Article MEASURE4, MML version 4.99.1005

:: MEASURE4:th 1
theorem
for b1, b2, b3 being Element of ExtREAL
      st 0. <= b1 & 0. <= b2 & 0. <= b3
   holds (b1 + b2) + b3 = b1 + (b2 + b3);

:: MEASURE4:th 2
theorem
for b1, b2, b3 being Element of ExtREAL
      st b1 <> -infty & b1 <> +infty
   holds    b2 + b1 <= b3
   iff
      b2 <= b3 - b1;

:: MEASURE4:th 4
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
for b5 being Element of b2
      st for b6 being Element of NAT holds
           b4 . b6 = b5 /\ (b3 . b6)
   holds union rng b4 = b5 /\ union rng b3;

:: MEASURE4:th 5
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 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))
   holds union rng b3 = union rng b5;

:: MEASURE4:th 6
theorem
for b1 being set holds
   bool b1 is non empty compl-closed sigma-multiplicative Element of bool bool b1;

:: MEASURE4:th 7
theorem
for b1 being set
for b2, b3 being Element of ExtREAL holds
ex b4 being Function-like quasi_total Relation of bool b1,ExtREAL st
   for b5 being Element of bool b1 holds
      (b5 = {} implies b4 . b5 = b2) & (b5 = {} or b4 . b5 = b3);

:: MEASURE4:th 8
theorem
for b1 being set holds
   ex b2 being Function-like quasi_total Relation of bool b1,ExtREAL st
      for b3 being Element of bool b1 holds
         b2 . b3 = 0.;

:: MEASURE4:th 11
theorem
for b1 being set holds
   ex b2 being Function-like quasi_total Relation of bool b1,ExtREAL st
      b2 is nonnegative &
       b2 . {} = 0. &
       (for b3, b4 being Element of bool b1
             st b3 c= b4
          holds b2 . b3 <= b2 . b4 &
           (for b5 being Function-like quasi_total Relation of NAT,bool b1 holds
              b2 . union rng b5 <= SUM (b2 * b5)));

:: MEASURE4:modenot 1 => MEASURE4:mode 1
definition
  let a1 be set;
  mode C_Measure of A1 -> Function-like quasi_total Relation of bool a1,ExtREAL means
    it is nonnegative &
     it . {} = 0. &
     (for b1, b2 being Element of bool a1
           st b1 c= b2
        holds it . b1 <= it . b2 &
         (for b3 being Function-like quasi_total Relation of NAT,bool a1 holds
            it . union rng b3 <= SUM (it * b3)));
end;

:: MEASURE4:dfs 1
definiens
  let a1 be set;
  let a2 be Function-like quasi_total Relation of bool a1,ExtREAL;
To prove
     a2 is C_Measure of a1
it is sufficient to prove
  thus a2 is nonnegative &
     a2 . {} = 0. &
     (for b1, b2 being Element of bool a1
           st b1 c= b2
        holds a2 . b1 <= a2 . b2 &
         (for b3 being Function-like quasi_total Relation of NAT,bool a1 holds
            a2 . union rng b3 <= SUM (a2 * b3)));

:: MEASURE4:def 2
theorem
for b1 being set
for b2 being Function-like quasi_total Relation of bool b1,ExtREAL holds
      b2 is C_Measure of b1
   iff
      b2 is nonnegative &
       b2 . {} = 0. &
       (for b3, b4 being Element of bool b1
             st b3 c= b4
          holds b2 . b3 <= b2 . b4 &
           (for b5 being Function-like quasi_total Relation of NAT,bool b1 holds
              b2 . union rng b5 <= SUM (b2 * b5)));

:: MEASURE4:funcnot 1 => MEASURE4:func 1
definition
  let a1 be set;
  let a2 be C_Measure of a1;
  func sigma_Field A2 -> non empty Element of bool bool a1 means
    for b1 being Element of bool a1 holds
          b1 in it
       iff
          for b2, b3 being Element of bool a1
                st b2 c= b1 & b3 c= a1 \ b1
             holds (a2 . b2) + (a2 . b3) <= a2 . (b2 \/ b3);
end;

:: MEASURE4:def 3
theorem
for b1 being set
for b2 being C_Measure of b1
for b3 being non empty Element of bool bool b1 holds
      b3 = sigma_Field b2
   iff
      for b4 being Element of bool b1 holds
            b4 in b3
         iff
            for b5, b6 being Element of bool b1
                  st b5 c= b4 & b6 c= b1 \ b4
               holds (b2 . b5) + (b2 . b6) <= b2 . (b5 \/ b6);

:: MEASURE4:th 12
theorem
for b1 being set
for b2 being C_Measure of b1
for b3, b4 being Element of bool b1 holds
b2 . (b3 \/ b4) <= (b2 . b3) + (b2 . b4);

:: MEASURE4:th 14
theorem
for b1 being set
for b2 being C_Measure of b1
for b3 being Element of bool b1 holds
      b3 in sigma_Field b2
   iff
      for b4, b5 being Element of bool b1
            st b4 c= b3 & b5 c= b1 \ b3
         holds (b2 . b4) + (b2 . b5) = b2 . (b4 \/ b5);

:: MEASURE4:th 15
theorem
for b1 being set
for b2 being C_Measure of b1
for b3, b4 being Element of bool b1
      st b3 in sigma_Field b2 & b4 in sigma_Field b2 & b4 misses b3
   holds b2 . (b3 \/ b4) = (b2 . b3) + (b2 . b4);

:: MEASURE4:th 16
theorem
for b1, b2 being set
for b3 being C_Measure of b2
      st b1 in sigma_Field b3
   holds b2 \ b1 in sigma_Field b3;

:: MEASURE4:th 17
theorem
for b1, b2, b3 being set
for b4 being C_Measure of b1
      st b2 in sigma_Field b4 & b3 in sigma_Field b4
   holds b2 \/ b3 in sigma_Field b4;

:: MEASURE4:th 18
theorem
for b1, b2, b3 being set
for b4 being C_Measure of b1
      st b2 in sigma_Field b4 & b3 in sigma_Field b4
   holds b2 /\ b3 in sigma_Field b4;

:: MEASURE4:th 19
theorem
for b1, b2, b3 being set
for b4 being C_Measure of b1
      st b2 in sigma_Field b4 & b3 in sigma_Field b4
   holds b2 \ b3 in sigma_Field b4;

:: MEASURE4:th 20
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
for b4 being Element of b2 holds
   ex b5 being Function-like quasi_total Relation of NAT,b2 st
      for b6 being Element of NAT holds
         b5 . b6 = b4 /\ (b3 . b6);

:: MEASURE4:th 21
theorem
for b1 being set
for b2 being C_Measure of b1 holds
   sigma_Field b2 is non empty compl-closed sigma-multiplicative Element of bool bool b1;

:: MEASURE4:funcreg 1
registration
  let a1 be set;
  let a2 be C_Measure of a1;
  cluster sigma_Field a2 -> non empty compl-closed sigma-additive;
end;

:: MEASURE4:funcnot 2 => MEASURE4: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;

:: MEASURE4:funcnot 3 => MEASURE4:func 3
definition
  let a1 be set;
  let a2 be C_Measure of a1;
  func sigma_Meas A2 -> Function-like quasi_total Relation of sigma_Field a2,ExtREAL means
    for b1 being Element of bool a1
          st b1 in sigma_Field a2
       holds it . b1 = a2 . b1;
end;

:: MEASURE4:def 4
theorem
for b1 being set
for b2 being C_Measure of b1
for b3 being Function-like quasi_total Relation of sigma_Field b2,ExtREAL holds
      b3 = sigma_Meas b2
   iff
      for b4 being Element of bool b1
            st b4 in sigma_Field b2
         holds b3 . b4 = b2 . b4;

:: MEASURE4:th 22
theorem
for b1 being set
for b2 being C_Measure of b1 holds
   sigma_Meas b2 is Measure of sigma_Field b2;

:: MEASURE4:th 23
theorem
for b1 being set
for b2 being C_Measure of b1 holds
   sigma_Meas b2 is sigma_Measure of sigma_Field b2;

:: MEASURE4:funcnot 4 => MEASURE4:func 4
definition
  let a1 be set;
  let a2 be C_Measure of a1;
  redefine func sigma_Meas a2 -> sigma_Measure of sigma_Field a2;
end;

:: MEASURE4:th 24
theorem
for b1 being set
for b2 being C_Measure of b1
for b3 being Element of bool b1
      st b2 . b3 = 0.
   holds b3 in sigma_Field b2;

:: MEASURE4:th 25
theorem
for b1 being set
for b2 being C_Measure of b1 holds
   sigma_Meas b2 is_complete sigma_Field b2;