@MISC{MEASURE9.MIZ,
  AUTHOR = {Endou, Noboru},
  TITLE = {Construction of Measure from Semialgebra of Sets},
  DAY = {14},
  MONTH = {August},
  YEAR ={2015},
  ADDRESS1 = {Gifu National College of Technology\\2236-2 Kamimakuwa, Motosu, Gifu, Japan},
  SECTION1 = {Join of many finite sequence},
  SECTION2 = {Extended real valued matrix},
  SECTION3 = {Definition of pre-measure},
  SECTION4 = {Pre-measure on semialgera and construction of measure},
  SUMMARY = {In our previous article \cite{MEASURE8.abs}, we showed complete additivity
  as a condition for extension of a measure. However, this condition is premised the existence
  of a sigma field and the measurer on it. In general, the existence of the measure on sigma
  field is not trivial. On the other hand, the existence of a measure on semi-algebra is
  easier than sigma field. Therefore, in this article we define a measure (pre-measure) on
  semi algebra and extend it to a measure on sigma field. Furthermore, we give a sigma
  measure as extension of the measure on sigma field. We follow \cite{Halmos74},
  \cite{Bogachev}, and \cite{Rao}.},
  NOTE = {This work was supported by JSPS KAKENHI 23500029.}}

@BOOK{Halmos74,
         AUTHOR = {Halmos, P.~R.},
          TITLE = {Measure Theory},
      PUBLISHER = {Springer-Verlag},
           YEAR = {1974}}

@BOOK{Bogachev,
  AUTHOR={Bogachev, V.I.},
  TITLE={Measure Theory},
  PUBLISHER={Springer},
  YEAR={2006}
  VOLUME={1}}

@BOOK{Rao,
  AUTHOR={Rao, M.M.},
  TITLE={Measure Theory and Integration},
  PUBLISHER={Marcel Dekker},
  EDITION={2nd}}

