Article MESFUNC7, MML version 4.99.1005

:: MESFUNC7:th 1
theorem
for b1 being non empty set
for b2, b3 being Function-like Relation of b1,ExtREAL
      st for b4 being Element of b1
              st b4 in dom b2
           holds b2 . b4 <= b3 . b4
   holds b3 - b2 is nonnegative;

:: MESFUNC7:th 2
theorem
for b1 being non empty set
for b2 being set
for b3 being Function-like Relation of b1,ExtREAL
for b4 being Element of REAL holds
   (b4 (#) b3) | b2 = b4 (#) (b3 | b2);

:: MESFUNC7: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 sigma_Measure of b2
for b4, b5 being Function-like Relation of b1,ExtREAL
      st b4 is_integrable_on b3 & b5 is_integrable_on b3 & b5 - b4 is nonnegative
   holds ex b6 being Element of b2 st
      b6 = (dom b4) /\ dom b5 &
       Integral(b3,b4 | b6) <= Integral(b3,b5 | b6);

:: MESFUNC7:exreg 1
registration
  let a1 be non empty set;
  cluster Relation-like Function-like ext-real-valued nonnegative Relation of a1,ExtREAL;
end;

:: MESFUNC7:funcnot 1 => MESFUNC7:func 1
definition
  let a1 be non empty set;
  let a2 be Function-like Relation of a1,ExtREAL;
  redefine func |.a2.| -> Function-like nonnegative Relation of a1,ExtREAL;
end;

:: MESFUNC7:th 4
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_integrable_on b3
   holds ex b5 being Function-like quasi_total Relation of NAT,b2 st
      (for b6 being Element of NAT holds
          b5 . b6 = (dom b4) /\ great_eq_dom(|.b4.|,R_EAL (1 / (b6 + 1)))) &
       (dom b4) \ eq_dom(b4,0.) = union rng b5 &
       (for b6 being Element of NAT holds
          b5 . b6 in b2 & b3 . (b5 . b6) < +infty);

:: MESFUNC7:attrnot 1 => VALUED_0:attr 2
notation
  let a1 be Relation-like set;
  synonym extreal-yielding for ext-real-valued;
end;

:: MESFUNC7:funcnot 2 => MESFUNC7:func 2
definition
  let a1 be natural set;
  let a2 be Element of ExtREAL;
  redefine func a1 |-> a2 -> FinSequence of ExtREAL;
end;

:: MESFUNC7:exreg 2
registration
  cluster Relation-like Function-like finite FinSequence-like ext-real-valued set;
end;

:: MESFUNC7:funcnot 3 => MESFUNC7:func 3
definition
  func multextreal -> Function-like quasi_total Relation of [:ExtREAL,ExtREAL:],ExtREAL means
    for b1, b2 being Element of ExtREAL holds
    it .(b1,b2) = b1 * b2;
end;

:: MESFUNC7:def 2
theorem
for b1 being Function-like quasi_total Relation of [:ExtREAL,ExtREAL:],ExtREAL holds
      b1 = multextreal
   iff
      for b2, b3 being Element of ExtREAL holds
      b1 .(b2,b3) = b2 * b3;

:: MESFUNC7:funcreg 1
registration
  cluster multextreal -> Function-like quasi_total commutative associative;
end;

:: MESFUNC7:th 5
theorem
the_unity_wrt multextreal = 1;

:: MESFUNC7:funcreg 2
registration
  cluster multextreal -> Function-like quasi_total having_a_unity;
end;

:: MESFUNC7:funcnot 4 => MESFUNC7:func 4
definition
  let a1 be Relation-like Function-like FinSequence-like ext-real-valued set;
  func Product A1 -> Element of ExtREAL means
    ex b1 being FinSequence of ExtREAL st
       b1 = a1 & it = multextreal "**" b1;
end;

:: MESFUNC7:def 3
theorem
for b1 being Relation-like Function-like FinSequence-like ext-real-valued set
for b2 being Element of ExtREAL holds
      b2 = Product b1
   iff
      ex b3 being FinSequence of ExtREAL st
         b3 = b1 & b2 = multextreal "**" b3;

:: MESFUNC7:funcreg 3
registration
  let a1 be Element of ExtREAL;
  let a2 be natural set;
  cluster a2 |-> a1 -> Relation-like Function-like FinSequence-like ext-real-valued;
end;

:: MESFUNC7:funcnot 5 => MESFUNC7:func 5
definition
  let a1 be Element of ExtREAL;
  let a2 be natural set;
  func A1 |^ A2 -> set equals
    Product (a2 |-> a1);
end;

:: MESFUNC7:def 4
theorem
for b1 being Element of ExtREAL
for b2 being natural set holds
   b1 |^ b2 = Product (b2 |-> b1);

:: MESFUNC7:funcnot 6 => MESFUNC7:func 6
definition
  let a1 be Element of ExtREAL;
  let a2 be natural set;
  redefine func a1 |^ a2 -> Element of ExtREAL;
end;

:: MESFUNC7:funcreg 4
registration
  cluster <*> ExtREAL -> ext-real-valued;
end;

:: MESFUNC7:funcreg 5
registration
  let a1 be Element of ExtREAL;
  cluster <*a1*> -> ext-real-valued;
end;

:: MESFUNC7:th 6
theorem
Product <*> ExtREAL = 1;

:: MESFUNC7:th 7
theorem
for b1 being Element of ExtREAL holds
   Product <*b1*> = b1;

:: MESFUNC7:funcreg 6
registration
  let a1, a2 be Relation-like Function-like FinSequence-like ext-real-valued set;
  cluster a1 ^ a2 -> Relation-like Function-like FinSequence-like ext-real-valued;
end;

:: MESFUNC7:th 8
theorem
for b1 being Relation-like Function-like FinSequence-like ext-real-valued set
for b2 being Element of ExtREAL holds
   Product (b1 ^ <*b2*>) = (Product b1) * b2;

:: MESFUNC7:th 9
theorem
for b1 being Element of ExtREAL holds
   b1 |^ 1 = b1;

:: MESFUNC7:th 10
theorem
for b1 being Element of ExtREAL
for b2 being natural set holds
   b1 |^ (b2 + 1) = (b1 |^ b2) * b1;

:: MESFUNC7:funcnot 7 => MESFUNC7:func 7
definition
  let a1 be natural set;
  let a2 be non empty set;
  let a3 be Function-like Relation of a2,ExtREAL;
  func A3 |^ A1 -> Function-like Relation of a2,ExtREAL means
    dom it = dom a3 &
     (for b1 being Element of a2
           st b1 in dom it
        holds it . b1 = (a3 . b1) |^ a1);
end;

:: MESFUNC7:def 5
theorem
for b1 being natural set
for b2 being non empty set
for b3, b4 being Function-like Relation of b2,ExtREAL holds
   b4 = b3 |^ b1
iff
   dom b4 = dom b3 &
    (for b5 being Element of b2
          st b5 in dom b4
       holds b4 . b5 = (b3 . b5) |^ b1);

:: MESFUNC7:th 11
theorem
for b1 being Element of ExtREAL
for b2 being real set
for b3 being natural set
      st b1 = b2
   holds b1 |^ b3 = b2 |^ b3;

:: MESFUNC7:th 12
theorem
for b1 being Element of ExtREAL
for b2 being natural set
      st 0 <= b1
   holds 0 <= b1 |^ b2;

:: MESFUNC7:th 13
theorem
for b1 being natural set
      st 1 <= b1
   holds +infty |^ b1 = +infty;

:: MESFUNC7:th 14
theorem
for b1 being natural set
for b2 being non empty set
for b3 being non empty compl-closed sigma-multiplicative Element of bool bool b2
for b4 being Function-like Relation of b2,ExtREAL
for b5 being Element of b3
      st b5 c= dom b4 & b4 is_measurable_on b5
   holds |.b4.| |^ b1 is_measurable_on b5;

:: MESFUNC7: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, b4 being Function-like Relation of b1,ExtREAL
for b5 being Element of b2
      st (dom b3) /\ dom b4 = b5 & 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;

:: MESFUNC7:th 16
theorem
for b1 being non empty set
for b2 being Function-like Relation of b1,ExtREAL
      st rng b2 is bounded
   holds b2 is real-valued;

:: MESFUNC7:th 17
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, b5 being Function-like Relation of b1,ExtREAL
for b6 being Element of b2
for b7 being non empty Element of bool ExtREAL
      st (dom b4) /\ dom b5 = b6 & rng b4 = b7 & b5 is real-valued & b4 is_measurable_on b6 & rng b4 is bounded & b5 is_integrable_on b3
   holds (b4 (#) b5) | b6 is_integrable_on b3 &
    (ex b8 being Element of REAL st
       inf b7 <= b8 &
        b8 <= sup b7 &
        Integral(b3,(b4 (#) |.b5.|) | b6) = (R_EAL b8) * Integral(b3,|.b5.| | b6));

:: MESFUNC7:th 18
theorem
for b1 being non empty set
for b2 being Function-like Relation of b1,ExtREAL
for b3 being set holds
   |.b2.| | b3 = |.b2 | b3.|;

:: MESFUNC7:th 19
theorem
for b1 being non empty set
for b2, b3 being Function-like Relation of b1,ExtREAL holds
dom (|.b2.| + |.b3.|) = (dom b2) /\ dom b3 &
 dom |.b2 + b3.| c= dom |.b2.|;

:: MESFUNC7:th 20
theorem
for b1 being non empty set
for b2, b3 being Function-like Relation of b1,ExtREAL holds
(|.b2.| | dom |.b2 + b3.|) + (|.b3.| | dom |.b2 + b3.|) = (|.b2.| + |.b3.|) | dom |.b2 + b3.|;

:: MESFUNC7:th 21
theorem
for b1 being non empty set
for b2, b3 being Function-like Relation of b1,ExtREAL
for b4 being set
      st b4 in dom |.b2 + b3.|
   holds |.b2 + b3.| . b4 <= (|.b2.| + |.b3.|) . b4;

:: MESFUNC7:th 22
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, b5 being Function-like Relation of b1,ExtREAL
      st b4 is_integrable_on b3 & b5 is_integrable_on b3
   holds ex b6 being Element of b2 st
      b6 = dom (b4 + b5) &
       Integral(b3,|.b4 + b5.| | b6) <= (Integral(b3,|.b4.| | b6)) + Integral(b3,|.b5.| | b6);

:: MESFUNC7:th 23
theorem
for b1 being non empty set
for b2 being set holds
   max+ chi(b2,b1) = chi(b2,b1);

:: MESFUNC7:th 24
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 Element of b2
      st b3 . b4 < +infty
   holds chi(b4,b1) is_integrable_on b3 &
    Integral(b3,chi(b4,b1)) = b3 . b4 &
    Integral(b3,(chi(b4,b1)) | b4) = b3 . b4;

:: MESFUNC7:th 25
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, b5 being Element of b2
      st b3 . (b4 /\ b5) < +infty
   holds Integral(b3,(chi(b4,b1)) | b5) = b3 . (b4 /\ b5);

:: MESFUNC7:th 26
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
for b5 being Element of b2
for b6, b7 being real set
      st b4 is_integrable_on b3 &
         b5 c= dom b4 &
         b3 . b5 < +infty &
         (for b8 being Element of b1
               st b8 in b5
            holds b6 <= b4 . b8 & b4 . b8 <= b7)
   holds (R_EAL b6) * (b3 . b5) <= Integral(b3,b4 | b5) &
    Integral(b3,b4 | b5) <= (R_EAL b7) * (b3 . b5);