@MISC{PROB_3.MIZ,
  AUTHOR = {Zhang, Bo and Yamazaki, Hiroshi and Nakamura, Yatsuka},
  TITLE = {Set Sequences and Monotone Class},
  DAY = {12},
  MONTH = {August},
  YEAR = {2005},
  ADDRESS1 = {Shinshu University\\Nagano, Japan},
  ADDRESS2 = {Shinshu University\\Nagano, Japan},
  ADDRESS3 = {Shinshu University\\Nagano, Japan},
  SUMMARY = {In this paper, we first defined the partial-union
  sequence, the partial-intersection sequence, and the
  partial-difference-union sequence of given sequence of subsets,
  and then proved the additive theorem of infinite sequences and
  sub-additive theorem of finite sequences for probability.
  Further, we defined the monotone class of families of subsets,
  and discussed about the relations between the monotone class and
  the $\sigma$-field which are generated by field of subsets of a
  given set.}}