Faculty of Engineering\\Shinshu University\\4-17-1 Wakasato, Nagano, Japan},
  SUMMARY = {This article contains some definitions and properties
  about a function spaces formed by partial functions defined over
  a measurable space. We formalized a function space, the so-called $L^1$
  space and proved that the space turns to be a normed space.
  A formalization of a real function space was given in \cite{FUNCSDOM.ABS}.
  The set of all function forms additive group, here addition is defined
  by point-wise addition of two functions. However it is not true for
  that of partial functions. The set of partial functions does not form
  an additive group due to lack of right zeroed condition. Therefore firstly
  we introduced a kind of a quasi-linear space, secondary introduced
  the definition of an equivalent relation of two functions are almost
  everywhere equal (=a.e.), thirdly formalized a linear space by taking
  the quotient of a quasi-linear space by the relation (=a.e.).}}

@BOOK{Halmos:1974,
 AUTHOR={Paul R. Halmos},
 TITLE={Measure Theory},
 PUBLISHER={Springer-Verlag},
 YEAR=1974}
