Article MESFUNC2, MML version 4.99.1005

:: MESFUNC2:attrnot 1 => VALUED_0:attr 3
notation
  let a1 be non empty set;
  let a2 be Function-like Relation of a1,ExtREAL;
  synonym finite for real-valued;
end;

:: MESFUNC2:attrnot 2 => MESFUNC2:attr 1
definition
  let a1 be non empty set;
  let a2 be Function-like Relation of a1,ExtREAL;
  redefine attr a2 is finite means
    for b1 being Element of a1
          st b1 in dom a2
       holds |.a2 . b1.| < +infty;
end;

:: MESFUNC2:dfs 1
definiens
  let a1 be non empty set;
  let a2 be Function-like Relation of a1,ExtREAL;
To prove
     a1 is real-valued
it is sufficient to prove
  thus for b1 being Element of a1
          st b1 in dom a2
       holds |.a2 . b1.| < +infty;

:: MESFUNC2:def 1
theorem
for b1 being non empty set
for b2 being Function-like Relation of b1,ExtREAL holds
      b2 is real-valued
   iff
      for b3 being Element of b1
            st b3 in dom b2
         holds |.b2 . b3.| < +infty;

:: MESFUNC2:th 1
theorem
for b1 being non empty set
for b2 being Function-like Relation of b1,ExtREAL holds
   b2 = 1 (#) b2;

:: MESFUNC2:th 2
theorem
for b1 being non empty set
for b2, b3 being Function-like Relation of b1,ExtREAL
      st (b2 is real-valued or b3 is real-valued)
   holds dom (b2 + b3) = (dom b2) /\ dom b3 &
    dom (b2 - b3) = (dom b2) /\ dom b3;

:: MESFUNC2: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, b4 being Function-like Relation of b1,ExtREAL
for b5 being Function-like quasi_total Relation of RAT,b2
for b6 being Element of REAL
for b7 being Element of b2
      st b3 is real-valued &
         b4 is real-valued &
         (for b8 being rational set holds
            b5 . b8 = (b7 /\ less_dom(b3,R_EAL b8)) /\ (b7 /\ less_dom(b4,R_EAL (b6 - b8))))
   holds b7 /\ less_dom(b3 + b4,R_EAL b6) = union rng b5;

:: MESFUNC2:th 4
theorem
ex b1 being Function-like quasi_total Relation of NAT,RAT st
   b1 is one-to-one & dom b1 = NAT & rng b1 = RAT;

:: MESFUNC2:th 5
theorem
for b1, b2, b3 being non empty set
for b4 being Function-like quasi_total Relation of b1,b3
      st b1,b2 are_equipotent
   holds ex b5 being Function-like quasi_total Relation of b2,b3 st
      rng b4 = rng b5;

:: MESFUNC2:th 6
theorem
for b1 being non empty set
for b2 being Element of REAL
for b3 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b4, b5 being Function-like Relation of b1,ExtREAL
for b6 being Element of b3
      st b4 is_measurable_on b6 & b5 is_measurable_on b6
   holds ex b7 being Function-like quasi_total Relation of RAT,b3 st
      for b8 being rational set holds
         b7 . b8 = (b6 /\ less_dom(b4,R_EAL b8)) /\ (b6 /\ less_dom(b5,R_EAL (b2 - b8)));

:: MESFUNC2:th 7
theorem
for b1 being non empty set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3, b4 being Function-like Relation of b1,ExtREAL
for b5 being Element of b2
      st b3 is real-valued & b4 is real-valued & b3 is_measurable_on b5 & b4 is_measurable_on b5
   holds b3 + b4 is_measurable_on b5;

:: MESFUNC2:th 9
theorem
for b1 being non empty set
for b2, b3 being Function-like Relation of b1,ExtREAL holds
b2 - b3 = b2 + - b3;

:: MESFUNC2:th 11
theorem
for b1 being non empty set
for b2 being Function-like Relation of b1,ExtREAL holds
   - b2 = (- 1) (#) b2;

:: MESFUNC2:th 12
theorem
for b1 being non empty set
for b2 being Function-like Relation of b1,ExtREAL
for b3 being Element of REAL
      st b2 is real-valued
   holds b3 (#) b2 is real-valued;

:: MESFUNC2: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, b4 being Function-like Relation of b1,ExtREAL
for b5 being Element of b2
      st b3 is real-valued & b4 is real-valued & b3 is_measurable_on b5 & b4 is_measurable_on b5 & b5 c= dom b4
   holds b3 - b4 is_measurable_on b5;

:: MESFUNC2:funcnot 1 => MESFUNC2:func 1
definition
  let a1 be non empty set;
  let a2 be Function-like Relation of a1,ExtREAL;
  func max+ A2 -> Function-like Relation of a1,ExtREAL means
    dom it = dom a2 &
     (for b1 being Element of a1
           st b1 in dom it
        holds it . b1 = max(a2 . b1,0.));
end;

:: MESFUNC2:def 2
theorem
for b1 being non empty set
for b2, b3 being Function-like Relation of b1,ExtREAL holds
   b3 = max+ b2
iff
   dom b3 = dom b2 &
    (for b4 being Element of b1
          st b4 in dom b3
       holds b3 . b4 = max(b2 . b4,0.));

:: MESFUNC2:funcnot 2 => MESFUNC2:func 2
definition
  let a1 be non empty set;
  let a2 be Function-like Relation of a1,ExtREAL;
  func max- A2 -> Function-like Relation of a1,ExtREAL means
    dom it = dom a2 &
     (for b1 being Element of a1
           st b1 in dom it
        holds it . b1 = max(- (a2 . b1),0.));
end;

:: MESFUNC2:def 3
theorem
for b1 being non empty set
for b2, b3 being Function-like Relation of b1,ExtREAL holds
   b3 = max- b2
iff
   dom b3 = dom b2 &
    (for b4 being Element of b1
          st b4 in dom b3
       holds b3 . b4 = max(- (b2 . b4),0.));

:: MESFUNC2:th 14
theorem
for b1 being non empty set
for b2 being Function-like Relation of b1,ExtREAL
for b3 being Element of b1 holds
   0. <= (max+ b2) . b3;

:: MESFUNC2:th 15
theorem
for b1 being non empty set
for b2 being Function-like Relation of b1,ExtREAL
for b3 being Element of b1 holds
   0. <= (max- b2) . b3;

:: MESFUNC2:th 16
theorem
for b1 being non empty set
for b2 being Function-like Relation of b1,ExtREAL holds
   max- b2 = max+ - b2;

:: MESFUNC2:th 17
theorem
for b1 being non empty set
for b2 being Function-like Relation of b1,ExtREAL
for b3 being Element of b1
      st 0. < (max+ b2) . b3
   holds (max- b2) . b3 = 0.;

:: MESFUNC2:th 18
theorem
for b1 being non empty set
for b2 being Function-like Relation of b1,ExtREAL
for b3 being Element of b1
      st 0. < (max- b2) . b3
   holds (max+ b2) . b3 = 0.;

:: MESFUNC2:th 19
theorem
for b1 being non empty set
for b2 being Function-like Relation of b1,ExtREAL holds
   dom b2 = dom ((max+ b2) - max- b2) &
    dom b2 = dom ((max+ b2) + max- b2);

:: MESFUNC2:th 20
theorem
for b1 being non empty set
for b2 being Function-like Relation of b1,ExtREAL
for b3 being Element of b1 holds
   ((max+ b2) . b3 = b2 . b3 or (max+ b2) . b3 = 0.) &
    ((max- b2) . b3 = - (b2 . b3) or (max- b2) . b3 = 0.);

:: MESFUNC2:th 21
theorem
for b1 being non empty set
for b2 being Function-like Relation of b1,ExtREAL
for b3 being Element of b1
      st (max+ b2) . b3 = b2 . b3
   holds (max- b2) . b3 = 0.;

:: MESFUNC2:th 22
theorem
for b1 being non empty set
for b2 being Function-like Relation of b1,ExtREAL
for b3 being Element of b1
      st b3 in dom b2 & (max+ b2) . b3 = 0.
   holds (max- b2) . b3 = - (b2 . b3);

:: MESFUNC2:th 23
theorem
for b1 being non empty set
for b2 being Function-like Relation of b1,ExtREAL
for b3 being Element of b1
      st (max- b2) . b3 = - (b2 . b3)
   holds (max+ b2) . b3 = 0.;

:: MESFUNC2:th 24
theorem
for b1 being non empty set
for b2 being Function-like Relation of b1,ExtREAL
for b3 being Element of b1
      st b3 in dom b2 & (max- b2) . b3 = 0.
   holds (max+ b2) . b3 = b2 . b3;

:: MESFUNC2:th 25
theorem
for b1 being non empty set
for b2 being Function-like Relation of b1,ExtREAL holds
   b2 = (max+ b2) - max- b2;

:: MESFUNC2:th 26
theorem
for b1 being non empty set
for b2 being Function-like Relation of b1,ExtREAL holds
   |.b2.| = (max+ b2) + max- b2;

:: MESFUNC2:th 27
theorem
for b1 being non empty set
for b2 being Function-like Relation of b1,ExtREAL
for b3 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b4 being Element of b3
      st b2 is_measurable_on b4
   holds max+ b2 is_measurable_on b4;

:: MESFUNC2:th 28
theorem
for b1 being non empty set
for b2 being Function-like Relation of b1,ExtREAL
for b3 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b4 being Element of b3
      st b2 is_measurable_on b4 & b4 c= dom b2
   holds max- b2 is_measurable_on b4;

:: MESFUNC2:th 29
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 Element of b2
      st b3 is_measurable_on b4 & b4 c= dom b3
   holds |.b3.| is_measurable_on b4;

:: MESFUNC2:funcnot 3 => MESFUNC2:func 3
definition
  let a1, a2 be set;
  redefine func chi(a1,a2) -> Function-like Relation of a2,ExtREAL;
end;

:: MESFUNC2:th 31
theorem
for b1 being non empty set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being Element of b2 holds
   chi(b3,b1) is real-valued;

:: MESFUNC2:th 32
theorem
for b1 being non empty set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3, b4 being Element of b2 holds
chi(b3,b1) is_measurable_on b4;

:: MESFUNC2:exreg 1
registration
  let a1 be set;
  let a2 be non empty compl-closed sigma-multiplicative Element of bool bool a1;
  cluster Relation-like Function-like finite FinSequence-like disjoint_valued FinSequence of a2;
end;

:: MESFUNC2:modenot 1
definition
  let a1 be set;
  let a2 be non empty compl-closed sigma-multiplicative Element of bool bool a1;
  mode Finite_Sep_Sequence of a2 is disjoint_valued FinSequence of a2;
end;

:: MESFUNC2:th 33
theorem
for b1 being non empty set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being Relation-like Function-like set
      st b3 is disjoint_valued FinSequence of b2
   holds ex b4 being Function-like quasi_total disjoint_valued Relation of NAT,b2 st
      union proj2 b3 = union rng b4 &
       (for b5 being natural set
             st b5 in proj1 b3
          holds b3 . b5 = b4 . b5) &
       (for b5 being natural set
             st not b5 in proj1 b3
          holds b4 . b5 = {});

:: MESFUNC2:th 34
theorem
for b1 being non empty set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being Relation-like Function-like set
      st b3 is disjoint_valued FinSequence of b2
   holds union proj2 b3 in b2;

:: MESFUNC2:prednot 1 => MESFUNC2: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;
  pred A3 is_simple_func_in A2 means
    a3 is real-valued &
     (ex b1 being disjoint_valued FinSequence of a2 st
        dom a3 = union rng b1 &
         (for b2 being natural set
         for b3, b4 being Element of a1
               st b2 in dom b1 & b3 in b1 . b2 & b4 in b1 . b2
            holds a3 . b3 = a3 . b4));
end;

:: MESFUNC2:dfs 4
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;
To prove
     a3 is_simple_func_in a2
it is sufficient to prove
  thus a3 is real-valued &
     (ex b1 being disjoint_valued FinSequence of a2 st
        dom a3 = union rng b1 &
         (for b2 being natural set
         for b3, b4 being Element of a1
               st b2 in dom b1 & b3 in b1 . b2 & b4 in b1 . b2
            holds a3 . b3 = a3 . b4));

:: MESFUNC2:def 5
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 holds
      b3 is_simple_func_in b2
   iff
      b3 is real-valued &
       (ex b4 being disjoint_valued FinSequence of b2 st
          dom b3 = union rng b4 &
           (for b5 being natural set
           for b6, b7 being Element of b1
                 st b5 in dom b4 & b6 in b4 . b5 & b7 in b4 . b5
              holds b3 . b6 = b3 . b7));

:: MESFUNC2:th 35
theorem
for b1 being non empty set
for b2 being Function-like Relation of b1,ExtREAL
      st b2 is real-valued
   holds rng b2 is Element of bool REAL;

:: MESFUNC2:th 36
theorem
for b1 being non empty set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being natural set
for b4 being Relation-like set
      st b4 is disjoint_valued FinSequence of b2
   holds b4 | Seg b3 is disjoint_valued FinSequence of b2;

:: MESFUNC2:th 37
theorem
for b1 being non empty set
for b2 being Function-like Relation of b1,ExtREAL
for b3 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b4 being Element of b3
      st b2 is_simple_func_in b3
   holds b2 is_measurable_on b4;