Article MESFUNC3, MML version 4.99.1005

:: MESFUNC3:th 1
theorem
for b1, b2 being natural set
for b3 being Function-like quasi_total Relation of [:Seg b1,Seg b2:],REAL
for b4, b5 being FinSequence of REAL
      st dom b4 = Seg b1 &
         (for b6 being natural set
               st b6 in dom b4
            holds ex b7 being FinSequence of REAL st
               dom b7 = Seg b2 &
                b4 . b6 = Sum b7 &
                (for b8 being natural set
                      st b8 in dom b7
                   holds b7 . b8 = b3 . [b6,b8])) &
         dom b5 = Seg b2 &
         (for b6 being natural set
               st b6 in dom b5
            holds ex b7 being FinSequence of REAL st
               dom b7 = Seg b1 &
                b5 . b6 = Sum b7 &
                (for b8 being natural set
                      st b8 in dom b7
                   holds b7 . b8 = b3 . [b8,b6]))
   holds Sum b4 = Sum b5;

:: MESFUNC3:th 2
theorem
for b1 being FinSequence of ExtREAL
for b2 being FinSequence of REAL
      st b1 = b2
   holds Sum b1 = Sum b2;

:: MESFUNC3:th 3
theorem
for b1 being non empty set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being Function-like Relation of b1,ExtREAL
      st b3 is_simple_func_in b2
   holds ex b4 being disjoint_valued FinSequence of b2 st
      ex b5 being FinSequence of ExtREAL st
         dom b3 = union rng b4 &
          dom b4 = dom b5 &
          (for b6 being natural set
             st b6 in dom b4
          for b7 being set
                st b7 in b4 . b6
             holds b3 . b7 = b5 . b6) &
          (for b6 being set
                st b6 in dom b3
             holds ex b7 being FinSequence of ExtREAL st
                dom b7 = dom b5 &
                 (for b8 being natural set
                       st b8 in dom b7
                    holds b7 . b8 = (b5 . b8) * ((chi(b4 . b8,b1)) . b6)));

:: MESFUNC3:th 4
theorem
for b1 being set
for b2 being FinSequence of b1 holds
      b2 is disjoint_valued
   iff
      for b3, b4 being natural set
            st b3 in dom b2 & b4 in dom b2 & b3 <> b4
         holds b2 . b3 misses b2 . b4;

:: MESFUNC3:th 5
theorem
for b1 being non empty set
for b2 being set
for b3 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b4 being disjoint_valued FinSequence of b3
for b5 being FinSequence of b3
      st dom b5 = dom b4 &
         (for b6 being natural set
               st b6 in dom b5
            holds b5 . b6 = b2 /\ (b4 . b6))
   holds b5 is disjoint_valued FinSequence of b3;

:: MESFUNC3:th 6
theorem
for b1 being non empty set
for b2 being set
for b3, b4 being FinSequence of b1
      st dom b4 = dom b3 &
         (for b5 being natural set
               st b5 in dom b4
            holds b4 . b5 = b2 /\ (b3 . b5))
   holds union rng b4 = b2 /\ union rng b3;

:: MESFUNC3:th 7
theorem
for b1 being set
for b2 being FinSequence of b1
for b3 being natural set
      st b3 in dom b2
   holds b2 . b3 c= union rng b2 &
    (b2 . b3) /\ union rng b2 = b2 . b3;

:: MESFUNC3:th 8
theorem
for b1 being non empty 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 disjoint_valued FinSequence of b2 holds
   dom b4 = dom (b3 * b4);

:: MESFUNC3:th 9
theorem
for b1 being non empty 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 disjoint_valued FinSequence of b2 holds
   b3 . union rng b4 = Sum (b3 * b4);

:: MESFUNC3:th 10
theorem
for b1, b2 being FinSequence of ExtREAL
for b3 being Element of ExtREAL
      st ((b3 = +infty or b3 = -infty) &
          (ex b4 being natural set st
             b4 in dom b1 & 0. <= b1 . b4) implies for b4 being natural set
               st b4 in dom b1
            holds 0. < b1 . b4) &
         dom b1 = dom b2 &
         (for b4 being natural set
               st b4 in dom b2
            holds b2 . b4 = b3 * (b1 . b4))
   holds Sum b2 = b3 * Sum b1;

:: MESFUNC3:th 11
theorem
for b1 being FinSequence of REAL holds
   b1 is FinSequence of ExtREAL;

:: MESFUNC3:prednot 1 => MESFUNC3:pred 1
definition
  let a1 be non empty set;
  let a2 be non empty compl-closed sigma-multiplicative Element of bool bool a1;
  let a3 be Function-like Relation of a1,ExtREAL;
  let a4 be disjoint_valued FinSequence of a2;
  let a5 be FinSequence of ExtREAL;
  pred A4,A5 are_Re-presentation_of A3 means
    dom a3 = union rng a4 &
     dom a4 = dom a5 &
     (for b1 being natural set
        st b1 in dom a4
     for b2 being set
           st b2 in a4 . b1
        holds a3 . b2 = a5 . b1);
end;

:: MESFUNC3:dfs 1
definiens
  let a1 be non empty set;
  let a2 be non empty compl-closed sigma-multiplicative Element of bool bool a1;
  let a3 be Function-like Relation of a1,ExtREAL;
  let a4 be disjoint_valued FinSequence of a2;
  let a5 be FinSequence of ExtREAL;
To prove
     a4,a5 are_Re-presentation_of a3
it is sufficient to prove
  thus dom a3 = union rng a4 &
     dom a4 = dom a5 &
     (for b1 being natural set
        st b1 in dom a4
     for b2 being set
           st b2 in a4 . b1
        holds a3 . b2 = a5 . b1);

:: MESFUNC3:def 1
theorem
for b1 being non empty set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being Function-like Relation of b1,ExtREAL
for b4 being disjoint_valued FinSequence of b2
for b5 being FinSequence of ExtREAL holds
      b4,b5 are_Re-presentation_of b3
   iff
      dom b3 = union rng b4 &
       dom b4 = dom b5 &
       (for b6 being natural set
          st b6 in dom b4
       for b7 being set
             st b7 in b4 . b6
          holds b3 . b7 = b5 . b6);

:: MESFUNC3:th 12
theorem
for b1 being non empty set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being Function-like Relation of b1,ExtREAL
      st b3 is_simple_func_in b2
   holds ex b4 being disjoint_valued FinSequence of b2 st
      ex b5 being FinSequence of ExtREAL st
         b4,b5 are_Re-presentation_of b3;

:: MESFUNC3:th 13
theorem
for b1 being non empty set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being disjoint_valued FinSequence of b2 holds
   ex b4 being disjoint_valued FinSequence of b2 st
      union rng b3 = union rng b4 &
       (for b5 being natural set
             st b5 in dom b4
          holds b4 . b5 <> {} &
           (ex b6 being natural set st
              b6 in dom b3 & b3 . b6 = b4 . b5));

:: MESFUNC3:th 14
theorem
for b1 being non empty set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being Function-like Relation of b1,ExtREAL
      st b3 is_simple_func_in b2 &
         (for b4 being set
               st b4 in dom b3
            holds 0. <= b3 . b4)
   holds ex b4 being disjoint_valued FinSequence of b2 st
      ex b5 being FinSequence of ExtREAL st
         b4,b5 are_Re-presentation_of b3 &
          b5 . 1 = 0. &
          (for b6 being natural set
                st 2 <= b6 & b6 in dom b5
             holds 0. < b5 . b6 & b5 . b6 < +infty);

:: MESFUNC3:th 15
theorem
for b1 being non empty set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being Function-like Relation of b1,ExtREAL
for b4 being disjoint_valued FinSequence of b2
for b5 being FinSequence of ExtREAL
for b6 being Element of b1
      st b4,b5 are_Re-presentation_of b3 & b6 in dom b3
   holds ex b7 being FinSequence of ExtREAL st
      dom b7 = dom b5 &
       (for b8 being natural set
             st b8 in dom b7
          holds b7 . b8 = (b5 . b8) * ((chi(b4 . b8,b1)) . b6)) &
       b3 . b6 = Sum b7;

:: MESFUNC3:th 16
theorem
for b1 being FinSequence of ExtREAL
for b2 being FinSequence of REAL
      st b1 = b2
   holds Sum b1 = Sum b2;

:: MESFUNC3:th 17
theorem
for b1 being FinSequence of ExtREAL
      st (for b2 being natural set
               st b2 in dom b1
            holds 0. <= b1 . b2) &
         (ex b2 being natural set st
            b2 in dom b1 & b1 . b2 = +infty)
   holds Sum b1 = +infty;

:: MESFUNC3:funcnot 1 => MESFUNC3:func 1
definition
  let a1 be non empty 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 Function-like Relation of a1,ExtREAL;
  assume a4 is_simple_func_in a2 &
     dom a4 <> {} &
     (for b1 being set
           st b1 in dom a4
        holds 0. <= a4 . b1);
  func integral(A1,A2,A3,A4) -> Element of ExtREAL means
    ex b1 being disjoint_valued FinSequence of a2 st
       ex b2, b3 being FinSequence of ExtREAL st
          b1,b2 are_Re-presentation_of a4 &
           b2 . 1 = 0. &
           (for b4 being natural set
                 st 2 <= b4 & b4 in dom b2
              holds 0. < b2 . b4 & b2 . b4 < +infty) &
           dom b3 = dom b1 &
           (for b4 being natural set
                 st b4 in dom b3
              holds b3 . b4 = (b2 . b4) * ((a3 * b1) . b4)) &
           it = Sum b3;
end;

:: MESFUNC3:def 2
theorem
for b1 being non empty 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 Relation of b1,ExtREAL
   st b4 is_simple_func_in b2 &
      dom b4 <> {} &
      (for b5 being set
            st b5 in dom b4
         holds 0. <= b4 . b5)
for b5 being Element of ExtREAL holds
      b5 = integral(b1,b2,b3,b4)
   iff
      ex b6 being disjoint_valued FinSequence of b2 st
         ex b7, b8 being FinSequence of ExtREAL st
            b6,b7 are_Re-presentation_of b4 &
             b7 . 1 = 0. &
             (for b9 being natural set
                   st 2 <= b9 & b9 in dom b7
                holds 0. < b7 . b9 & b7 . b9 < +infty) &
             dom b8 = dom b6 &
             (for b9 being natural set
                   st b9 in dom b8
                holds b8 . b9 = (b7 . b9) * ((b3 * b6) . b9)) &
             b5 = Sum b8;

:: MESFUNC3:th 18
theorem
for b1 being FinSequence of ExtREAL
for b2, b3 being Element of ExtREAL
      st b3 = len b1 &
         (for b4 being natural set
               st b4 in dom b1
            holds b1 . b4 = b2)
   holds Sum b1 = b3 * b2;