Article MEASURE2, MML version 4.99.1005

:: MEASURE2:th 1
theorem
for b1 being set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being sigma_Measure of b2
for b4 being Function-like quasi_total Relation of NAT,b2 holds
   b3 * b4 is nonnegative;

:: MEASURE2:modenot 1 => MEASURE2:mode 1
definition
  let a1 be set;
  let a2 be non empty compl-closed sigma-multiplicative Element of bool bool a1;
  mode N_Measure_fam of A2 -> non empty countable Element of bool bool a1 means
    it c= a2;
end;

:: MEASURE2:dfs 1
definiens
  let a1 be set;
  let a2 be non empty compl-closed sigma-multiplicative Element of bool bool a1;
  let a3 be non empty countable Element of bool bool a1;
To prove
     a3 is N_Measure_fam of a2
it is sufficient to prove
  thus a3 c= a2;

:: MEASURE2:def 1
theorem
for b1 being set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being non empty countable Element of bool bool b1 holds
      b3 is N_Measure_fam of b2
   iff
      b3 c= b2;

:: MEASURE2:th 3
theorem
for b1 being set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being N_Measure_fam of b2 holds
   meet b3 in b2 & union b3 in b2;

:: MEASURE2:funcnot 1 => MEASURE2:func 1
definition
  let a1 be set;
  let a2 be non empty compl-closed sigma-multiplicative Element of bool bool a1;
  let a3 be N_Measure_fam of a2;
  redefine func meet a3 -> Element of a2;
end;

:: MEASURE2:funcnot 2 => MEASURE2:func 2
definition
  let a1 be set;
  let a2 be non empty compl-closed sigma-multiplicative Element of bool bool a1;
  let a3 be N_Measure_fam of a2;
  redefine func union a3 -> Element of a2;
end;

:: MEASURE2:th 4
theorem
for b1 being set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being Function-like quasi_total Relation of NAT,b2 holds
   ex b4 being Function-like quasi_total Relation of NAT,b2 st
      b4 . 0 = b3 . 0 &
       (for b5 being Element of NAT holds
          b4 . (b5 + 1) = (b3 . (b5 + 1)) \ (b3 . b5));

:: MEASURE2:th 5
theorem
for b1 being set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being Function-like quasi_total Relation of NAT,b2 holds
   ex b4 being Function-like quasi_total Relation of NAT,b2 st
      b4 . 0 = b3 . 0 &
       (for b5 being Element of NAT holds
          b4 . (b5 + 1) = (b3 . (b5 + 1)) \/ (b4 . b5));

:: MEASURE2:th 6
theorem
for b1 being set
for b2 being non empty Element of bool bool b1
for b3, b4 being Function-like quasi_total Relation of NAT,b2
   st b4 . 0 = b3 . 0 &
      (for b5 being Element of NAT holds
         b4 . (b5 + 1) = (b3 . (b5 + 1)) \/ (b4 . b5))
for b5 being set
for b6 being Element of NAT holds
      b5 in b4 . b6
   iff
      ex b7 being Element of NAT st
         b7 <= b6 & b5 in b3 . b7;

:: MEASURE2:th 7
theorem
for b1 being set
for b2 being non empty Element of bool bool b1
for b3, b4 being Function-like quasi_total Relation of NAT,b2
   st b4 . 0 = b3 . 0 &
      (for b5 being Element of NAT holds
         b4 . (b5 + 1) = (b3 . (b5 + 1)) \/ (b4 . b5))
for b5, b6 being Element of NAT
      st b5 < b6
   holds b4 . b5 c= b4 . b6;

:: MEASURE2:th 8
theorem
for b1 being set
for b2 being non empty Element of bool bool b1
for b3, b4, b5 being Function-like quasi_total Relation of NAT,b2
   st b4 . 0 = b3 . 0 &
      (for b6 being Element of NAT holds
         b4 . (b6 + 1) = (b3 . (b6 + 1)) \/ (b4 . b6)) &
      b5 . 0 = b3 . 0 &
      (for b6 being Element of NAT holds
         b5 . (b6 + 1) = (b3 . (b6 + 1)) \ (b4 . b6))
for b6, b7 being Element of NAT
      st b6 <= b7
   holds b5 . b6 c= b4 . b7;

:: MEASURE2:th 9
theorem
for b1 being set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3, b4 being Function-like quasi_total Relation of NAT,b2 holds
ex b5 being Function-like quasi_total Relation of NAT,b2 st
   b5 . 0 = b3 . 0 &
    (for b6 being Element of NAT holds
       b5 . (b6 + 1) = (b3 . (b6 + 1)) \ (b4 . b6));

:: MEASURE2:th 10
theorem
for b1 being set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being Function-like quasi_total Relation of NAT,b2 holds
   ex b4 being Function-like quasi_total Relation of NAT,b2 st
      b4 . 0 = {} &
       (for b5 being Element of NAT holds
          b4 . (b5 + 1) = (b3 . 0) \ (b3 . b5));

:: MEASURE2:th 11
theorem
for b1 being set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3, b4, b5 being Function-like quasi_total Relation of NAT,b2
   st b4 . 0 = b3 . 0 &
      (for b6 being Element of NAT holds
         b4 . (b6 + 1) = (b3 . (b6 + 1)) \/ (b4 . b6)) &
      b5 . 0 = b3 . 0 &
      (for b6 being Element of NAT holds
         b5 . (b6 + 1) = (b3 . (b6 + 1)) \ (b4 . b6))
for b6, b7 being Element of NAT
      st b6 < b7
   holds b5 . b6 misses b5 . b7;

:: MEASURE2:th 13
theorem
for b1 being set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being sigma_Measure of b2
for b4 being N_Measure_fam of b2
for b5 being Function-like quasi_total Relation of NAT,b2
      st b4 = rng b5
   holds b3 . union b4 <= SUM (b3 * b5);

:: MEASURE2:th 14
theorem
for b1 being set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being N_Measure_fam of b2 holds
   ex b4 being Function-like quasi_total Relation of NAT,b2 st
      b3 = rng b4;

:: MEASURE2:th 15
theorem
for b1, b2 being Relation-like Function-like set
   st b2 . 0 = {} &
      (for b3 being Element of NAT holds
         b2 . (b3 + 1) = (b1 . 0) \ (b1 . b3) &
          b1 . (b3 + 1) c= b1 . b3)
for b3 being Element of NAT holds
   b2 . b3 c= b2 . (b3 + 1);

:: MEASURE2:th 16
theorem
for b1 being set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being sigma_Measure of b2
for b4 being N_Measure_fam of b2
      st for b5 being set
              st b5 in b4
           holds b5 is measure_zero of b3
   holds union b4 is measure_zero of b3;

:: MEASURE2:th 17
theorem
for b1 being set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being sigma_Measure of b2
for b4 being N_Measure_fam of b2
      st ex b5 being set st
           b5 in b4 & b5 is measure_zero of b3
   holds meet b4 is measure_zero of b3;

:: MEASURE2:th 18
theorem
for b1 being set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being sigma_Measure of b2
for b4 being N_Measure_fam of b2
      st for b5 being set
              st b5 in b4
           holds b5 is measure_zero of b3
   holds meet b4 is measure_zero of b3;

:: MEASURE2:attrnot 1 => MEASURE2:attr 1
definition
  let a1 be set;
  let a2 be non empty compl-closed sigma-multiplicative Element of bool bool a1;
  let a3 be N_Measure_fam of a2;
  attr a3 is non-decreasing means
    ex b1 being Function-like quasi_total Relation of NAT,a2 st
       a3 = rng b1 &
        (for b2 being Element of NAT holds
           b1 . b2 c= b1 . (b2 + 1));
end;

:: MEASURE2:dfs 2
definiens
  let a1 be set;
  let a2 be non empty compl-closed sigma-multiplicative Element of bool bool a1;
  let a3 be N_Measure_fam of a2;
To prove
     a3 is non-decreasing
it is sufficient to prove
  thus ex b1 being Function-like quasi_total Relation of NAT,a2 st
       a3 = rng b1 &
        (for b2 being Element of NAT holds
           b1 . b2 c= b1 . (b2 + 1));

:: MEASURE2:def 2
theorem
for b1 being set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being N_Measure_fam of b2 holds
      b3 is non-decreasing(b1, b2)
   iff
      ex b4 being Function-like quasi_total Relation of NAT,b2 st
         b3 = rng b4 &
          (for b5 being Element of NAT holds
             b4 . b5 c= b4 . (b5 + 1));

:: MEASURE2:exreg 1
registration
  let a1 be set;
  let a2 be non empty compl-closed sigma-multiplicative Element of bool bool a1;
  cluster non empty countable non-decreasing N_Measure_fam of a2;
end;

:: MEASURE2:attrnot 2 => MEASURE2:attr 2
definition
  let a1 be set;
  let a2 be non empty compl-closed sigma-multiplicative Element of bool bool a1;
  let a3 be N_Measure_fam of a2;
  attr a3 is non-increasing means
    ex b1 being Function-like quasi_total Relation of NAT,a2 st
       a3 = rng b1 &
        (for b2 being Element of NAT holds
           b1 . (b2 + 1) c= b1 . b2);
end;

:: MEASURE2:dfs 3
definiens
  let a1 be set;
  let a2 be non empty compl-closed sigma-multiplicative Element of bool bool a1;
  let a3 be N_Measure_fam of a2;
To prove
     a3 is non-increasing
it is sufficient to prove
  thus ex b1 being Function-like quasi_total Relation of NAT,a2 st
       a3 = rng b1 &
        (for b2 being Element of NAT holds
           b1 . (b2 + 1) c= b1 . b2);

:: MEASURE2:def 3
theorem
for b1 being set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being N_Measure_fam of b2 holds
      b3 is non-increasing(b1, b2)
   iff
      ex b4 being Function-like quasi_total Relation of NAT,b2 st
         b3 = rng b4 &
          (for b5 being Element of NAT holds
             b4 . (b5 + 1) c= b4 . b5);

:: MEASURE2:exreg 2
registration
  let a1 be set;
  let a2 be non empty compl-closed sigma-multiplicative Element of bool bool a1;
  cluster non empty countable non-increasing N_Measure_fam of a2;
end;

:: MEASURE2:th 21
theorem
for b1 being set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3, b4 being Function-like quasi_total Relation of NAT,b2
      st b4 . 0 = {} &
         (for b5 being Element of NAT holds
            b4 . (b5 + 1) = (b3 . 0) \ (b3 . b5) &
             b3 . (b5 + 1) c= b3 . b5)
   holds rng b4 is non-decreasing N_Measure_fam of b2;

:: MEASURE2:th 22
theorem
for b1 being Relation-like Function-like set
   st for b2 being Element of NAT holds
        b1 . b2 c= b1 . (b2 + 1)
for b2, b3 being Element of NAT
      st b3 <= b2
   holds b1 . b3 c= b1 . b2;

:: MEASURE2:th 23
theorem
for b1, b2 being Relation-like Function-like set
   st b2 . 0 = b1 . 0 &
      (for b3 being Element of NAT holds
         b2 . (b3 + 1) = (b1 . (b3 + 1)) \ (b1 . b3) &
          b1 . b3 c= b1 . (b3 + 1))
for b3, b4 being Element of NAT
      st b3 < b4
   holds b2 . b3 misses b2 . b4;

:: MEASURE2:th 24
theorem
for b1 being set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3, b4 being Function-like quasi_total Relation of NAT,b2
      st b4 . 0 = b3 . 0 &
         (for b5 being Element of NAT holds
            b4 . (b5 + 1) = (b3 . (b5 + 1)) \ (b3 . b5) &
             b3 . b5 c= b3 . (b5 + 1))
   holds union rng b4 = union rng b3;

:: MEASURE2:th 25
theorem
for b1 being set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3, b4 being Function-like quasi_total Relation of NAT,b2
      st b4 . 0 = b3 . 0 &
         (for b5 being Element of NAT holds
            b4 . (b5 + 1) = (b3 . (b5 + 1)) \ (b3 . b5) &
             b3 . b5 c= b3 . (b5 + 1))
   holds b4 is Function-like quasi_total disjoint_valued Relation of NAT,b2;

:: MEASURE2:th 26
theorem
for b1 being set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3, b4 being Function-like quasi_total Relation of NAT,b2
      st b4 . 0 = b3 . 0 &
         (for b5 being Element of NAT holds
            b4 . (b5 + 1) = (b3 . (b5 + 1)) \ (b3 . b5) &
             b3 . b5 c= b3 . (b5 + 1))
   holds b3 . 0 = b4 . 0 &
    (for b5 being Element of NAT holds
       b3 . (b5 + 1) = (b4 . (b5 + 1)) \/ (b3 . b5));

:: MEASURE2:th 27
theorem
for b1 being set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being sigma_Measure of b2
for b4 being Function-like quasi_total Relation of NAT,b2
      st for b5 being Element of NAT holds
           b4 . b5 c= b4 . (b5 + 1)
   holds b3 . union rng b4 = sup rng (b3 * b4);