Article MEASURE3, MML version 4.99.1005

:: MEASURE3:th 1
theorem
for b1 being Element of ExtREAL
      st -infty < b1 & b1 < +infty
   holds b1 is Element of REAL;

:: MEASURE3:th 2
theorem
for b1 being ext-real set
      st b1 <> -infty & b1 <> +infty
   holds b1 is Element of REAL;

:: MEASURE3:th 3
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,ExtREAL
      st for b3 being Element of NAT holds
           (Ser b1) . b3 <= (Ser b2) . b3
   holds SUM b1 <= SUM b2;

:: MEASURE3:th 4
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,ExtREAL
      st for b3 being Element of NAT holds
           (Ser b1) . b3 = (Ser b2) . b3
   holds SUM b1 = SUM b2;

:: MEASURE3:modenot 1 => MEASURE2:mode 1
notation
  let a1 be set;
  let a2 be non empty compl-closed sigma-multiplicative Element of bool bool a1;
  synonym N_Sub_fam of a2 for N_Measure_fam of a2;
end;

:: MEASURE3:funcnot 1 => MEASURE3:func 1
definition
  let a1 be set;
  let a2 be non empty compl-closed sigma-multiplicative Element of bool bool a1;
  let a3 be Function-like quasi_total Relation of NAT,a2;
  redefine func rng a3 -> N_Measure_fam of a2;
end;

:: MEASURE3:th 5
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
for b5 being Element of b2
      st meet rng b4 c= b5 &
         (for b6 being Element of NAT holds
            b5 c= b4 . b6)
   holds b3 . b5 = b3 . meet rng b4;

:: MEASURE3:th 6
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 b3 . 0 = {} &
         (for b5 being Element of NAT holds
            b3 . (b5 + 1) = (b4 . 0) \ (b4 . b5) &
             b4 . (b5 + 1) c= b4 . b5)
   holds union rng b3 = (b4 . 0) \ meet rng b4;

:: MEASURE3:th 7
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 b3 . 0 = {} &
         (for b5 being Element of NAT holds
            b3 . (b5 + 1) = (b4 . 0) \ (b4 . b5) &
             b4 . (b5 + 1) c= b4 . b5)
   holds meet rng b4 = (b4 . 0) \ union rng b3;

:: MEASURE3:th 8
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, b5 being Function-like quasi_total Relation of NAT,b2
      st b3 . (b5 . 0) < +infty &
         b4 . 0 = {} &
         (for b6 being Element of NAT holds
            b4 . (b6 + 1) = (b5 . 0) \ (b5 . b6) &
             b5 . (b6 + 1) c= b5 . b6)
   holds b3 . meet rng b5 = (b3 . (b5 . 0)) - (b3 . union rng b4);

:: MEASURE3:th 9
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, b5 being Function-like quasi_total Relation of NAT,b2
      st b3 . (b5 . 0) < +infty &
         b4 . 0 = {} &
         (for b6 being Element of NAT holds
            b4 . (b6 + 1) = (b5 . 0) \ (b5 . b6) &
             b5 . (b6 + 1) c= b5 . b6)
   holds b3 . union rng b4 = (b3 . (b5 . 0)) - (b3 . meet rng b5);

:: MEASURE3:th 10
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, b5 being Function-like quasi_total Relation of NAT,b2
      st b3 . (b5 . 0) < +infty &
         b4 . 0 = {} &
         (for b6 being Element of NAT holds
            b4 . (b6 + 1) = (b5 . 0) \ (b5 . b6) &
             b5 . (b6 + 1) c= b5 . b6)
   holds b3 . meet rng b5 = (b3 . (b5 . 0)) - sup rng (b3 * b4);

:: MEASURE3:th 11
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, b5 being Function-like quasi_total Relation of NAT,b2
      st b3 . (b5 . 0) < +infty &
         b4 . 0 = {} &
         (for b6 being Element of NAT holds
            b4 . (b6 + 1) = (b5 . 0) \ (b5 . b6) &
             b5 . (b6 + 1) c= b5 . b6)
   holds sup rng (b3 * b4) is Element of REAL & b3 . (b5 . 0) is Element of REAL & inf rng (b3 * b5) is Element of REAL;

:: MEASURE3:th 12
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, b5 being Function-like quasi_total Relation of NAT,b2
      st b3 . (b5 . 0) < +infty &
         b4 . 0 = {} &
         (for b6 being Element of NAT holds
            b4 . (b6 + 1) = (b5 . 0) \ (b5 . b6) &
             b5 . (b6 + 1) c= b5 . b6)
   holds sup rng (b3 * b4) = (b3 . (b5 . 0)) - inf rng (b3 * b5);

:: MEASURE3: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, b5 being Function-like quasi_total Relation of NAT,b2
      st b3 . (b5 . 0) < +infty &
         b4 . 0 = {} &
         (for b6 being Element of NAT holds
            b4 . (b6 + 1) = (b5 . 0) \ (b5 . b6) &
             b5 . (b6 + 1) c= b5 . b6)
   holds inf rng (b3 * b5) = (b3 . (b5 . 0)) - sup rng (b3 * b4);

:: MEASURE3:th 14
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 + 1) c= b4 . b5) &
         b3 . (b4 . 0) < +infty
   holds b3 . meet rng b4 = inf rng (b3 * b4);

:: MEASURE3:th 15
theorem
for b1 being set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being Measure of b2
for b4 being Function-like quasi_total disjoint_valued Relation of NAT,b2 holds
   SUM (b3 * b4) <= b3 . union rng b4;

:: MEASURE3:th 17
theorem
for b1 being set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being Measure of b2
      st for b4 being Function-like quasi_total disjoint_valued Relation of NAT,b2 holds
           b3 . union rng b4 <= SUM (b3 * b4)
   holds b3 is sigma_Measure of b2;

:: MEASURE3:prednot 1 => MEASURE3:pred 1
definition
  let a1 be set;
  let a2 be non empty compl-closed sigma-multiplicative Element of bool bool a1;
  let a3 be sigma_Measure of a2;
  pred A3 is_complete A2 means
    for b1 being Element of bool a1
    for b2 being set
          st b2 in a2 & b1 c= b2 & a3 . b2 = 0.
       holds b1 in a2;
end;

:: MEASURE3:dfs 1
definiens
  let a1 be set;
  let a2 be non empty compl-closed sigma-multiplicative Element of bool bool a1;
  let a3 be sigma_Measure of a2;
To prove
     a3 is_complete a2
it is sufficient to prove
  thus for b1 being Element of bool a1
    for b2 being set
          st b2 in a2 & b1 c= b2 & a3 . b2 = 0.
       holds b1 in a2;

:: MEASURE3:def 2
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 holds
      b3 is_complete b2
   iff
      for b4 being Element of bool b1
      for b5 being set
            st b5 in b2 & b4 c= b5 & b3 . b5 = 0.
         holds b4 in b2;

:: MEASURE3:modenot 2 => MEASURE3:mode 1
definition
  let a1 be set;
  let a2 be non empty compl-closed sigma-multiplicative Element of bool bool a1;
  let a3 be sigma_Measure of a2;
  mode thin of A3 -> Element of bool a1 means
    ex b1 being set st
       b1 in a2 & it c= b1 & a3 . b1 = 0.;
end;

:: MEASURE3:dfs 2
definiens
  let a1 be set;
  let a2 be non empty compl-closed sigma-multiplicative Element of bool bool a1;
  let a3 be sigma_Measure of a2;
  let a4 be Element of bool a1;
To prove
     a4 is thin of a3
it is sufficient to prove
  thus ex b1 being set st
       b1 in a2 & a4 c= b1 & a3 . b1 = 0.;

:: MEASURE3:def 3
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 Element of bool b1 holds
      b4 is thin of b3
   iff
      ex b5 being set st
         b5 in b2 & b4 c= b5 & b3 . b5 = 0.;

:: MEASURE3:funcnot 2 => MEASURE3:func 2
definition
  let a1 be set;
  let a2 be non empty compl-closed sigma-multiplicative Element of bool bool a1;
  let a3 be sigma_Measure of a2;
  func COM(A2,A3) -> non empty Element of bool bool a1 means
    for b1 being set holds
          b1 in it
       iff
          ex b2 being set st
             b2 in a2 &
              (ex b3 being thin of a3 st
                 b1 = b2 \/ b3);
end;

:: MEASURE3:def 4
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 non empty Element of bool bool b1 holds
      b4 = COM(b2,b3)
   iff
      for b5 being set holds
            b5 in b4
         iff
            ex b6 being set st
               b6 in b2 &
                (ex b7 being thin of b3 st
                   b5 = b6 \/ b7);

:: MEASURE3:funcnot 3 => MEASURE3:func 3
definition
  let a1 be set;
  let a2 be non empty compl-closed sigma-multiplicative Element of bool bool a1;
  let a3 be sigma_Measure of a2;
  let a4 be Element of COM(a2,a3);
  func MeasPart A4 -> non empty Element of bool bool a1 means
    for b1 being set holds
          b1 in it
       iff
          b1 in a2 & b1 c= a4 & a4 \ b1 is thin of a3;
end;

:: MEASURE3:def 5
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 Element of COM(b2,b3)
for b5 being non empty Element of bool bool b1 holds
      b5 = MeasPart b4
   iff
      for b6 being set holds
            b6 in b5
         iff
            b6 in b2 & b6 c= b4 & b4 \ b6 is thin of b3;

:: MEASURE3: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 Function-like quasi_total Relation of NAT,COM(b2,b3) holds
   ex b5 being Function-like quasi_total Relation of NAT,b2 st
      for b6 being Element of NAT holds
         b5 . b6 in MeasPart (b4 . b6);

:: MEASURE3:th 19
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,COM(b2,b3)
for b5 being Function-like quasi_total Relation of NAT,b2 holds
   ex b6 being Function-like quasi_total Relation of NAT,bool b1 st
      for b7 being Element of NAT holds
         b6 . b7 = (b4 . b7) \ (b5 . b7);

:: MEASURE3:th 20
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,bool b1
      st for b5 being Element of NAT holds
           b4 . b5 is thin of b3
   holds ex b5 being Function-like quasi_total Relation of NAT,b2 st
      for b6 being Element of NAT holds
         b4 . b6 c= b5 . b6 & b3 . (b5 . b6) = 0.;

:: MEASURE3:th 21
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 non empty Element of bool bool b1
      st for b5 being set holds
              b5 in b4
           iff
              ex b6 being set st
                 b6 in b2 &
                  (ex b7 being thin of b3 st
                     b5 = b6 \/ b7)
   holds b4 is non empty compl-closed sigma-multiplicative Element of bool bool b1;

:: MEASURE3:funcreg 1
registration
  let a1 be set;
  let a2 be non empty compl-closed sigma-multiplicative Element of bool bool a1;
  let a3 be sigma_Measure of a2;
  cluster COM(a2,a3) -> non empty compl-closed sigma-additive;
end;

:: MEASURE3:th 22
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, b5 being set
   st b4 in b2 & b5 in b2
for b6, b7 being thin of b3
      st b4 \/ b6 = b5 \/ b7
   holds b3 . b4 = b3 . b5;

:: MEASURE3:funcnot 4 => MEASURE3:func 4
definition
  let a1 be set;
  let a2 be non empty compl-closed sigma-multiplicative Element of bool bool a1;
  let a3 be sigma_Measure of a2;
  func COM A3 -> sigma_Measure of COM(a2,a3) means
    for b1 being set
       st b1 in a2
    for b2 being thin of a3 holds
       it . (b1 \/ b2) = a3 . b1;
end;

:: MEASURE3:def 6
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 sigma_Measure of COM(b2,b3) holds
      b4 = COM b3
   iff
      for b5 being set
         st b5 in b2
      for b6 being thin of b3 holds
         b4 . (b5 \/ b6) = b3 . b5;

:: MEASURE3:th 23
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 holds
   COM b3 is_complete COM(b2,b3);