Article MESFUNC8, MML version 4.99.1005

:: MESFUNC8:th 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 sigma_Measure of b2
for b4 being Function-like quasi_total Relation of NAT,b2
for b5 being natural set holds
   {b6 where b6 is Element of b1: for b7 being natural set
         st b5 <= b7
      holds b6 in b4 . b7} is Element of b2;

:: MESFUNC8:th 2
theorem
for b1 being non empty set
for b2 being Function-like quasi_total Relation of NAT,bool b1
for b3 being Element of NAT holds
   (superior_setsequence b2) . b3 = union rng (b2 ^\ b3) &
    (inferior_setsequence b2) . b3 = meet rng (b2 ^\ b3);

:: MESFUNC8: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 being SetSequence of b2 holds
   ex b5 being Function-like quasi_total Relation of NAT,b2 st
      b5 = @inferior_setsequence b4 &
       b3 . lim_inf b4 = sup rng (b3 * b5);

:: MESFUNC8: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 SetSequence of b2
      st b3 . Union b4 < +infty
   holds ex b5 being Function-like quasi_total Relation of NAT,b2 st
      b5 = @superior_setsequence b4 &
       b3 . lim_sup b4 = inf rng (b3 * b5);

:: MESFUNC8:th 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 sigma_Measure of b2
for b4 being SetSequence of b2
      st b4 is convergent(b1, b2)
   holds ex b5 being Function-like quasi_total Relation of NAT,b2 st
      b5 = @inferior_setsequence b4 &
       b3 . lim b4 = sup rng (b3 * b5);

:: MESFUNC8:th 6
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 SetSequence of b2
      st b4 is convergent(b1, b2) & b3 . Union b4 < +infty
   holds ex b5 being Function-like quasi_total Relation of NAT,b2 st
      b5 = @superior_setsequence b4 &
       b3 . lim b4 = inf rng (b3 * b5);

:: MESFUNC8:attrnot 1 => MESFUNC8:attr 1
definition
  let a1, a2 be set;
  let a3 be Functional_Sequence of a1,a2;
  attr a3 is with_the_same_dom means
    proj2 a3 is with_common_domain;
end;

:: MESFUNC8:dfs 1
definiens
  let a1, a2 be set;
  let a3 be Functional_Sequence of a1,a2;
To prove
     a3 is with_the_same_dom
it is sufficient to prove
  thus proj2 a3 is with_common_domain;

:: MESFUNC8:def 1
theorem
for b1, b2 being set
for b3 being Functional_Sequence of b1,b2 holds
      b3 is with_the_same_dom(b1, b2)
   iff
      proj2 b3 is with_common_domain;

:: MESFUNC8:attrnot 2 => MESFUNC8:attr 1
definition
  let a1, a2 be set;
  let a3 be Functional_Sequence of a1,a2;
  attr a3 is with_the_same_dom means
    for b1, b2 being natural set holds
    dom (a3 . b1) = dom (a3 . b2);
end;

:: MESFUNC8:dfs 2
definiens
  let a1, a2 be set;
  let a3 be Functional_Sequence of a1,a2;
To prove
     a3 is with_the_same_dom
it is sufficient to prove
  thus for b1, b2 being natural set holds
    dom (a3 . b1) = dom (a3 . b2);

:: MESFUNC8:def 2
theorem
for b1, b2 being set
for b3 being Functional_Sequence of b1,b2 holds
      b3 is with_the_same_dom(b1, b2)
   iff
      for b4, b5 being natural set holds
      dom (b3 . b4) = dom (b3 . b5);

:: MESFUNC8:exreg 1
registration
  let a1, a2 be set;
  cluster Relation-like Function-like with_the_same_dom Functional_Sequence of a1,a2;
end;

:: MESFUNC8:funcnot 1 => MESFUNC8:func 1
definition
  let a1 be non empty set;
  let a2 be Functional_Sequence of a1,ExtREAL;
  func inf A2 -> Function-like Relation of a1,ExtREAL means
    dom it = dom (a2 . 0) &
     (for b1 being Element of a1
           st b1 in dom it
        holds it . b1 = inf (a2 # b1));
end;

:: MESFUNC8:def 3
theorem
for b1 being non empty set
for b2 being Functional_Sequence of b1,ExtREAL
for b3 being Function-like Relation of b1,ExtREAL holds
      b3 = inf b2
   iff
      dom b3 = dom (b2 . 0) &
       (for b4 being Element of b1
             st b4 in dom b3
          holds b3 . b4 = inf (b2 # b4));

:: MESFUNC8:funcnot 2 => MESFUNC8:func 2
definition
  let a1 be non empty set;
  let a2 be Functional_Sequence of a1,ExtREAL;
  func sup A2 -> Function-like Relation of a1,ExtREAL means
    dom it = dom (a2 . 0) &
     (for b1 being Element of a1
           st b1 in dom it
        holds it . b1 = sup (a2 # b1));
end;

:: MESFUNC8:def 4
theorem
for b1 being non empty set
for b2 being Functional_Sequence of b1,ExtREAL
for b3 being Function-like Relation of b1,ExtREAL holds
      b3 = sup b2
   iff
      dom b3 = dom (b2 . 0) &
       (for b4 being Element of b1
             st b4 in dom b3
          holds b3 . b4 = sup (b2 # b4));

:: MESFUNC8:funcnot 3 => MESFUNC8:func 3
definition
  let a1 be non empty set;
  let a2 be Functional_Sequence of a1,ExtREAL;
  func inferior_realsequence A2 -> with_the_same_dom Functional_Sequence of a1,ExtREAL means
    for b1 being natural set holds
       dom (it . b1) = dom (a2 . 0) &
        (for b2 being Element of a1
              st b2 in dom (it . b1)
           holds (it . b1) . b2 = (inferior_realsequence (a2 # b2)) . b1);
end;

:: MESFUNC8:def 5
theorem
for b1 being non empty set
for b2 being Functional_Sequence of b1,ExtREAL
for b3 being with_the_same_dom Functional_Sequence of b1,ExtREAL holds
      b3 = inferior_realsequence b2
   iff
      for b4 being natural set holds
         dom (b3 . b4) = dom (b2 . 0) &
          (for b5 being Element of b1
                st b5 in dom (b3 . b4)
             holds (b3 . b4) . b5 = (inferior_realsequence (b2 # b5)) . b4);

:: MESFUNC8:funcnot 4 => MESFUNC8:func 4
definition
  let a1 be non empty set;
  let a2 be Functional_Sequence of a1,ExtREAL;
  func superior_realsequence A2 -> with_the_same_dom Functional_Sequence of a1,ExtREAL means
    for b1 being natural set holds
       dom (it . b1) = dom (a2 . 0) &
        (for b2 being Element of a1
              st b2 in dom (it . b1)
           holds (it . b1) . b2 = (superior_realsequence (a2 # b2)) . b1);
end;

:: MESFUNC8:def 6
theorem
for b1 being non empty set
for b2 being Functional_Sequence of b1,ExtREAL
for b3 being with_the_same_dom Functional_Sequence of b1,ExtREAL holds
      b3 = superior_realsequence b2
   iff
      for b4 being natural set holds
         dom (b3 . b4) = dom (b2 . 0) &
          (for b5 being Element of b1
                st b5 in dom (b3 . b4)
             holds (b3 . b4) . b5 = (superior_realsequence (b2 # b5)) . b4);

:: MESFUNC8:th 7
theorem
for b1 being non empty set
for b2 being Functional_Sequence of b1,ExtREAL
for b3 being Element of b1
      st b3 in dom (b2 . 0)
   holds (inferior_realsequence b2) # b3 = inferior_realsequence (b2 # b3);

:: MESFUNC8:modenot 1 => MESFUNC8:mode 1
definition
  let a1, a2 be set;
  redefine mode Functional_Sequence of a1,a2 -> Function-like quasi_total Relation of NAT,PFuncs(a1,a2);
end;

:: MESFUNC8:funcnot 5 => MESFUNC8:func 5
definition
  let a1, a2 be set;
  let a3 be Functional_Sequence of a1,a2;
  let a4 be Element of NAT;
  func A3 ^\ A4 -> Functional_Sequence of a1,a2 equals
    a3 ^\ a4;
end;

:: MESFUNC8:def 7
theorem
for b1, b2 being set
for b3 being Functional_Sequence of b1,b2
for b4 being Element of NAT holds
   b3 ^\ b4 = b3 ^\ b4;

:: MESFUNC8:funcreg 1
registration
  let a1, a2 be set;
  let a3 be with_the_same_dom Functional_Sequence of a1,a2;
  let a4 be Element of NAT;
  cluster a3 ^\ a4 -> with_the_same_dom;
end;

:: MESFUNC8:th 8
theorem
for b1 being non empty set
for b2 being with_the_same_dom Functional_Sequence of b1,ExtREAL
for b3 being Element of NAT holds
   (inferior_realsequence b2) . b3 = inf (b2 ^\ b3);

:: MESFUNC8:th 9
theorem
for b1 being non empty set
for b2 being with_the_same_dom Functional_Sequence of b1,ExtREAL
for b3 being Element of NAT holds
   (superior_realsequence b2) . b3 = sup (b2 ^\ b3);

:: MESFUNC8:th 10
theorem
for b1 being non empty set
for b2 being Functional_Sequence of b1,ExtREAL
for b3 being Element of b1
      st b3 in dom (b2 . 0)
   holds (superior_realsequence b2) # b3 = superior_realsequence (b2 # b3);

:: MESFUNC8:funcnot 6 => MESFUNC8:func 6
definition
  let a1 be non empty set;
  let a2 be Functional_Sequence of a1,ExtREAL;
  func lim_inf A2 -> Function-like Relation of a1,ExtREAL means
    dom it = dom (a2 . 0) &
     (for b1 being Element of a1
           st b1 in dom it
        holds it . b1 = lim_inf (a2 # b1));
end;

:: MESFUNC8:def 8
theorem
for b1 being non empty set
for b2 being Functional_Sequence of b1,ExtREAL
for b3 being Function-like Relation of b1,ExtREAL holds
      b3 = lim_inf b2
   iff
      dom b3 = dom (b2 . 0) &
       (for b4 being Element of b1
             st b4 in dom b3
          holds b3 . b4 = lim_inf (b2 # b4));

:: MESFUNC8:funcnot 7 => MESFUNC8:func 7
definition
  let a1 be non empty set;
  let a2 be Functional_Sequence of a1,ExtREAL;
  func lim_sup A2 -> Function-like Relation of a1,ExtREAL means
    dom it = dom (a2 . 0) &
     (for b1 being Element of a1
           st b1 in dom it
        holds it . b1 = lim_sup (a2 # b1));
end;

:: MESFUNC8:def 9
theorem
for b1 being non empty set
for b2 being Functional_Sequence of b1,ExtREAL
for b3 being Function-like Relation of b1,ExtREAL holds
      b3 = lim_sup b2
   iff
      dom b3 = dom (b2 . 0) &
       (for b4 being Element of b1
             st b4 in dom b3
          holds b3 . b4 = lim_sup (b2 # b4));

:: MESFUNC8:th 11
theorem
for b1 being non empty set
for b2 being Functional_Sequence of b1,ExtREAL holds
   (for b3 being Element of b1
          st b3 in dom lim_inf b2
       holds (lim_inf b2) . b3 = sup inferior_realsequence (b2 # b3) &
        (lim_inf b2) . b3 = sup ((inferior_realsequence b2) # b3) &
        (lim_inf b2) . b3 = (sup inferior_realsequence b2) . b3) &
    lim_inf b2 = sup inferior_realsequence b2;

:: MESFUNC8:th 12
theorem
for b1 being non empty set
for b2 being Functional_Sequence of b1,ExtREAL holds
   (for b3 being Element of b1
          st b3 in dom lim_sup b2
       holds (lim_sup b2) . b3 = inf superior_realsequence (b2 # b3) &
        (lim_sup b2) . b3 = inf ((superior_realsequence b2) # b3) &
        (lim_sup b2) . b3 = (inf superior_realsequence b2) . b3) &
    lim_sup b2 = inf superior_realsequence b2;

:: MESFUNC8:th 13
theorem
for b1 being non empty set
for b2 being Functional_Sequence of b1,ExtREAL
for b3 being Element of b1
      st b3 in dom (b2 . 0)
   holds    b2 # b3 is convergent
   iff
      (lim_sup b2) . b3 = (lim_inf b2) . b3;

:: MESFUNC8:funcnot 8 => MESFUNC8:func 8
definition
  let a1 be non empty set;
  let a2 be Functional_Sequence of a1,ExtREAL;
  func lim A2 -> Function-like Relation of a1,ExtREAL means
    dom it = dom (a2 . 0) &
     (for b1 being Element of a1
           st b1 in dom it
        holds it . b1 = lim (a2 # b1));
end;

:: MESFUNC8:def 10
theorem
for b1 being non empty set
for b2 being Functional_Sequence of b1,ExtREAL
for b3 being Function-like Relation of b1,ExtREAL holds
      b3 = lim b2
   iff
      dom b3 = dom (b2 . 0) &
       (for b4 being Element of b1
             st b4 in dom b3
          holds b3 . b4 = lim (b2 # b4));

:: MESFUNC8:th 14
theorem
for b1 being non empty set
for b2 being Functional_Sequence of b1,ExtREAL
for b3 being Element of b1
      st b3 in dom lim b2 & b2 # b3 is convergent
   holds (lim b2) . b3 = (lim_sup b2) . b3 & (lim b2) . b3 = (lim_inf b2) . b3;

:: MESFUNC8: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 with_the_same_dom Functional_Sequence of b1,ExtREAL
for b4 being SetSequence of b2
for b5 being real set
      st for b6 being natural set holds
           b4 . b6 = (dom (b3 . 0)) /\ great_dom(b3 . b6,R_EAL b5)
   holds union rng b4 = (dom (b3 . 0)) /\ great_dom(sup b3,R_EAL b5);

:: MESFUNC8:th 16
theorem
for b1 being non empty set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being with_the_same_dom Functional_Sequence of b1,ExtREAL
for b4 being SetSequence of b2
for b5 being real set
      st for b6 being natural set holds
           b4 . b6 = (dom (b3 . 0)) /\ great_eq_dom(b3 . b6,R_EAL b5)
   holds meet rng b4 = (dom (b3 . 0)) /\ great_eq_dom(inf b3,R_EAL b5);

:: MESFUNC8: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 with_the_same_dom Functional_Sequence of b1,ExtREAL
for b4 being SetSequence of b2
for b5 being real set
   st for b6 being natural set holds
        b4 . b6 = (dom (b3 . 0)) /\ great_dom(b3 . b6,R_EAL b5)
for b6 being natural set holds
   (superior_setsequence b4) . b6 = (dom (b3 . 0)) /\ great_dom((superior_realsequence b3) . b6,R_EAL b5);

:: MESFUNC8:th 18
theorem
for b1 being non empty set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being with_the_same_dom Functional_Sequence of b1,ExtREAL
for b4 being SetSequence of b2
for b5 being real set
   st for b6 being natural set holds
        b4 . b6 = (dom (b3 . 0)) /\ great_eq_dom(b3 . b6,R_EAL b5)
for b6 being natural set holds
   (inferior_setsequence b4) . b6 = (dom (b3 . 0)) /\ great_eq_dom((inferior_realsequence b3) . b6,R_EAL b5);

:: MESFUNC8:th 19
theorem
for b1 being non empty set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being with_the_same_dom Functional_Sequence of b1,ExtREAL
for b4 being Element of b2
   st dom (b3 . 0) = b4 &
      (for b5 being natural set holds
         b3 . b5 is_measurable_on b4)
for b5 being natural set holds
   (superior_realsequence b3) . b5 is_measurable_on b4;

:: MESFUNC8:th 20
theorem
for b1 being non empty set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being with_the_same_dom Functional_Sequence of b1,ExtREAL
for b4 being Element of b2
   st dom (b3 . 0) = b4 &
      (for b5 being natural set holds
         b3 . b5 is_measurable_on b4)
for b5 being natural set holds
   (inferior_realsequence b3) . b5 is_measurable_on b4;

:: MESFUNC8:th 21
theorem
for b1 being non empty set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being Functional_Sequence of b1,ExtREAL
for b4 being SetSequence of b2
for b5 being real set
      st for b6 being natural set holds
           b4 . b6 = (dom (b3 . 0)) /\ great_eq_dom((superior_realsequence b3) . b6,R_EAL b5)
   holds meet b4 = (dom (b3 . 0)) /\ great_eq_dom(lim_sup b3,R_EAL b5);

:: MESFUNC8: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 Functional_Sequence of b1,ExtREAL
for b4 being SetSequence of b2
for b5 being real set
      st for b6 being natural set holds
           b4 . b6 = (dom (b3 . 0)) /\ great_dom((inferior_realsequence b3) . b6,R_EAL b5)
   holds union rng b4 = (dom (b3 . 0)) /\ great_dom(lim_inf b3,R_EAL b5);

:: MESFUNC8:th 23
theorem
for b1 being non empty set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being with_the_same_dom Functional_Sequence of b1,ExtREAL
for b4 being Element of b2
      st dom (b3 . 0) = b4 &
         (for b5 being natural set holds
            b3 . b5 is_measurable_on b4)
   holds lim_sup b3 is_measurable_on b4;

:: MESFUNC8: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 with_the_same_dom Functional_Sequence of b1,ExtREAL
for b4 being Element of b2
      st dom (b3 . 0) = b4 &
         (for b5 being natural set holds
            b3 . b5 is_measurable_on b4)
   holds lim_inf b3 is_measurable_on b4;

:: MESFUNC8: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 with_the_same_dom Functional_Sequence of b1,ExtREAL
for b4 being Element of b2
      st dom (b3 . 0) = b4 &
         (for b5 being natural set holds
            b3 . b5 is_measurable_on b4) &
         (for b5 being Element of b1
               st b5 in b4
            holds b3 # b5 is convergent)
   holds lim b3 is_measurable_on b4;

:: MESFUNC8: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 with_the_same_dom Functional_Sequence of b1,ExtREAL
for b4 being Function-like Relation of b1,ExtREAL
for b5 being Element of b2
      st dom (b3 . 0) = b5 &
         (for b6 being natural set holds
            b3 . b6 is_measurable_on b5) &
         dom b4 = b5 &
         (for b6 being Element of b1
               st b6 in b5
            holds b3 # b6 is convergent & b4 . b6 = lim (b3 # b6))
   holds b4 is_measurable_on b5;

:: MESFUNC8:th 27
theorem
for b1 being non empty set
for b2 being Functional_Sequence of b1,ExtREAL
for b3 being Function-like Relation of b1,ExtREAL
      st for b4 being Element of b1
              st b4 in dom b3
           holds b2 # b4 is convergent_to_finite_number & b3 . b4 = lim (b2 # b4)
   holds b3 is real-valued;

:: MESFUNC8:th 28
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 with_the_same_dom Functional_Sequence of b1,ExtREAL
for b5 being Function-like Relation of b1,ExtREAL
for b6 being Element of b2
   st b3 . b6 < +infty &
      dom (b4 . 0) = b6 &
      (for b7 being natural set holds
         b4 . b7 is_measurable_on b6 & b4 . b7 is real-valued) &
      dom b5 = b6 &
      (for b7 being Element of b1
            st b7 in b6
         holds b4 # b7 is convergent_to_finite_number & b5 . b7 = lim (b4 # b7))
for b7, b8 being real set
      st 0 < b7 & 0 < b8
   holds ex b9 being Element of b2 st
      ex b10 being natural set st
         b9 c= b6 &
          b3 . b9 < b7 &
          (for b11 being natural set
             st b10 < b11
          for b12 being Element of b1
                st b12 in b6 \ b9
             holds |.((b4 . b11) . b12) - (b5 . b12).| < b8);

:: MESFUNC8:th 29
theorem
for b1, b2 being non empty set
for b3 being set
for b4, b5 being Function-like quasi_total Relation of b1,b2
      st for b6 being Element of b1 holds
           b5 . b6 = b3 \ (b4 . b6)
   holds union rng b5 = b3 \ meet rng b4;

:: MESFUNC8:th 30
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 with_the_same_dom Functional_Sequence of b1,ExtREAL
for b5 being Function-like Relation of b1,ExtREAL
for b6 being Element of b2
   st dom (b4 . 0) = b6 &
      (for b7 being natural set holds
         b4 . b7 is_measurable_on b6) &
      b3 . b6 < +infty &
      (for b7 being natural set holds
         ex b8 being Element of b2 st
            b8 c= b6 &
             b3 . (b6 \ b8) = 0 &
             (for b9 being Element of b1
                   st b9 in b8
                holds |.(b4 . b7) . b9.| < +infty)) &
      (ex b7 being Element of b2 st
         b7 c= b6 &
          b3 . (b6 \ b7) = 0 &
          (for b8 being Element of b1
                st b8 in b6
             holds b4 # b8 is convergent_to_finite_number) &
          dom b5 = b6 &
          (for b8 being Element of b1
                st b8 in b7
             holds b5 . b8 = lim (b4 # b8)))
for b7 being real set
      st 0 < b7
   holds ex b8 being Element of b2 st
      b8 c= b6 &
       b3 . (b6 \ b8) <= b7 &
       (for b9 being real set
             st 0 < b9
          holds ex b10 being natural set st
             for b11 being natural set
                st b10 < b11
             for b12 being Element of b1
                   st b12 in b8
                holds |.((b4 . b11) . b12) - (b5 . b12).| < b9);