Article MEASURE1, MML version 4.99.1005

:: MEASURE1:th 1
theorem
for b1, b2 being set holds
union {b1,b2,{}} = union {b1,b2};

:: MEASURE1:th 4
theorem
for b1, b2, b3, b4 being Element of ExtREAL
      st 0. <= b1 & 0. <= b3 & b1 <= b2 & b3 <= b4
   holds b1 + b3 <= b2 + b4;

:: MEASURE1:th 5
theorem
for b1, b2, b3 being Element of ExtREAL
      st 0. <= b2 & 0. <= b3 & b1 = b2 + b3 & b2 < +infty
   holds b3 = b1 - b2;

:: MEASURE1:th 7
theorem
for b1 being set
for b2, b3 being Element of bool b1 holds
{b2,b3} is Element of bool bool b1;

:: MEASURE1:th 8
theorem
for b1 being set
for b2, b3, b4 being Element of bool b1 holds
{b2,b3,b4} is non empty Element of bool bool b1;

:: MEASURE1:th 9
theorem
for b1 being set holds
   {{}} is Element of bool bool b1;

:: MEASURE1:sch 1
scheme MEASURE1:sch 1
{F1 -> set}:
ex b1 being non empty Element of bool bool F1() st
   for b2 being set holds
         b2 in b1
      iff
         b2 c= F1() & P1[b2]
provided
   ex b1 being set st
      b1 c= F1() & P1[b1];


:: MEASURE1:funcnot 1 => SETFAM_1:func 7
notation
  let a1 be set;
  let a2 be non empty Element of bool bool a1;
  synonym a1 \ a2 for COMPLEMENT a2;
end;

:: MEASURE1:funcreg 1
registration
  let a1 be set;
  let a2 be non empty Element of bool bool a1;
  cluster COMPLEMENT a2 -> non empty;
end;

:: MEASURE1:th 15
theorem
for b1 being set
for b2 being non empty Element of bool bool b1 holds
   meet b2 = b1 \ union COMPLEMENT b2 & union b2 = b1 \ meet COMPLEMENT b2;

:: MEASURE1:attrnot 1 => PROB_1:attr 1
definition
  let a1 be set;
  let a2 be Element of bool bool a1;
  attr a2 is compl-closed means
    for b1 being set
          st b1 in a2
       holds a1 \ b1 in a2;
end;

:: MEASURE1:dfs 1
definiens
  let a1 be set;
  let a2 be Element of bool bool a1;
To prove
     a2 is compl-closed
it is sufficient to prove
  thus for b1 being set
          st b1 in a2
       holds a1 \ b1 in a2;

:: MEASURE1:def 3
theorem
for b1 being set
for b2 being Element of bool bool b1 holds
      b2 is compl-closed(b1)
   iff
      for b3 being set
            st b3 in b2
         holds b1 \ b3 in b2;

:: MEASURE1:th 16
theorem
for b1 being set
for b2 being Element of bool bool b1
      st b2 is cup-closed & b2 is compl-closed(b1)
   holds b2 is cap-closed;

:: MEASURE1:condreg 1
registration
  let a1 be set;
  cluster cup-closed compl-closed -> cap-closed (Element of bool bool a1);
end;

:: MEASURE1:condreg 2
registration
  let a1 be set;
  cluster cap-closed compl-closed -> cup-closed (Element of bool bool a1);
end;

:: MEASURE1:th 17
theorem
for b1 being set
for b2 being non empty cap-closed compl-closed Element of bool bool b1 holds
   b2 = COMPLEMENT b2;

:: MEASURE1:th 18
theorem
for b1, b2 being set holds
   b2 is non empty cap-closed compl-closed Element of bool bool b1
iff
   ex b3 being non empty Element of bool bool b1 st
      b2 = b3 &
       (for b4 being set
             st b4 in b3
          holds b1 \ b4 in b3 &
           (for b5, b6 being set
                 st b5 in b3 & b6 in b3
              holds b5 \/ b6 in b3));

:: MEASURE1:th 19
theorem
for b1 being set
for b2 being non empty Element of bool bool b1 holds
      b2 is non empty cap-closed compl-closed Element of bool bool b1
   iff
      for b3 being set
            st b3 in b2
         holds b1 \ b3 in b2 &
          (for b4, b5 being set
                st b4 in b2 & b5 in b2
             holds b4 /\ b5 in b2);

:: MEASURE1:th 20
theorem
for b1 being set
for b2 being non empty cap-closed compl-closed Element of bool bool b1
for b3, b4 being set
      st b3 in b2 & b4 in b2
   holds b3 \ b4 in b2;

:: MEASURE1:th 21
theorem
for b1 being set
for b2 being non empty cap-closed compl-closed Element of bool bool b1 holds
   {} in b2 & b1 in b2;

:: MEASURE1:attrnot 2 => MEASURE1:attr 1
definition
  let a1 be non empty set;
  let a2 be Function-like quasi_total Relation of a1,ExtREAL;
  redefine attr a2 is nonnegative means
    for b1 being Element of a1 holds
       0. <= a2 . b1;
end;

:: MEASURE1:dfs 2
definiens
  let a1 be non empty set;
  let a2 be Function-like quasi_total Relation of a1,ExtREAL;
To prove
     a1 is nonnegative
it is sufficient to prove
  thus for b1 being Element of a1 holds
       0. <= a2 . b1;

:: MEASURE1:def 4
theorem
for b1 being non empty set
for b2 being Function-like quasi_total Relation of b1,ExtREAL holds
      b2 is nonnegative
   iff
      for b3 being Element of b1 holds
         0. <= b2 . b3;

:: MEASURE1:th 23
theorem
for b1 being set
for b2 being non empty cap-closed compl-closed Element of bool bool b1 holds
   ex b3 being Function-like quasi_total Relation of b2,ExtREAL st
      b3 is nonnegative &
       b3 . {} = 0. &
       (for b4, b5 being Element of b2
             st b4 misses b5
          holds b3 . (b4 \/ b5) = (b3 . b4) + (b3 . b5));

:: MEASURE1:modenot 1 => MEASURE1:mode 1
definition
  let a1 be set;
  let a2 be non empty cap-closed compl-closed Element of bool bool a1;
  mode Measure of A2 -> Function-like quasi_total Relation of a2,ExtREAL means
    it is nonnegative &
     it . {} = 0. &
     (for b1, b2 being Element of a2
           st b1 misses b2
        holds it . (b1 \/ b2) = (it . b1) + (it . b2));
end;

:: MEASURE1:dfs 3
definiens
  let a1 be set;
  let a2 be non empty cap-closed compl-closed Element of bool bool a1;
  let a3 be Function-like quasi_total Relation of a2,ExtREAL;
To prove
     a3 is Measure of a2
it is sufficient to prove
  thus a3 is nonnegative &
     a3 . {} = 0. &
     (for b1, b2 being Element of a2
           st b1 misses b2
        holds a3 . (b1 \/ b2) = (a3 . b1) + (a3 . b2));

:: MEASURE1:def 5
theorem
for b1 being set
for b2 being non empty cap-closed compl-closed Element of bool bool b1
for b3 being Function-like quasi_total Relation of b2,ExtREAL holds
      b3 is Measure of b2
   iff
      b3 is nonnegative &
       b3 . {} = 0. &
       (for b4, b5 being Element of b2
             st b4 misses b5
          holds b3 . (b4 \/ b5) = (b3 . b4) + (b3 . b5));

:: MEASURE1:th 25
theorem
for b1 being set
for b2 being non empty cap-closed compl-closed Element of bool bool b1
for b3 being Measure of b2
for b4, b5 being Element of b2
      st b4 c= b5
   holds b3 . b4 <= b3 . b5;

:: MEASURE1:th 26
theorem
for b1 being set
for b2 being non empty cap-closed compl-closed Element of bool bool b1
for b3 being Measure of b2
for b4, b5 being Element of b2
      st b4 c= b5 & b3 . b4 < +infty
   holds b3 . (b5 \ b4) = (b3 . b5) - (b3 . b4);

:: MEASURE1:exreg 1
registration
  let a1 be set;
  cluster non empty cap-closed compl-closed Element of bool bool a1;
end;

:: MEASURE1:funcnot 2 => MEASURE1:func 1
definition
  let a1 be set;
  let a2 be non empty cup-closed Element of bool bool a1;
  let a3, a4 be Element of a2;
  redefine func a3 \/ a4 -> Element of a2;
  commutativity;
::  for a1 being set
::  for a2 being non empty cup-closed Element of bool bool a1
::  for a3, a4 being Element of a2 holds
::  a3 \/ a4 = a4 \/ a3;
  idempotence;
::  for a1 being set
::  for a2 being non empty cup-closed Element of bool bool a1
::  for a3 being Element of a2 holds
::     a3 \/ a3 = a3;
end;

:: MEASURE1:funcnot 3 => MEASURE1:func 2
definition
  let a1 be set;
  let a2 be non empty cap-closed compl-closed Element of bool bool a1;
  let a3, a4 be Element of a2;
  redefine func a3 /\ a4 -> Element of a2;
  commutativity;
::  for a1 being set
::  for a2 being non empty cap-closed compl-closed Element of bool bool a1
::  for a3, a4 being Element of a2 holds
::  a3 /\ a4 = a4 /\ a3;
  idempotence;
::  for a1 being set
::  for a2 being non empty cap-closed compl-closed Element of bool bool a1
::  for a3 being Element of a2 holds
::     a3 /\ a3 = a3;
end;

:: MEASURE1:funcnot 4 => MEASURE1:func 3
definition
  let a1 be set;
  let a2 be non empty cap-closed compl-closed Element of bool bool a1;
  let a3, a4 be Element of a2;
  redefine func a3 \ a4 -> Element of a2;
end;

:: MEASURE1:th 27
theorem
for b1 being set
for b2 being non empty cap-closed compl-closed Element of bool bool b1
for b3 being Measure of b2
for b4, b5 being Element of b2 holds
b3 . (b4 \/ b5) <= (b3 . b4) + (b3 . b5);

:: MEASURE1:th 29
theorem
for b1 being set
for b2 being non empty cap-closed compl-closed Element of bool bool b1
for b3 being Measure of b2 holds
   {} in b2 &
    b1 in b2 &
    (for b4, b5 being set
          st b4 in b2 & b5 in b2
       holds b1 \ b4 in b2 & b4 \/ b5 in b2 & b4 /\ b5 in b2);

:: MEASURE1:modenot 2 => MEASURE1:mode 2
definition
  let a1 be set;
  let a2 be non empty cap-closed compl-closed Element of bool bool a1;
  let a3 be Measure of a2;
  mode measure_zero of A3 -> Element of a2 means
    a3 . it = 0.;
end;

:: MEASURE1:dfs 4
definiens
  let a1 be set;
  let a2 be non empty cap-closed compl-closed Element of bool bool a1;
  let a3 be Measure of a2;
  let a4 be Element of a2;
To prove
     a4 is measure_zero of a3
it is sufficient to prove
  thus a3 . a4 = 0.;

:: MEASURE1:def 7
theorem
for b1 being set
for b2 being non empty cap-closed compl-closed Element of bool bool b1
for b3 being Measure of b2
for b4 being Element of b2 holds
      b4 is measure_zero of b3
   iff
      b3 . b4 = 0.;

:: MEASURE1:th 31
theorem
for b1 being set
for b2 being non empty cap-closed compl-closed Element of bool bool b1
for b3 being Measure of b2
for b4 being Element of b2
for b5 being measure_zero of b3
      st b4 c= b5
   holds b4 is measure_zero of b3;

:: MEASURE1:th 32
theorem
for b1 being set
for b2 being non empty cap-closed compl-closed Element of bool bool b1
for b3 being Measure of b2
for b4, b5 being measure_zero of b3 holds
b4 \/ b5 is measure_zero of b3 & b4 /\ b5 is measure_zero of b3 & b4 \ b5 is measure_zero of b3;

:: MEASURE1:th 33
theorem
for b1 being set
for b2 being non empty cap-closed compl-closed Element of bool bool b1
for b3 being Measure of b2
for b4 being Element of b2
for b5 being measure_zero of b3 holds
   b3 . (b4 \/ b5) = b3 . b4 & b3 . (b4 /\ b5) = 0. & b3 . (b4 \ b5) = b3 . b4;

:: MEASURE1:th 34
theorem
for b1 being set
for b2 being Element of bool b1 holds
   ex b3 being Function-like quasi_total Relation of NAT,bool b1 st
      rng b3 = {b2};

:: MEASURE1:th 35
theorem
for b1 being set holds
   ex b2 being Function-like quasi_total Relation of NAT,{b1} st
      for b3 being Element of NAT holds
         b2 . b3 = b1;

:: MEASURE1:exreg 2
registration
  let a1 be set;
  cluster non empty countable Element of bool bool a1;
end;

:: MEASURE1:modenot 3
definition
  let a1 be set;
  mode N_Sub_set_fam of a1 is non empty countable Element of bool bool a1;
end;

:: MEASURE1:th 38
theorem
for b1 being set
for b2, b3, b4 being Element of bool b1 holds
ex b5 being Function-like quasi_total Relation of NAT,bool b1 st
   rng b5 = {b2,b3,b4} &
    b5 . 0 = b2 &
    b5 . 1 = b3 &
    (for b6 being Element of NAT
          st 1 < b6
       holds b5 . b6 = b4);

:: MEASURE1:th 39
theorem
for b1 being set
for b2, b3 being Element of bool b1 holds
{b2,b3,{}} is non empty countable Element of bool bool b1;

:: MEASURE1:th 40
theorem
for b1 being set
for b2, b3 being Element of bool b1 holds
ex b4 being Function-like quasi_total Relation of NAT,bool b1 st
   rng b4 = {b2,b3} &
    b4 . 0 = b2 &
    (for b5 being Element of NAT
          st 0 < b5
       holds b4 . b5 = b3);

:: MEASURE1:th 41
theorem
for b1 being set
for b2, b3 being Element of bool b1 holds
{b2,b3} is non empty countable Element of bool bool b1;

:: MEASURE1:th 42
theorem
for b1 being set
for b2 being non empty countable Element of bool bool b1 holds
   COMPLEMENT b2 is non empty countable Element of bool bool b1;

:: MEASURE1:attrnot 3 => MEASURE1:attr 2
definition
  let a1 be set;
  let a2 be Element of bool bool a1;
  attr a2 is sigma-additive means
    for b1 being non empty countable Element of bool bool a1
          st b1 c= a2
       holds union b1 in a2;
end;

:: MEASURE1:dfs 5
definiens
  let a1 be set;
  let a2 be Element of bool bool a1;
To prove
     a2 is sigma-additive
it is sufficient to prove
  thus for b1 being non empty countable Element of bool bool a1
          st b1 c= a2
       holds union b1 in a2;

:: MEASURE1:def 9
theorem
for b1 being set
for b2 being Element of bool bool b1 holds
      b2 is sigma-additive(b1)
   iff
      for b3 being non empty countable Element of bool bool b1
            st b3 c= b2
         holds union b3 in b2;

:: MEASURE1:exreg 3
registration
  let a1 be set;
  cluster non empty compl-closed sigma-additive Element of bool bool a1;
end;

:: MEASURE1:condreg 3
registration
  let a1 be set;
  cluster compl-closed sigma-multiplicative -> sigma-additive (Element of bool bool a1);
end;

:: MEASURE1:condreg 4
registration
  let a1 be set;
  cluster compl-closed sigma-additive -> sigma-multiplicative (Element of bool bool a1);
end;

:: MEASURE1:condreg 5
registration
  let a1 be set;
  cluster non empty compl-closed sigma-multiplicative -> (Element of bool bool a1);
end;

:: MEASURE1:th 49
theorem
for b1 being set
for b2 being non empty Element of bool bool b1 holds
      (for b3 being set
             st b3 in b2
          holds b1 \ b3 in b2) &
       (for b3 being non empty countable Element of bool bool b1
             st b3 c= b2
          holds meet b3 in b2)
   iff
      b2 is non empty compl-closed sigma-multiplicative Element of bool bool b1;

:: MEASURE1:exreg 4
registration
  let a1 be set;
  let a2 be non empty compl-closed sigma-multiplicative Element of bool bool a1;
  cluster Relation-like Function-like non empty total quasi_total disjoint_valued Relation of NAT,a2;
end;

:: MEASURE1:modenot 4
definition
  let a1 be set;
  let a2 be non empty compl-closed sigma-multiplicative Element of bool bool a1;
  mode Sep_Sequence of a2 is Function-like quasi_total disjoint_valued Relation of NAT,a2;
end;

:: MEASURE1:funcnot 5 => MEASURE1:func 4
definition
  let a1 be set;
  let a2 be non empty compl-closed sigma-multiplicative Element of bool bool a1;
  let a3 be Function-like quasi_total Relation of NAT,a2;
  redefine func rng a3 -> non empty Element of bool bool a1;
end;

:: MEASURE1:th 52
theorem
for b1 being set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being Function-like quasi_total Relation of NAT,b2 holds
   rng b3 is non empty countable Element of bool bool b1;

:: MEASURE1:th 53
theorem
for b1 being set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being Function-like quasi_total Relation of NAT,b2 holds
   union rng b3 is Element of b2;

:: MEASURE1:th 54
theorem
for b1, b2 being non empty set
for b3 being Function-like quasi_total Relation of b1,b2
for b4 being Function-like quasi_total Relation of b2,ExtREAL
      st b4 is nonnegative
   holds b4 * b3 is nonnegative;

:: MEASURE1:th 55
theorem
for b1 being set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3, b4 being Element of ExtREAL holds
ex b5 being Function-like quasi_total Relation of b2,ExtREAL st
   for b6 being Element of b2 holds
      (b6 = {} implies b5 . b6 = b3) & (b6 = {} or b5 . b6 = b4);

:: MEASURE1:th 56
theorem
for b1 being set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1 holds
   ex b3 being Function-like quasi_total Relation of b2,ExtREAL st
      for b4 being Element of b2 holds
         (b4 = {} implies b3 . b4 = 0.) & (b4 = {} or b3 . b4 = +infty);

:: MEASURE1:th 57
theorem
for b1 being set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1 holds
   ex b3 being Function-like quasi_total Relation of b2,ExtREAL st
      for b4 being Element of b2 holds
         b3 . b4 = 0.;

:: MEASURE1:th 58
theorem
for b1 being set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1 holds
   ex b3 being Function-like quasi_total Relation of b2,ExtREAL st
      b3 is nonnegative &
       b3 . {} = 0. &
       (for b4 being Function-like quasi_total disjoint_valued Relation of NAT,b2 holds
          SUM (b3 * b4) = b3 . union rng b4);

:: MEASURE1:modenot 5 => MEASURE1:mode 3
definition
  let a1 be set;
  let a2 be non empty compl-closed sigma-multiplicative Element of bool bool a1;
  mode sigma_Measure of A2 -> Function-like quasi_total Relation of a2,ExtREAL means
    it is nonnegative &
     it . {} = 0. &
     (for b1 being Function-like quasi_total disjoint_valued Relation of NAT,a2 holds
        SUM (it * b1) = it . union rng b1);
end;

:: MEASURE1:dfs 6
definiens
  let a1 be set;
  let a2 be non empty compl-closed sigma-multiplicative Element of bool bool a1;
  let a3 be Function-like quasi_total Relation of a2,ExtREAL;
To prove
     a3 is sigma_Measure of a2
it is sufficient to prove
  thus a3 is nonnegative &
     a3 . {} = 0. &
     (for b1 being Function-like quasi_total disjoint_valued Relation of NAT,a2 holds
        SUM (a3 * b1) = a3 . union rng b1);

:: MEASURE1:def 11
theorem
for b1 being set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being Function-like quasi_total Relation of b2,ExtREAL holds
      b3 is sigma_Measure of b2
   iff
      b3 is nonnegative &
       b3 . {} = 0. &
       (for b4 being Function-like quasi_total disjoint_valued Relation of NAT,b2 holds
          SUM (b3 * b4) = b3 . union rng b4);

:: MEASURE1:condreg 6
registration
  let a1 be set;
  cluster non empty compl-closed sigma-additive -> cup-closed (Element of bool bool a1);
end;

:: MEASURE1:th 60
theorem
for b1 being set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being sigma_Measure of b2 holds
   b3 is Measure of b2;

:: MEASURE1:th 61
theorem
for b1 being 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 b4 misses b5
   holds b3 . (b4 \/ b5) = (b3 . b4) + (b3 . b5);

:: MEASURE1:th 62
theorem
for b1 being 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 b4 c= b5
   holds b3 . b4 <= b3 . b5;

:: MEASURE1:th 63
theorem
for b1 being 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 b4 c= b5 & b3 . b4 < +infty
   holds b3 . (b5 \ b4) = (b3 . b5) - (b3 . b4);

:: MEASURE1:th 64
theorem
for b1 being 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 holds
b3 . (b4 \/ b5) <= (b3 . b4) + (b3 . b5);

:: MEASURE1:th 66
theorem
for b1 being set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being sigma_Measure of b2 holds
   {} in b2 &
    b1 in b2 &
    (for b4, b5 being set
          st b4 in b2 & b5 in b2
       holds b1 \ b4 in b2 & b4 \/ b5 in b2 & b4 /\ b5 in b2);

:: MEASURE1:th 67
theorem
for b1 being 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 non empty countable Element of bool bool b1
      st for b5 being set
              st b5 in b4
           holds b5 in b2
   holds union b4 in b2 & meet b4 in b2;

:: MEASURE1:modenot 6 => MEASURE1:mode 4
definition
  let a1 be set;
  let a2 be non empty compl-closed sigma-multiplicative Element of bool bool a1;
  let a3 be sigma_Measure of a2;
  mode measure_zero of A3 -> Element of a2 means
    a3 . it = 0.;
end;

:: MEASURE1:dfs 7
definiens
  let a1 be 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 Element of a2;
To prove
     a4 is measure_zero of a3
it is sufficient to prove
  thus a3 . a4 = 0.;

:: MEASURE1:def 13
theorem
for b1 being 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 holds
      b4 is measure_zero of b3
   iff
      b3 . b4 = 0.;

:: MEASURE1:th 69
theorem
for b1 being 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
for b5 being measure_zero of b3
      st b4 c= b5
   holds b4 is measure_zero of b3;

:: MEASURE1:th 70
theorem
for b1 being 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 measure_zero of b3 holds
b4 \/ b5 is measure_zero of b3 & b4 /\ b5 is measure_zero of b3 & b4 \ b5 is measure_zero of b3;

:: MEASURE1:th 71
theorem
for b1 being 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
for b5 being measure_zero of b3 holds
   b3 . (b4 \/ b5) = b3 . b4 & b3 . (b4 /\ b5) = 0. & b3 . (b4 \ b5) = b3 . b4;