Article MEASURE7, MML version 4.99.1005

:: MEASURE7:th 1
theorem
for b1 being Function-like quasi_total Relation of NAT,ExtREAL
      st for b2 being Element of NAT holds
           b1 . b2 = 0.
   holds SUM b1 = 0.;

:: MEASURE7:th 2
theorem
for b1 being Function-like quasi_total Relation of NAT,ExtREAL
   st b1 is nonnegative
for b2 being Element of NAT holds
   b1 . b2 <= (Ser b1) . b2;

:: MEASURE7:th 3
theorem
for b1, b2, b3 being Function-like quasi_total Relation of NAT,ExtREAL
   st b2 is nonnegative &
      b3 is nonnegative &
      (for b4 being Element of NAT holds
         b1 . b4 = (b2 . b4) + (b3 . b4))
for b4 being Element of NAT holds
   (Ser b1) . b4 = ((Ser b2) . b4) + ((Ser b3) . b4);

:: MEASURE7:th 4
theorem
for b1, b2, b3 being Function-like quasi_total Relation of NAT,ExtREAL
      st (for b4 being Element of NAT holds
            b1 . b4 = (b2 . b4) + (b3 . b4)) &
         b2 is nonnegative &
         b3 is nonnegative
   holds SUM b1 <= (SUM b2) + SUM b3;

:: MEASURE7:th 6
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,ExtREAL
   st b1 is nonnegative &
      (for b3 being Element of NAT holds
         b1 . b3 <= b2 . b3)
for b3 being Element of NAT holds
   (Ser b1) . b3 <= SUM b2;

:: MEASURE7:th 7
theorem
for b1 being Function-like quasi_total Relation of NAT,ExtREAL
   st b1 is nonnegative
for b2 being Element of NAT holds
   (Ser b1) . b2 <= SUM b1;

:: MEASURE7:funcnot 1 => MEASURE7:func 1
definition
  let a1 be non empty set;
  let a2 be Function-like quasi_total Relation of a1,ExtREAL;
  func On A2 -> Function-like quasi_total Relation of NAT,ExtREAL means
    for b1 being Element of NAT holds
       (b1 in a1 implies it . b1 = a2 . b1) & (b1 in a1 or it . b1 = 0.);
end;

:: MEASURE7:def 1
theorem
for b1 being non empty set
for b2 being Function-like quasi_total Relation of b1,ExtREAL
for b3 being Function-like quasi_total Relation of NAT,ExtREAL holds
      b3 = On b2
   iff
      for b4 being Element of NAT holds
         (b4 in b1 implies b3 . b4 = b2 . b4) & (b4 in b1 or b3 . b4 = 0.);

:: MEASURE7:th 8
theorem
for b1 being non empty set
for b2 being Function-like quasi_total Relation of b1,ExtREAL
      st b2 is nonnegative
   holds On b2 is nonnegative;

:: MEASURE7:th 9
theorem
for b1 being Function-like quasi_total Relation of NAT,ExtREAL
   st b1 is nonnegative
for b2, b3 being Element of NAT
      st b2 <= b3
   holds (Ser b1) . b2 <= (Ser b1) . b3;

:: MEASURE7:th 10
theorem
for b1 being Element of NAT
for b2 being Function-like quasi_total Relation of NAT,ExtREAL
      st for b3 being Element of NAT
              st b3 <> b1
           holds b2 . b3 = 0.
   holds (for b3 being Element of NAT
          st b3 < b1
       holds (Ser b2) . b3 = 0.) &
    (for b3 being Element of NAT
          st b1 <= b3
       holds (Ser b2) . b3 = b2 . b1);

:: MEASURE7:th 11
theorem
for b1 being Function-like quasi_total Relation of NAT,ExtREAL
   st b1 is nonnegative
for b2 being non empty Element of bool NAT
for b3 being Function-like quasi_total Relation of b2,NAT
      st b3 is one-to-one
   holds SUM On (b1 * b3) <= SUM b1;

:: MEASURE7:th 12
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,ExtREAL
   st b1 is nonnegative & b2 is nonnegative
for b3 being non empty Element of bool NAT
for b4 being Function-like quasi_total Relation of b3,NAT
      st b4 is one-to-one &
         (for b5 being Element of NAT holds
            (b5 in b3 implies b1 . b5 = (b2 * b4) . b5) &
             (b5 in b3 or b1 . b5 = 0.))
   holds SUM b1 <= SUM b2;

:: MEASURE7:modenot 1 => MEASURE7:mode 1
definition
  let a1 be Element of bool REAL;
  mode Interval_Covering of A1 -> Function-like quasi_total Relation of NAT,bool REAL means
    a1 c= union rng it &
     (for b1 being Element of NAT holds
        it . b1 is interval Element of bool REAL);
end;

:: MEASURE7:dfs 2
definiens
  let a1 be Element of bool REAL;
  let a2 be Function-like quasi_total Relation of NAT,bool REAL;
To prove
     a2 is Interval_Covering of a1
it is sufficient to prove
  thus a1 c= union rng a2 &
     (for b1 being Element of NAT holds
        a2 . b1 is interval Element of bool REAL);

:: MEASURE7:def 2
theorem
for b1 being Element of bool REAL
for b2 being Function-like quasi_total Relation of NAT,bool REAL holds
      b2 is Interval_Covering of b1
   iff
      b1 c= union rng b2 &
       (for b3 being Element of NAT holds
          b2 . b3 is interval Element of bool REAL);

:: MEASURE7:funcnot 2 => MEASURE7:func 2
definition
  let a1 be Element of bool REAL;
  let a2 be Interval_Covering of a1;
  let a3 be Element of NAT;
  redefine func a2 . a3 -> interval Element of bool REAL;
end;

:: MEASURE7:modenot 2 => MEASURE7:mode 2
definition
  let a1 be Function-like quasi_total Relation of NAT,bool REAL;
  mode Interval_Covering of A1 -> Function-like quasi_total Relation of NAT,Funcs(NAT,bool REAL) means
    for b1 being Element of NAT holds
       it . b1 is Interval_Covering of a1 . b1;
end;

:: MEASURE7:dfs 3
definiens
  let a1 be Function-like quasi_total Relation of NAT,bool REAL;
  let a2 be Function-like quasi_total Relation of NAT,Funcs(NAT,bool REAL);
To prove
     a2 is Interval_Covering of a1
it is sufficient to prove
  thus for b1 being Element of NAT holds
       a2 . b1 is Interval_Covering of a1 . b1;

:: MEASURE7:def 3
theorem
for b1 being Function-like quasi_total Relation of NAT,bool REAL
for b2 being Function-like quasi_total Relation of NAT,Funcs(NAT,bool REAL) holds
      b2 is Interval_Covering of b1
   iff
      for b3 being Element of NAT holds
         b2 . b3 is Interval_Covering of b1 . b3;

:: MEASURE7:funcnot 3 => MEASURE7:func 3
definition
  let a1 be Element of bool REAL;
  let a2 be Interval_Covering of a1;
  func A2 vol -> Function-like quasi_total Relation of NAT,ExtREAL means
    for b1 being Element of NAT holds
       it . b1 = vol (a2 . b1);
end;

:: MEASURE7:def 4
theorem
for b1 being Element of bool REAL
for b2 being Interval_Covering of b1
for b3 being Function-like quasi_total Relation of NAT,ExtREAL holds
      b3 = b2 vol
   iff
      for b4 being Element of NAT holds
         b3 . b4 = vol (b2 . b4);

:: MEASURE7:th 13
theorem
for b1 being Element of bool REAL
for b2 being Interval_Covering of b1 holds
   b2 vol is nonnegative;

:: MEASURE7:funcnot 4 => MEASURE7:func 4
definition
  let a1 be Function-like quasi_total Relation of NAT,bool REAL;
  let a2 be Interval_Covering of a1;
  let a3 be Element of NAT;
  redefine func a2 . a3 -> Interval_Covering of a1 . a3;
end;

:: MEASURE7:funcnot 5 => MEASURE7:func 5
definition
  let a1 be Function-like quasi_total Relation of NAT,bool REAL;
  let a2 be Interval_Covering of a1;
  func A2 vol -> Function-like quasi_total Relation of NAT,Funcs(NAT,ExtREAL) means
    for b1 being Element of NAT holds
       it . b1 = (a2 . b1) vol;
end;

:: MEASURE7:def 5
theorem
for b1 being Function-like quasi_total Relation of NAT,bool REAL
for b2 being Interval_Covering of b1
for b3 being Function-like quasi_total Relation of NAT,Funcs(NAT,ExtREAL) holds
      b3 = b2 vol
   iff
      for b4 being Element of NAT holds
         b3 . b4 = (b2 . b4) vol;

:: MEASURE7:funcnot 6 => MEASURE7:func 6
definition
  let a1 be Element of bool REAL;
  let a2 be Interval_Covering of a1;
  func vol A2 -> Element of ExtREAL equals
    SUM (a2 vol);
end;

:: MEASURE7:def 6
theorem
for b1 being Element of bool REAL
for b2 being Interval_Covering of b1 holds
   vol b2 = SUM (b2 vol);

:: MEASURE7:funcnot 7 => MEASURE7:func 7
definition
  let a1 be Function-like quasi_total Relation of NAT,bool REAL;
  let a2 be Interval_Covering of a1;
  func vol A2 -> Function-like quasi_total Relation of NAT,ExtREAL means
    for b1 being Element of NAT holds
       it . b1 = vol (a2 . b1);
end;

:: MEASURE7:def 7
theorem
for b1 being Function-like quasi_total Relation of NAT,bool REAL
for b2 being Interval_Covering of b1
for b3 being Function-like quasi_total Relation of NAT,ExtREAL holds
      b3 = vol b2
   iff
      for b4 being Element of NAT holds
         b3 . b4 = vol (b2 . b4);

:: MEASURE7:th 14
theorem
for b1 being Function-like quasi_total Relation of NAT,bool REAL
for b2 being Interval_Covering of b1
for b3 being Element of NAT holds
   0. <= (vol b2) . b3;

:: MEASURE7:funcnot 8 => MEASURE7:func 8
definition
  let a1 be Element of bool REAL;
  func Svc A1 -> Element of bool ExtREAL means
    for b1 being Element of ExtREAL holds
          b1 in it
       iff
          ex b2 being Interval_Covering of a1 st
             b1 = vol b2;
end;

:: MEASURE7:def 8
theorem
for b1 being Element of bool REAL
for b2 being Element of bool ExtREAL holds
      b2 = Svc b1
   iff
      for b3 being Element of ExtREAL holds
            b3 in b2
         iff
            ex b4 being Interval_Covering of b1 st
               b3 = vol b4;

:: MEASURE7:funcreg 1
registration
  let a1 be Element of bool REAL;
  cluster Svc a1 -> non empty;
end;

:: MEASURE7:funcnot 9 => MEASURE7:func 9
definition
  let a1 be Element of bool REAL;
  func COMPLEX A1 -> Element of ExtREAL equals
    inf Svc a1;
end;

:: MEASURE7:def 9
theorem
for b1 being Element of bool REAL holds
   COMPLEX b1 = inf Svc b1;

:: MEASURE7:funcnot 10 => MEASURE7:func 10
definition
  func OS_Meas -> Function-like quasi_total Relation of bool REAL,ExtREAL means
    for b1 being Element of bool REAL holds
       it . b1 = inf Svc b1;
end;

:: MEASURE7:def 10
theorem
for b1 being Function-like quasi_total Relation of bool REAL,ExtREAL holds
      b1 = OS_Meas
   iff
      for b2 being Element of bool REAL holds
         b1 . b2 = inf Svc b2;

:: MEASURE7:funcnot 11 => MEASURE7:func 11
definition
  let a1 be Function-like quasi_total Relation of NAT,[:NAT,NAT:];
  redefine func pr1 A1 -> Function-like quasi_total Relation of NAT,NAT means
    for b1 being Element of NAT holds
       ex b2 being Element of NAT st
          a1 . b1 = [it . b1,b2];
end;

:: MEASURE7:def 11
theorem
for b1 being Function-like quasi_total Relation of NAT,[:NAT,NAT:]
for b2 being Function-like quasi_total Relation of NAT,NAT holds
      b2 = pr1 b1
   iff
      for b3 being Element of NAT holds
         ex b4 being Element of NAT st
            b1 . b3 = [b2 . b3,b4];

:: MEASURE7:funcnot 12 => MEASURE7:func 12
definition
  let a1 be Function-like quasi_total Relation of NAT,[:NAT,NAT:];
  redefine func pr2 A1 -> Function-like quasi_total Relation of NAT,NAT means
    for b1 being Element of NAT holds
       a1 . b1 = [(pr1 a1) . b1,it . b1];
end;

:: MEASURE7:def 12
theorem
for b1 being Function-like quasi_total Relation of NAT,[:NAT,NAT:]
for b2 being Function-like quasi_total Relation of NAT,NAT holds
      b2 = pr2 b1
   iff
      for b3 being Element of NAT holds
         b1 . b3 = [(pr1 b1) . b3,b2 . b3];

:: MEASURE7:funcnot 13 => MEASURE7:func 13
definition
  let a1 be Function-like quasi_total Relation of NAT,bool REAL;
  let a2 be Interval_Covering of a1;
  let a3 be Function-like quasi_total Relation of NAT,[:NAT,NAT:];
  assume rng a3 = [:NAT,NAT:];
  func On(A2,A3) -> Interval_Covering of union rng a1 means
    for b1 being Element of NAT holds
       it . b1 = (a2 . ((pr1 a3) . b1)) . ((pr2 a3) . b1);
end;

:: MEASURE7:def 13
theorem
for b1 being Function-like quasi_total Relation of NAT,bool REAL
for b2 being Interval_Covering of b1
for b3 being Function-like quasi_total Relation of NAT,[:NAT,NAT:]
   st rng b3 = [:NAT,NAT:]
for b4 being Interval_Covering of union rng b1 holds
      b4 = On(b2,b3)
   iff
      for b5 being Element of NAT holds
         b4 . b5 = (b2 . ((pr1 b3) . b5)) . ((pr2 b3) . b5);

:: MEASURE7:th 15
theorem
for b1 being Function-like quasi_total Relation of NAT,[:NAT,NAT:]
   st b1 is one-to-one & rng b1 = [:NAT,NAT:]
for b2 being Element of NAT holds
   ex b3 being Element of NAT st
      for b4 being Function-like quasi_total Relation of NAT,bool REAL
      for b5 being Interval_Covering of b4 holds
         (Ser ((On(b5,b1)) vol)) . b2 <= (Ser vol b5) . b3;

:: MEASURE7:th 16
theorem
for b1 being Function-like quasi_total Relation of NAT,bool REAL
for b2 being Interval_Covering of b1 holds
   inf Svc union rng b1 <= SUM vol b2;

:: MEASURE7:th 17
theorem
OS_Meas is C_Measure of REAL;

:: MEASURE7:funcnot 14 => MEASURE7:func 14
definition
  redefine func OS_Meas -> C_Measure of REAL;
end;

:: MEASURE7:funcnot 15 => MEASURE7:func 15
definition
  func Lmi_sigmaFIELD -> non empty compl-closed sigma-multiplicative Element of bool bool REAL equals
    sigma_Field OS_Meas;
end;

:: MEASURE7:def 14
theorem
Lmi_sigmaFIELD = sigma_Field OS_Meas;

:: MEASURE7:funcnot 16 => MEASURE7:func 16
definition
  func L_mi -> sigma_Measure of Lmi_sigmaFIELD equals
    sigma_Meas OS_Meas;
end;

:: MEASURE7:def 15
theorem
L_mi = sigma_Meas OS_Meas;