The Mizar article:

The $\sigma$-additive Measure Theory

by
Jozef Bialas

Received October 15, 1990

Copyright (c) 1990 Association of Mizar Users

MML identifier: MEASURE1
[ MML identifier index ]


environ

 vocabulary TARSKI, BOOLE, SUPINF_1, RLVECT_1, ARYTM_3, ARYTM_1, SETFAM_1,
      PROB_1, SUBSET_1, FINSUB_1, FUNCT_1, SUPINF_2, RELAT_1, FUNCOP_1, CARD_4,
      PROB_2, MEASURE1;
 notation TARSKI, XBOOLE_0, ENUMSET1, ZFMISC_1, SUBSET_1, NUMBERS, XREAL_0,
      REAL_1, RELAT_1, FINSUB_1, FUNCT_1, FUNCT_2, NAT_1, SETFAM_1, FUNCOP_1,
      PROB_1, PROB_2, SUPINF_1, SUPINF_2;
 constructors ENUMSET1, NAT_1, REAL_1, SUPINF_2, FUNCOP_1, FINSUB_1, PROB_2,
      MEMBERED, XBOOLE_0;
 clusters SUBSET_1, SUPINF_1, RELSET_1, ARYTM_3, PROB_1, MEMBERED, ZFMISC_1,
      XBOOLE_0, NUMBERS, ORDINAL2;
 requirements NUMERALS, REAL, SUBSET, BOOLE, ARITHM;
 definitions SUPINF_2, TARSKI, FINSUB_1, PROB_2, PROB_1, XBOOLE_0;
 theorems TARSKI, FUNCT_1, FUNCT_2, ENUMSET1, ZFMISC_1, NAT_1, SETFAM_1,
      SUPINF_1, SUPINF_2, CQC_THE1, RELAT_1, RELSET_1, FUNCOP_1, FINSUB_1,
      PROB_2, PROB_1, SUBSET_1, XBOOLE_0, XBOOLE_1, XCMPLX_0, XCMPLX_1;
 schemes FUNCT_2, XBOOLE_0;

begin :: Some useful theorems about sets and R_eal numbers

 reserve X for set;

theorem Th1:
  for X,Y being set holds union {X,Y,{}} = union {X,Y}
proof
   let X,Y be set;
   thus union {X,Y,{}} = union ({X,Y} \/ {{}}) by ENUMSET1:43
                .= union {X,Y} \/ union {{}} by ZFMISC_1:96
                .= union {X,Y} \/ {} by ZFMISC_1:31
                .= union {X,Y};
end;

canceled 2;

theorem Th4:
   for x,y,s,t being R_eal holds
   0.<=' x & 0.<=' s & x <=' y & s <=' t implies
   x + s <=' y + t
proof
   let x,y,s,t be R_eal;
   assume
A1:0. <=' x & 0.<=' s & x <=' y & s <=' t;
     not ((x = +infty & s = -infty) or (x = -infty & s = +infty)) &
   not ((y = +infty & t = -infty) or (y = -infty & t = +infty))
   proof
        0.<=' y & 0.<=' t by A1,SUPINF_1:29;
      hence thesis by A1,SUPINF_1:17,SUPINF_2:def 1;
   end;
   hence thesis by A1,SUPINF_2:14;
end;

theorem Th5:
   for x,y,z being R_eal holds
   0.<=' y & 0.<=' z & x = y + z & y <' +infty implies z = x - y
proof
   let x,y,z be R_eal;
   assume
A1:0. <=' y & 0.<=' z & x = y + z & y <' +infty;
   then 0. + 0. <=' y + z by Th4;
then A2:0. <=' x by A1,SUPINF_2:18;
     (y + z) - y = z
   proof
   A3:not x = -infty & not y = -infty & not z = -infty
        by A1,A2,SUPINF_1:17,SUPINF_2:def 1;
        x in REAL \/ {-infty,+infty} & y in REAL \/ {-infty,+infty}
        by SUPINF_1:def 5;
then A4:  (x in REAL or x in {-infty,+infty}) & (y in REAL or
       y in {-infty,+infty}) by XBOOLE_0:def 2;
A5:  (x in REAL & y in REAL) implies (y + z) - y = z
      proof
         assume x in REAL & y in REAL;
         then reconsider a = y, b = z as Real by A1,A3,SUPINF_2:12;
           y + z = a + b by SUPINF_2:1;
         then (y + z) - y = (a + b) - a by SUPINF_2:5
                   .= (b - a) + a by XCMPLX_1:29
                   .= (b + (-a)) + a by XCMPLX_0:def 8
                   .= b + ((-a) + a) by XCMPLX_1:1
                   .= b + 0 by XCMPLX_0:def 6
                   .= z;
         hence thesis;
      end;
        (x = +infty & y in REAL) implies (y + z) - y = z
      proof
         assume
      A6:x = +infty & y in REAL;
         then (y + z) - y = +infty by A1,SUPINF_2:6
                    .= z by A1,A6,SUPINF_2:8;
         hence thesis;
      end;
      hence thesis by A1,A3,A4,A5,TARSKI:def 2;
   end;
   hence thesis by A1;
end;

theorem
     for A being Subset of X holds
   {A} is Subset-Family of X
proof
   let A be Subset of X;
     {A} c= bool X by ZFMISC_1:37;
   hence thesis by SETFAM_1:def 7;
end;

theorem Th7:
   for A,B being Subset of X holds
   {A,B} is Subset-Family of X
proof
   let A,B be Subset of X;
   set C = {A,B};
     C c= bool X
   proof
      let x be set;
      assume x in C;
      then x = A or x = B by TARSKI:def 2;
      hence thesis;
   end;
   hence thesis by SETFAM_1:def 7;
end;

theorem Th8:
   for A,B,C being Subset of X holds
   {A,B,C} is non empty Subset-Family of X
proof
   let A,B,C be Subset of X;
   set D = {A,B,C};
     D c= bool X
   proof
      let x be set;
      assume x in D;
      then x = A or x = B or x = C by ENUMSET1:def 1;
      hence thesis;
   end;
   hence thesis by ENUMSET1:def 1,SETFAM_1:def 7;
end;

theorem Th9:
   {{}} is Subset-Family of X
proof
     {} c= X by XBOOLE_1:2;
   then {{}} c= bool X by ZFMISC_1:37;
   hence thesis by SETFAM_1:def 7;
end;

scheme DomsetFamEx {A() -> set,P[set]}:
   ex F being non empty Subset-Family of A() st
   for B being set holds B in F iff (B c= A() & P[B])
   provided A1: ex B being set st B c= A() & P[B]
proof
   defpred X[set] means ex Z being set st $1 = Z & P[Z];
   consider X being set such that
A2:for x being set holds x in X iff x in bool A() & X[x] from Separation;
   consider B being set such that A3: B c= A() & P[B] by A1;
     X c= bool A()
   proof
     let x be set; assume x in X;
     hence thesis by A2;
   end;
   then reconsider X as non empty Subset-Family of A() by A2,A3,SETFAM_1:def 7;
   take X;
     for B being set holds B in X iff (B c= A() & P[B])
   proof
      let B be set;
      thus B in X implies B c= A() & P[B]
      proof
         assume
      A4:B in X;
         then ex Z being set st B = Z & P[Z] by A2;
         hence thesis by A4;
      end;
      assume B c= A() & P[B];
      hence B in X by A2;
   end;
  hence thesis;
end;

definition let X be set;
   let S be non empty Subset-Family of X;
 canceled;

   func X\S -> Subset-Family of X means :Def2:
     for A being set holds
      A in it iff ex B being set st B in S & A = X \ B;
existence
proof
   defpred P[set] means ex B being set st B in S & $1 = X \ B;
A1:ex C being set st C c= X & P[C]
   proof consider C0 being Element of S;
      take X \ C0;
      thus thesis by XBOOLE_1:36;
   end;
     ex R being non empty Subset-Family of X st
   for A being set holds A in R iff A c= X & P[A] from DomsetFamEx(A1);
   then consider R being Subset-Family of X such that
A2:for A being set holds A in R iff (A c= X &
   (ex B being set st B in S & A = X \ B));
   take R;
   let A be set;
   thus A in R implies (ex B being set st B in S & A = X \ B) by A2;
   given B being set such that A3: B in S & A = X \ B;
     A c= X by A3,XBOOLE_1:36;
   hence A in R by A2,A3;
end;
uniqueness
proof
   let R1,R2 be Subset-Family of X such that
A4:for A being set holds A in R1 iff
      (ex B being set st B in S & A = X \ B) and
A5:for A being set holds A in R2 iff
      (ex B being set st B in S & A = X \ B);
     R1 = R2
   proof
        for A being set holds A in R1 iff A in R2
      proof
         let A be set;
         hereby assume A in R1;
            then ex B being set st B in S & A = X \ B by A4;
            hence A in R2 by A5;
         end;
         assume A in R2;
         then ex B being set st B in S & A = X \ B by A5;
         hence thesis by A4;
      end;
      hence thesis by TARSKI:2;
   end;
   hence thesis;
end;
end;

definition let X be set;
   let S be non empty Subset-Family of X;
   cluster X\S -> non empty;
   coherence
   proof
     consider x being set such that A1: x in S by XBOOLE_0:def 1;
       X\x in X\S by A1,Def2;
     hence thesis;
   end;
end;

canceled 4;

theorem Th14:
   for S being non empty Subset-Family of X holds
   S = X \ (X \ S)
proof
   let S be non empty Subset-Family of X;
A1:S c= X \ (X \ S)
   proof
      let A be set;
      assume
   A2:A in S;
   then A3:X\A in X \ S by Def2;
        X \ (X \ A) = X /\ A by XBOOLE_1:48;
      then A = X \ (X\A) by A2,XBOOLE_1:28;
      hence thesis by A3,Def2;
   end;
     X \ (X \ S) c= S
   proof
      let A be set;
      assume A in X \ (X \ S);
      then consider B being set such that
   A4:B in X \ S & A = X \ B by Def2;
      consider C being set such that
   A5:C in S & B = X \ C by A4,Def2;
    A = X /\ C by A4,A5,XBOOLE_1:48; hence thesis by A5,XBOOLE_1:28;
   end;
   hence thesis by A1,XBOOLE_0:def 10;
end;

theorem Th15:
   for S being non empty Subset-Family of X holds
   meet S = X \ union (X \ S) & union S = X \ meet (X \ S)
proof
   let S be non empty Subset-Family of X;
   thus meet S = X \ union (X \ S)
   proof
      thus meet S c= X \ union (X \ S)
      proof
         let x be set;
         assume
A1:      x in meet S;
           not x in union (X \ S)
         proof
           assume x in union (X \ S);
           then consider Z being set such that
        A2:x in Z & Z in X \ S by TARSKI:def 4;
           consider B being set such that
        A3:B in S & Z = X \ B by A2,Def2;
             not x in B by A2,A3,XBOOLE_0:def 4;
           hence thesis by A1,A3,SETFAM_1:def 1;
         end;
         hence thesis by A1,XBOOLE_0:def 4;
      end;
      thus X \ union (X \ S) c= meet S
      proof
         let x be set;
         assume x in X \ union (X \ S);
      then A4:x in X & not x in union (X \ S) by XBOOLE_0:def 4;
           for Y being set st Y in S holds x in Y
         proof
            let Y be set;
            assume that
         A5:Y in S and
         A6:not x in Y;
         A7:x in X\Y by A4,A6,XBOOLE_0:def 4;
              X\Y in X \ S by A5,Def2;
            hence thesis by A4,A7,TARSKI:def 4;
         end;
         hence thesis by SETFAM_1:def 1;
      end;
   end;
   thus union S c= X \ meet (X \ S)
   proof
      let x be set;
      assume x in union S;
      then consider Y being set such that
   A8:x in Y & Y in S by TARSKI:def 4;
        not x in meet (X \ S)
      proof
         assume
     A9:x in meet (X \ S);
         set Z = X \ Y;
     A10:Z in X \ S by A8,Def2;
           not x in Z by A8,XBOOLE_0:def 4;
         hence thesis by A9,A10,SETFAM_1:def 1;
      end;
      hence thesis by A8,XBOOLE_0:def 4;
   end;
   let x be set;
   assume x in X \ meet (X \ S);
then A11:x in X & not x in meet (X \ S) by XBOOLE_0:def 4;
   then consider Z being set such that
A12:Z in X \ S & not x in Z by SETFAM_1:def 1;
     ex Y being set st Y in S & x in Y
   proof
      set Y = X \ Z;
        Y in X \ (X \ S) by A12,Def2;
  then A13:Y in S by Th14;
        x in Y by A11,A12,XBOOLE_0:def 4;
      hence thesis by A13;
   end;
   hence thesis by TARSKI:def 4;
end;

::
::        Field Subset of X and nonnegative measure
::

definition
   let X be set;
   let IT be Subset-Family of X;
   redefine attr IT is compl-closed means :Def3:
    for A being set holds A in IT implies X\A in IT;
   compatibility
   proof
     hereby assume A1: IT is compl-closed;
       let A be set;
       assume A2: A in IT;
       then reconsider A' = A as Subset of X;
         A'` in IT by A1,A2,PROB_1:def 1;
       hence X\A in IT by SUBSET_1:def 5;
     end;
     assume A3: for A being set holds A in IT implies X\A in IT;
     let A be Subset of X;
     assume A in IT;
     then X\A in IT by A3;
     hence thesis by SUBSET_1:def 5;
   end;
end;

theorem Th16:
  for X being set,
      F being Subset-Family of X st
    F is cup-closed compl-closed holds F is cap-closed
proof
  let X be set;
  let S be Subset-Family of X;
  assume A1: S is cup-closed compl-closed;
  let A,B be set;
  assume
 A2:A in S & B in S;
    then X \ A in S & X \ B in S by A1,Def3;
 then A3:(X \ A) \/ (X \ B) in S by A1,FINSUB_1:def 1;
      A /\ B c= A by XBOOLE_1:17;
    then A /\ B c= X by A2,XBOOLE_1:1;
    then A /\ B = X /\ (A /\ B) by XBOOLE_1:28
         .= X \ (X \ (A /\ B)) by XBOOLE_1:48
         .= X \ ((X \ A) \/ (X \ B)) by XBOOLE_1:54;
    hence thesis by A1,A3,Def3;
end;

definition let X be set;
  cluster cup-closed compl-closed -> cap-closed Subset-Family of X;
  coherence by Th16;
  cluster cap-closed compl-closed -> cup-closed Subset-Family of X;
  coherence
  proof
    let S be Subset-Family of X;
    assume A1: S is cap-closed compl-closed;
    let A,B be set;
    assume
    A2:A in S & B in S;
    then X \ A in S & X \ B in S by A1,Def3;
    then A3:(X \ A) /\ (X \ B) in S by A1,FINSUB_1:def 2;
      A \/ B c= X by A2,XBOOLE_1:8;
    then A \/ B = X /\ (A \/ B) by XBOOLE_1:28
         .= X \ (X \ (A \/ B)) by XBOOLE_1:48
         .= X \ ((X \ A) /\ (X \ B)) by XBOOLE_1:53;
    hence thesis by A1,A3,Def3;
  end;
end;

theorem Th17:
   for S being Field_Subset of X holds S = X \ S
proof
   let S be Field_Subset of X;
     for A being set holds A in S iff A in X \ S
   proof
      let A be set;
      hereby assume
      A1:A in S;
         then A2:X \ A in S by Def3;
           A = X \ (X \ A)
         proof
              for x be set holds x in A iff x in X\(X\A)
            proof
               let x be set;
               hereby assume x in A;
                 then x in X & not x in X\A by A1,XBOOLE_0:def 4;
                 hence x in X\(X\A) by XBOOLE_0:def 4;
               end;
               assume x in X\(X\A);
               then x in X & not x in X\A by XBOOLE_0:def 4;
               hence thesis by XBOOLE_0:def 4;
            end;
            hence thesis by TARSKI:2;
         end;
         hence A in X \ S by A2,Def2;
      end;
      assume A in X \ S;
      then ex B being set st B in S & A = X \ B by Def2;
      hence thesis by Def3;
   end;
   hence thesis by TARSKI:2;
end;

theorem
     for M being set holds
   M is Field_Subset of X iff
   ex S being non empty Subset-Family of X st M = S &
   for A being set st A in S holds X\A in S &
   for A,B being set st A in S & B in S holds A \/ B in S
proof
   let M be set;
   hereby assume
A1:  M is Field_Subset of X;
     then reconsider S = M as non empty Subset-Family of X;
     take S;
     thus M = S;
     thus for A being set st A in S holds X\A in S &
     for A,B being set st (A in S & B in S) holds A \/ B in S
       by A1,Def3,FINSUB_1:def 1;
   end;
   given S being non empty Subset-Family of X such that
A2: M = S & for A being set st A in S holds X\A in S &
   for A,B being set st A in S & B in S holds A \/ B in S;
A3:S is cup-closed
   proof
     let A, B be set; assume A in S & B in S;
     hence A \/ B in S by A2;
   end;
     S is compl-closed
   proof
     let A be set; assume A in S;
     hence X\A in S by A2;
   end;
   hence thesis by A2,A3,Th16;
end;

theorem Th19:
   for S being non empty Subset-Family of X holds
   S is Field_Subset of X iff
   for A being set st A in S holds X\A in S &
   for A,B being set st A in S & B in S holds A /\ B in S
proof
   let S be non empty Subset-Family of X;
   hereby assume A1: S is Field_Subset of X;
       for A,B being set holds A in S & B in S implies A /\ B in S
     proof
       let A,B be set;
       assume
    A2:A in S & B in S;
       then X \ A in S & X \ B in S by A1,Def3;
    then A3:(X \ A) \/ (X \ B) in S by A1,FINSUB_1:def 1;
         A /\ B c= A by XBOOLE_1:17;
       then A /\ B c= X by A2,XBOOLE_1:1;
       then A /\ B = X /\ (A /\ B) by XBOOLE_1:28
            .= X \ (X \ (A /\ B)) by XBOOLE_1:48
            .= X \ ((X \ A) \/ (X \ B)) by XBOOLE_1:54;
       hence thesis by A1,A3,Def3;
     end;
     hence for A being set st A in S holds X\A in S &
     for A,B being set st A in S & B in S holds A /\ B in S by A1,Def3;
   end;
   assume A4: for A being set st A in S holds X\A in S &
   for A,B being set st A in S & B in S holds A /\ B in S;
   then A5:for A being set st A in S holds X\A in S;
     for A,B being set st A in S & B in S holds A \/ B in S
   proof
     let A,B be set;
     assume
     A6:A in S & B in S;
     then X \ A in S & X \ B in S by A4;
     then A7:(X \ A) /\ (X \ B) in S by A4;
       A \/ B c= X by A6,XBOOLE_1:8;
     then A \/ B = X /\ (A \/ B) by XBOOLE_1:28
          .= X \ (X \ (A \/ B)) by XBOOLE_1:48
          .= X \ ((X \ A) /\ (X \ B)) by XBOOLE_1:53;
     hence thesis by A4,A7;
   end;
   then S is cup-closed compl-closed by A5,Def3,FINSUB_1:def 1;
   hence thesis by Th16;
end;

theorem Th20:
   for S being Field_Subset of X holds
   for A,B being set st A in S & B in S holds A \ B in S
proof
   let S be Field_Subset of X;
   let A,B be set;
   assume
   A1:A in S & B in S;
then A2:X \ B in S by Th19;
     A /\ (X \ B) = (A /\ X) \ B by XBOOLE_1:49
              .= A \ B by A1,XBOOLE_1:28;
   hence thesis by A1,A2,Th19;
end;

theorem
     for S being Field_Subset of X holds {} in S & X in S by PROB_1:10,11;

definition
   let S be non empty set;
   let F be Function of S,ExtREAL;
   let A be Element of S;
   redefine func F.A -> R_eal;
coherence by FUNCT_2:7;
end;

definition
   let X be non empty set,
       F be Function of X,ExtREAL;
   redefine attr F is nonnegative means
:Def4:for A being Element of X holds
         0. <=' F.A;
  compatibility by SUPINF_2:58;
end;

canceled;

theorem Th23:
   for S being Field_Subset of X holds
   ex M being Function of S,ExtREAL st
   M is nonnegative & M.{} = 0.&
   for A,B being Element of S st A misses B holds M.(A \/ B) = M.A + M.B
proof
   let S be Field_Subset of X;
   deffunc F(Element of S) = 0.;
     ex f being Function of S,ExtREAL st for x being Element of S holds
   f.x = F(x) from LambdaD;
   then consider M being Function of S,ExtREAL such that
A1:for x being Element of S holds M.x = 0.;
A2:M is nonnegative
   proof
        for x being Element of S holds 0.<=' M.x by A1;
      hence thesis by Def4;
   end;
A3:M.{} = 0.
   proof
      reconsider A = {} as Element of S by PROB_1:10;
        M.A = 0.by A1;
      hence thesis;
   end;
A4:for A,B being Element of S st
   A misses B holds M.(A \/ B) = M.A + M.B
   proof
      let A,B be Element of S;
      assume A misses B;
   A5:M.A = 0.& M.B = 0.by A1;
      reconsider A,B as set;
A6:   A \/ B is Element of S by FINSUB_1:def 1;
      reconsider A,B as Element of S;
        0.= M.A + M.B by A5,SUPINF_2:18;
      hence thesis by A1,A6;
   end;
   take M;
   thus thesis by A2,A3,A4;
end;

definition
   let X be set,
       S be Field_Subset of X;
   mode Measure of S -> Function of S,ExtREAL means
:Def5:it is nonnegative & it.{} = 0. &
         for A,B being Element of S st A misses B holds
          it.(A \/ B) = it.A + it.B;
existence by Th23;
end;

canceled;

theorem Th25:
   for S being Field_Subset of X,
       M being Measure of S,
       A,B being Element of S holds
   A c= B implies M.A <=' M.B
proof
   let S be Field_Subset of X,
       M be Measure of S,
       A,B be Element of S;
   assume
A1:A c= B;
   reconsider C = B \ A as Element of S by Th20;
     A misses C by XBOOLE_1:79;
   then M.(A \/ C) = M.A + M.C by Def5;
   then A2:M.B = M.A + M.C by A1,XBOOLE_1:45;
     M is nonnegative by Def5; then 0.<=' M.A & 0.<=' M.C by Def4;
   hence thesis by A2,SUPINF_2:20;
end;

theorem Th26:
   for S being Field_Subset of X,
       M being Measure of S,
       A,B being Element of S holds
   A c= B & M.A <' +infty implies M.(B \ A) = M.B - M.A
proof
   let S be Field_Subset of X,
       M be Measure of S,
       A,B be Element of S;
   assume
A1:A c= B & M.A <' +infty;
   reconsider C = B \ A as Element of S by Th20;
     A misses C by XBOOLE_1:79;
   then M.(A \/ C) = M.A + M.C by Def5;
   then A2:M.B = M.A + M.C by A1,XBOOLE_1:45;
     M is nonnegative by Def5; then 0.<=' M.A & 0.<=' M.C & M.A <' +infty
   by A1,Def4;
   hence thesis by A2,Th5;
end;

definition let X be set;
   cluster non empty compl-closed cap-closed Subset-Family of X;
  existence
   proof
     consider S being non empty compl-closed cap-closed Subset-Family of X;
    take S;
    thus thesis;
   end;
end;

definition
   let X be set,
       S be non empty cup-closed Subset-Family of X,
       A,B be Element of S;
  redefine func A \/ B -> Element of S;
coherence by FINSUB_1:def 1;
end;

definition
   let X be set,
       S be Field_Subset of X,
       A,B be Element of S;
  redefine func A /\ B -> Element of S;
coherence by Th19;
  redefine func A \ B -> Element of S;
coherence by Th20;
end;

theorem Th27:
   for S being Field_Subset of X,
       M being Measure of S,
       A,B being Element of S holds
   M.(A \/ B) <=' M.A + M.B
proof
   let S be Field_Subset of X,
       M be Measure of S,
       A,B be Element of S;
   set C = B \ A;
A1:A misses C by XBOOLE_1:79;
A2:M.(A \/ B) = M.(A \/ C) by XBOOLE_1:39
            .= M.A + M.C by A1,Def5;
A3:C c= B by XBOOLE_1:36;
     M is nonnegative by Def5;
   then 0.<=' M.A & 0.<=' M.C & M.A <=' M.A & M.C <=' M.B by A3,Def4,Th25;
   hence thesis by A2,Th4;
end;

definition
   let X be set,
       S be Field_Subset of X,
       M be Measure of S,
       A be set;
pred A is_measurable M means
:Def6:A in S;
end;

canceled;

theorem
     for S being Field_Subset of X,
       M being Measure of S holds
   {} is_measurable M & X is_measurable M &
   for A,B being set st
   A is_measurable M & B is_measurable M holds
   X \ A is_measurable M & A \/ B is_measurable M & A /\ B is_measurable M
proof
   let S be Field_Subset of X,
       M be Measure of S;
A1:{} in S & X in S by PROB_1:10,11;
     for A,B being set st
   A is_measurable M & B is_measurable M holds
   X \ A is_measurable M & A \/ B is_measurable M & A /\ B is_measurable M
   proof
      let A,B be set;
      assume A is_measurable M & B is_measurable M;
      then A in S & B in S by Def6;
      then X \ A in S & A \/ B in S & A /\ B in S & A \ B in S
        by Th19,Th20,FINSUB_1:def 1;
      hence thesis by Def6;
   end;
   hence thesis by A1,Def6;
end;

definition
   let X be set,
       S be Field_Subset of X,
       M be Measure of S;
   mode measure_zero of M -> Element of S means
:Def7:M.it = 0.;
existence
proof
   reconsider A = {} as Element of S by PROB_1:10;
   take A;
   thus thesis by Def5;
end;
end;

canceled;

theorem
     for S being Field_Subset of X,
       M being Measure of S,
       A being Element of S,
       B being measure_zero of M st
   A c= B holds A is measure_zero of M
proof
   let S be Field_Subset of X,
       M be Measure of S,
       A be Element of S,
       B be measure_zero of M;
   assume
A1:A c= B;
     M is nonnegative by Def5;
   then 0.<=' M.A & M.A <=' M.B by A1,Def4,Th25;
   then 0.<=' M.A & M.A <=' 0.by Def7;
   then M.A = 0.by SUPINF_1:22;
   hence thesis by Def7;
end;

theorem
     for S being Field_Subset of X,
       M being Measure of S,
       A,B being measure_zero of M holds
   A \/ B is measure_zero of M & A /\ B is measure_zero of M &
    A \ B is measure_zero of M
proof
   let S be Field_Subset of X,
       M be Measure of S,
       A,B be measure_zero of M;
A1:M.A = 0.& M.B = 0.by Def7;
     M is nonnegative by Def5;
   then A2: 0.<=' M.(A \/ B) & 0.<=' M.(A /\ B) & 0.<=' M.(A \ B) by Def4;
     M.(A \/ B) <=' 0.+ 0.by A1,Th27;
   then M.(A \/ B) <=' 0.by SUPINF_2:18;
then A3:M.(A \/ B) = 0.by A2,SUPINF_1:22;
     A /\ B c= A by XBOOLE_1:17;
   then M.(A /\ B) <=' 0.by A1,Th25;
then A4:M.(A /\ B) = 0.by A2,SUPINF_1:22;
     A \ B c= A by XBOOLE_1:36;
   then M.(A \ B) <=' 0.by A1,Th25;
   then M.(A \ B) = 0.by A2,SUPINF_1:22;
   hence thesis by A3,A4,Def7;
end;

theorem
     for S being Field_Subset of X,
       M being Measure of S,
       A being Element of S,
       B being measure_zero of M holds
   M.(A \/ B) = M.A & M.(A /\ B) = 0.& M.(A \ B) = M.A
proof
   let S be Field_Subset of X,
       M be Measure of S,
       A be Element of S,
       B be measure_zero of M;
A1:M.B = 0.by Def7;
  M is nonnegative by Def5;
then A2: 0.<=' M.(A \/ B) & 0.<=' M.(A /\ B) & 0.<=' M.(A \ B) by Def4;
     M.(A \/ B) <=' M.A + 0.by A1,Th27;
then A3:M.(A \/ B) <=' M.A by SUPINF_2:18;
     A c= A \/ B by XBOOLE_1:7; then A4: M.A <=' M.(A \/ B) by Th25;
     A /\ B c= B by XBOOLE_1:17;
then A5:   M.(A /\ B) <=' 0.by A1,Th25;
then A6:M.(A /\ B) = 0.by A2,SUPINF_1:22;
     A \ B c= A by XBOOLE_1:36;
then A7:M.(A \ B) <=' M.A by Th25;
     M.A = M.((A /\ B) \/ (A \ B)) by XBOOLE_1:51; then M.A <='
   0.+ M.(A \ B) by A6,Th27;
   then M.A <=' M.(A \ B) by SUPINF_2:18;
   hence thesis by A2,A3,A4,A5,A7,SUPINF_1:22;
end;

::
::       sigma_Field Subset of X and sigma_additive nonnegative measure
::

theorem Th34:
   for A being Subset of X
   ex F being Function of NAT,bool X st rng F = {A}
proof
   let A be Subset of X;
   reconsider F = NAT --> A as Function of NAT, bool X by FUNCOP_1:57;
   take F;
   thus thesis by FUNCOP_1:14;
end;

theorem Th35:
   for A being set
   ex F being Function of NAT,{A} st
   for n being Nat holds F.n = A
proof
   let A be set;
     A in {A} by TARSKI:def 1;
   then reconsider F = NAT --> A as Function of NAT, {A} by FUNCOP_1:57;
   take F;
   thus thesis by TARSKI:def 1;
end;

definition let X be set;
   cluster non empty countable Subset-Family of X;
existence
proof
A1:{} is Subset of X by XBOOLE_1:2;
   reconsider S = {{}} as Subset-Family of X by Th9;
   take S;
   thus S is non empty;
   thus S is empty or ex F being Function of NAT,bool X st S = rng F
      by A1,Th34;
end;
end;

definition let X be set;
   mode N_Sub_set_fam of X is non empty countable Subset-Family of X;
end;

canceled 2;

theorem Th38:
   for A,B,C being Subset of X
   ex F being Function of NAT,bool X st rng F = {A,B,C} &
   F.0 = A & F.1 = B & for n being Nat st 1 < n holds F.n = C
proof
   let A,B,C be Subset of X;
   defpred P0[Nat,set] means ($1 = 0 implies $2 = A) &
                          ($1 = 1 implies $2 = B) &
                          (1 < $1 implies $2 = C);
   defpred P[set] means ex n being Nat st
                     ex X being set st
                     ($1 = [n,X] & P0[n,X]);
     ex GRAF being set st for x being set holds x in GRAF iff
   x in [:NAT,bool X:] & P[x] from Separation;
   then consider GRAF being set such that
A1:for x being set holds x in GRAF iff
   x in [:NAT,bool X:] & P[x];
A2:for x being set holds x in GRAF iff (x = [0,A] or x = [1,B] or
   (ex n being Nat st (x = [n,C] & 1 < n)))
   proof
      let x be set;
   A3:x in GRAF implies (x = [0,A] or x = [1,B] or
      (ex n being Nat st (x = [n,C] & 1 < n)))
      proof
         assume x in GRAF;
         then consider n being Nat such that
      A4:ex X0 being set st x = [n,X0] & P0[n,X0] by A1;
           n = 0 or n = 1 or 1 < n by CQC_THE1:2;
         hence thesis by A4;
      end;
        (x = [0,A] or x = [1,B] or (ex n being Nat st (x = [n,C] & 1 < n)))
      implies x in GRAF
      proof
         assume
      A5:x = [0,A] or x = [1,B] or (ex n being Nat st (x = [n,C] & 1 < n));
      A6:x = [0,A] implies x in GRAF
         proof
            assume x = [0,A];
            then x in [:NAT,bool X:] & P[x] by ZFMISC_1:106;
            hence thesis by A1;
         end;
      A7:x = [1,B] implies x in GRAF
         proof
            assume x = [1,B];
            then x in [:NAT,bool X:] & P[x] by ZFMISC_1:106;
            hence thesis by A1;
         end;
        (ex n being Nat st (x = [n,C] & 1 < n)) implies x in GRAF
         proof
            given n being Nat such that A8: x = [n,C] & 1 < n;
              P0[n,C] by A8;
            then x in [:NAT,bool X:] & P[x] by A8,ZFMISC_1:106;
            hence thesis by A1;
         end;
         hence thesis by A5,A6,A7;
      end;
      hence thesis by A3;
   end;
A9:for p being set st p in GRAF ex x,y being set st [x,y] = p
   proof
      let p be set;
      assume p in GRAF;
      then p = [0,A] or p = [1,B] or (ex n being Nat st p = [n,C] & 1 < n)
      by A2;
      hence thesis;
   end;
     for x,y1,y2 being set st [x,y1] in GRAF & [x,y2] in GRAF holds y1 = y2
   proof
      let x,y1,y2 be set;
      assume
   A10:[x,y1] in GRAF & [x,y2] in GRAF;
   then A11:[x,y1] in [:NAT,bool X:] & P[[x,y1]] by A1;
   A12:[x,y2] in [:NAT,bool X:] & P[[x,y2]] by A1,A10;
      then reconsider x as Nat by ZFMISC_1:106;
    per cases by CQC_THE1:2;
    suppose
A13:   x = 0;
     then y1 = A by A11,ZFMISC_1:33;
      hence thesis by A12,A13,ZFMISC_1:33;
    suppose
A14:   x = 1;
     then y1 = B by A11,ZFMISC_1:33;
      hence thesis by A12,A14,ZFMISC_1:33;
    suppose
A15:   x > 1;
     then y1 = C by A11,ZFMISC_1:33;
      hence thesis by A12,A15,ZFMISC_1:33;
   end;
   then reconsider F = GRAF as Function by A9,FUNCT_1:2;
A16:dom F = NAT
   proof
        for x being set holds x in NAT iff ex y being set st [x,y] in F
      proof
         let x be set;
         thus x in NAT implies ex y being set st [x,y] in F
         proof
            assume x in NAT; then reconsider x as Nat;
        A17:x = 0 implies ex y being set st [x,y] in F
            proof
               assume A18:x = 0;
               take A;
               thus thesis by A2,A18;
            end;
        A19:x = 1 implies ex y being set st [x,y] in F
            proof
               assume A20:x = 1;
               take B;
               thus thesis by A2,A20;
            end;
              1 < x implies ex y being set st [x,y] in F
            proof
               assume
           A21:1 < x;
               take C;
               thus thesis by A2,A21;
            end;
            hence thesis by A17,A19,CQC_THE1:2;
         end;
         given y being set such that
     A22:[x,y] in F;
           [x,y] in [:NAT,bool X:] by A1,A22;
         hence thesis by ZFMISC_1:106;
      end;
      hence thesis by RELAT_1:def 4;
   end;
A23:rng F = {A,B,C}
   proof
        for y being set holds y in {A,B,C} iff
      ex x being set st x in dom F & y = F.x
      proof
         let y be set;
         thus y in {A,B,C} implies ex x being set st (x in dom F & y = F.x)
         proof
           assume
A24:       y in {A,B,C};
           per cases by A24,ENUMSET1:def 1;
           suppose
            y = A;
           then A25:[0,y] in F by A2;
               take 0;
               thus thesis by A25,FUNCT_1:8;
           suppose
            y = B;
           then A26:[1,y] in F by A2;
               take 1;
               thus thesis by A26,FUNCT_1:8;
           suppose
            y = C;
           then A27:[2,y] in F by A2;
              take 2;
               thus thesis by A27,FUNCT_1:8;
         end;
         given x being set such that
     A28:x in dom F & y = F.x;
         reconsider x as Nat by A16,A28;
         per cases by CQC_THE1:2;
         suppose
            A29:x = 0; [0,A] in F by A2; then A = F.x by A29,FUNCT_1:8;
                hence thesis by A28,ENUMSET1:def 1;
         suppose
            A30:x = 1; [1,B] in F by A2; then B = F.x by A30,FUNCT_1:8;
                hence thesis by A28,ENUMSET1:def 1;
         suppose
              1 < x; then [x,C] in F by A2; then y = C by A28,FUNCT_1:8;
                hence thesis by ENUMSET1:def 1;
     end;
     hence thesis by FUNCT_1:def 5;
   end;
     rng F c= bool X
   proof
      let a be set; assume a in rng F;
      then a = A or a = B or a = C by A23,ENUMSET1:def 1;
      hence thesis;
   end;
   then reconsider F as Function of NAT,bool X by A16,FUNCT_2:4;
   take F;
     F.0 = A & F.1 = B & for n being Nat st 1 < n holds F.n = C
   proof
A31:   [0,A] in F by A2;
A32:   [1,B] in F by A2;
         for n being Nat st 1 < n holds F.n = C
       proof
          let n be Nat;
          assume 1 < n; then [n,C] in F by A2;
          hence thesis by FUNCT_1:8;
       end;
       hence thesis by A31,A32,FUNCT_1:8;
   end;
   hence thesis by A23;
end;

theorem
     for A,B being Subset of X holds
   {A,B,{}} is N_Sub_set_fam of X
proof
   let A,B be Subset of X;
     {} is Subset of X by XBOOLE_1:2;
   then consider C being Subset of X such that
A1:C = {};
     ex F being Function of NAT,bool X st rng F = {A,B,C} &
   F.0 = A & F.1 = B & for n being Nat st 1 < n holds F.n = C by Th38;
   hence thesis by A1,Th8,SUPINF_2:def 14;
end;

theorem Th40:
   for A,B being Subset of X
   ex F being Function of NAT,bool X st rng F = {A,B} &
   F.0 = A & for n being Nat st 0 < n holds F.n = B
proof
   let A,B be Subset of X;
   consider F being Function of NAT,bool X such that
A1:rng F = {A,B,B} &
   F.0 = A & F.1 = B & for n being Nat st 1 < n holds F.n = B by Th38;
A2:{A,B,B} = {B,B,A} by ENUMSET1:100
          .= {A,B} by ENUMSET1:70;
A3:for n being Nat st 0 < n holds F.n = B
   proof
      let n be Nat;
      assume 0 < n;
      then n = 1 or 1 < n by CQC_THE1:2;
      hence thesis by A1;
   end;
   take F;
   thus thesis by A1,A2,A3;
end;

theorem Th41:
   for A,B being Subset of X holds
   {A,B} is N_Sub_set_fam of X
proof
   let A,B be Subset of X;
     ex F being Function of NAT,bool X st (rng F = {A,B} &
   F.0 = A & for n being Nat st 0 < n holds F.n = B) by Th40;
   hence thesis by Th7,SUPINF_2:def 14;
end;

theorem Th42:
   for S being N_Sub_set_fam of X holds
   X \ S is N_Sub_set_fam of X
proof
   let S be N_Sub_set_fam of X;
   consider F being Function of NAT,bool X such that
A1:S = rng F by SUPINF_2:def 14;
A2:NAT = dom F by FUNCT_2:def 1;
A3:for n being set st n in NAT ex Y being set st Y in S & Y=F.n
   proof
      let n be set;
      assume
   A4:n in NAT;
      take F.n;
      thus thesis by A1,A4,FUNCT_2:6;
   end;
   defpred P[set,set] means ex V being set st V = F.$1 & $2 = X \ V;
A5:for n being set st n in NAT ex y being set st y in X\S & P[n,y]
   proof
      let n be set;
      assume n in NAT;
      then consider V being set such that
  A6: V in S & V=F.n by A3;
      take X \ V;
      thus thesis by A6,Def2;
   end;
A7:ex G being Function of NAT,X \ S st for n being set st n in NAT holds
    P[n,G.n] from FuncEx1(A5);
     ex G being Function of NAT,bool X st X \ S = rng G
   proof
       consider G being Function of NAT,X \ S such that
   A8:for n being set st n in NAT holds
       (ex V being set st V = F.n & G.n = X \ V) by A7;
   A9:dom G = NAT & rng G c= X \ S by FUNCT_2:def 1,RELSET_1:12;
A10:   X \ S c= rng G
       proof
         let x be set;
         assume x in X \ S;
         then consider B being set such that
     A11:B in S & x = X \ B by Def2;
         consider n being set such that
     A12:n in NAT & F.n = B by A1,A2,A11,FUNCT_1:def 5;
           ex V being set st V = F.n & G.n = X \ V by A8,A12;
         hence thesis by A9,A11,A12,FUNCT_1:def 5;
       end;
       reconsider G as Function of NAT,bool X by FUNCT_2:9;
       take G;
       thus thesis by A9,A10,XBOOLE_0:def 10;
   end;
   hence thesis by SUPINF_2:def 14;
end;

definition
   let X be set;
   let IT be Subset-Family of X;
 canceled;

   attr IT is sigma_Field_Subset-like means :Def9:
    for M being N_Sub_set_fam of X st M c= IT holds union M in IT;
end;

definition let X be set;
   cluster non empty compl-closed sigma_Field_Subset-like Subset-Family of X;
existence
proof
   reconsider S = {{},X} as non empty Subset-Family of X by PROB_1:14;
   take S;
   thus S is non empty;
   thus for A being set holds A in S implies X\A in S
   proof
      let A be set;
      assume
A1:   A in S;
A2:   A = {} implies X\A in S by TARSKI:def 2;
        A = X implies X\A in S
      proof
         assume A = X;
         then X\A = {} by XBOOLE_1:37;
         hence thesis by TARSKI:def 2;
      end;
      hence thesis by A1,A2,TARSKI:def 2;
   end;
   let M be N_Sub_set_fam of X;
   assume
A3:M c= S;
A4:X in M implies union M in S
   proof
      assume X in M;
      then X c= union M by ZFMISC_1:92;
      then X = union M by XBOOLE_0:def 10;
      hence thesis by TARSKI:def 2;
   end;
     not X in M implies union M in S
   proof
      assume not X in M;
      then for A being set st
      A in M holds A c= {} by A3,TARSKI:def 2;
      then union M c= {} by ZFMISC_1:94;
      then union M = {} by XBOOLE_1:3;
      hence thesis by TARSKI:def 2;
   end;
   hence thesis by A4;
end;
end;

definition let X be set;
   mode sigma_Field_Subset of X is
     compl-closed sigma_Field_Subset-like (non empty Subset-Family of X);
end;

Lm1:
   for S being non empty Subset-Family of X st
   S is sigma_Field_Subset of X holds S is Field_Subset of X
proof
   let S be non empty Subset-Family of X;
   assume
A1:S is sigma_Field_Subset of X;
     for A,B being set st A in S & B in S holds A \/ B in S
   proof
      let A,B be set;
      assume
   A2:A in S & B in S;
      then reconsider A,B as Subset of X;
   A3:{A,B} is N_Sub_set_fam of X by Th41;
   A4:union{A,B} = A \/ B by ZFMISC_1:93;
        {A,B} c= S
      proof
           for x being set holds x in {A,B} implies x in S by A2,TARSKI:def 2;
         hence thesis by TARSKI:def 3;
      end;
      hence thesis by A1,A3,A4,Def9;
   end;
   then S is cup-closed by FINSUB_1:def 1;
   hence thesis by A1,Th16;
end;

canceled 2;

theorem Th45:
   for S being sigma_Field_Subset of X holds {} in S & X in S
proof
   let S be sigma_Field_Subset of X;
     S is Field_Subset of X by Lm1;
   hence thesis by PROB_1:10,11;
end;

definition let X be set;
 cluster -> non empty sigma_Field_Subset of X;
 coherence;
end;

theorem Th46:
   for S being sigma_Field_Subset of X,
       A,B being set st A in S & B in S holds A \/ B in S & A /\ B in S
proof
   let S be sigma_Field_Subset of X;
   let A,B be set;
   assume
A1:A in S & B in S;
     S is Field_Subset of X by Lm1;
   hence thesis by A1,Th19,FINSUB_1:def 1;
end;

theorem Th47:
   for S being sigma_Field_Subset of X,
       A,B being set st A in S & B in S holds A \ B in S
proof
   let S be sigma_Field_Subset of X;
   let A,B be set;
   assume
A1:A in S & B in S;
     S is Field_Subset of X by Lm1;
   hence thesis by A1,Th20;
end;

theorem
     for S being sigma_Field_Subset of X holds S = X \ S
proof
   let S be sigma_Field_Subset of X;
     S is Field_Subset of X by Lm1;
   hence thesis by Th17;
end;

theorem Th49:
   for S being non empty Subset-Family of X holds
   ((for A being set holds A in S implies X\A in S) &
   (for M being N_Sub_set_fam of X st M c= S holds meet M in S)) iff
   S is sigma_Field_Subset of X
proof
   let S be non empty Subset-Family of X;
   hereby assume
A1:  (for A being set holds A in S implies X\A in S) &
     (for M being N_Sub_set_fam of X st M c= S holds meet M in S);
        for M being N_Sub_set_fam of X st M c= S holds union M in S
      proof
          let M be N_Sub_set_fam of X;
          assume
      A2: M c= S;
      A3: X \ M is N_Sub_set_fam of X by Th42;
            X \ M c= S
          proof
             let y be set;
             assume y in X \ M;
             then consider B being set such that
          A4:B in M & y = X \ B by Def2;
             thus thesis by A1,A2,A4;
          end;
          then meet (X \ M) in S by A1,A3;
          then X \ meet (X \ M) in S by A1;
          hence thesis by Th15;
      end;
      hence S is sigma_Field_Subset of X by A1,Def3,Def9;
    end;
    assume A5: S is sigma_Field_Subset of X;
      for M being N_Sub_set_fam of X st M c= S holds meet M in S
    proof
        let M be N_Sub_set_fam of X;
        assume
    A6: M c= S;
    A7: X \ M is N_Sub_set_fam of X by Th42;
          X \ M c= S
        proof
           let y be set;
           assume y in X \ M;
           then consider B being set such that
        A8:B in M & y = X \ B by Def2;
           thus thesis by A5,A6,A8,Def3;
        end;
        then union (X \ M) in S by A5,A7,Def9;
        then X \ union (X \ M) in S by A5,Def3;
        hence thesis by Th15;
    end;
    hence thesis by A5,Def3;
end;

definition
   let X be set;
   let S be sigma_Field_Subset of X;
   cluster disjoint_valued Function of NAT, S;
existence
proof
   consider F being Function of NAT,{{}} such that
A1:for n being Nat holds F.n = {} by Th35;
     {} in S by Th45; then {{}} c= S by ZFMISC_1:37;
   then reconsider F as Function of NAT,S by FUNCT_2:9;
   take F;
A2:for n being set holds F.n = {}
   proof
     let n be set;
     per cases;
     suppose n in dom F; then n in NAT by FUNCT_2:def 1; hence thesis by A1;
     suppose not n in dom F; hence thesis by FUNCT_1:def 4;
   end;
   thus for x,y being set st x <> y holds F.x misses F.y
   proof
     let x,y be set;
       F.x = {} & F.y = {} by A2;
     hence thesis by XBOOLE_1:65;
   end;
end;
end;

definition
   let X be set;
   let S be sigma_Field_Subset of X;
   mode Sep_Sequence of S is disjoint_valued Function of NAT, S;
end;

definition
   let X be set;
   let S be sigma_Field_Subset of X;
   let F be Function of NAT,S;
   redefine func rng F -> non empty Subset-Family of X;
coherence
proof
     0 in NAT;
   then rng F <> {} by FUNCT_2:6;
   then consider x being set such that A1: x in rng F by XBOOLE_0:def 1;
  rng F c= S by RELSET_1:12;
   then rng F c= bool X by XBOOLE_1:1;
   hence thesis by A1,SETFAM_1:def 7;
end;
end;

canceled 2;

theorem Th52:
   for S being sigma_Field_Subset of X,
       F being Function of NAT,S holds
   rng F is N_Sub_set_fam of X
proof
   let S be sigma_Field_Subset of X;
   let F be Function of NAT,S;
     ex H being Function of NAT,bool X st rng F = rng H
   proof
        rng F c= bool X;
      then reconsider F as Function of NAT,bool X by FUNCT_2:8;
      take F;
      thus thesis;
   end;
   hence thesis by SUPINF_2:def 14;
end;

theorem Th53:
   for S being sigma_Field_Subset of X,
       F being Function of NAT,S holds
   union rng F is Element of S
proof
   let S be sigma_Field_Subset of X,
       F be Function of NAT,S;
A1:rng F is N_Sub_set_fam of X by Th52;
     rng F c= S by RELSET_1:12;
   hence thesis by A1,Def9;
end;

theorem Th54:
   for Y,S being non empty set,
       F being Function of Y,S,
       M being Function of S,ExtREAL st
   M is nonnegative holds M*F is nonnegative
proof
   let Y,S be non empty set;
   let F be Function of Y,S;
   let M be Function of S,ExtREAL;
   assume
A1:M is nonnegative;
     for n being Element of Y holds 0.<=' (M*F).n
   proof
      let n be Element of Y;
        dom (M*F) = Y & rng (M*F) c= ExtREAL by FUNCT_2:def 1,RELSET_1:12;
      then (M*F).n = M.(F.n) by FUNCT_1:22;
      hence thesis by A1,Def4;
   end;
   hence thesis by SUPINF_2:58;
end;

theorem Th55:
   for S being sigma_Field_Subset of X,
       a,b being R_eal holds
   ex M being Function of S,ExtREAL st
   for A being Element of S holds
   (A = {} implies M.A = a) & (A <> {} implies M.A = b)
proof
   let S be sigma_Field_Subset of X,
       a,b be R_eal;
   defpred P[set,set] means ($1 = {} implies $2 = a) &
                            ($1 <> {} implies $2 = b);
   A1:for x being set st x in S ex y being set st y in ExtREAL & P[x,y]
      proof
         let x be set;
           x <> {} implies ex y being set st y in ExtREAL & P[x,y];
         hence thesis;
      end;
        ex F being Function of S,ExtREAL st for x being set st x in S holds
      P[x,F.x] from FuncEx1(A1);
      then consider M being Function of S,ExtREAL such that
   A2:for x being set st x in S holds P[x,M.x];
      take M;
      thus thesis by A2;
end;

theorem
     for S being sigma_Field_Subset of X
   ex M being Function of S,ExtREAL st
   for A being Element of S holds
   (A = {} implies M.A = 0.) & (A <> {} implies M.A = +infty) by Th55;

theorem Th57:
   for S being sigma_Field_Subset of X
   ex M being Function of S,ExtREAL st
   for A being Element of S holds M.A = 0.
proof
   let S be sigma_Field_Subset of X;
   consider M being Function of S,ExtREAL such that
A1:for A being Element of S holds
   (A = {} implies M.A = 0.) & (A <> {} implies M.A = 0.) by Th55;
A2:for A being Element of S holds M.A = 0.
   proof
      let A be Element of S;
        A = {} implies M.A = 0.by A1;
      hence thesis by A1;
   end;
   take M;
   thus thesis by A2;
end;

theorem Th58:
   for S being sigma_Field_Subset of X
   ex M being Function of S,ExtREAL st
   M is nonnegative & M.{} = 0. &
   for F being Sep_Sequence of S holds
   SUM(M*F) = M.(union rng F)
proof
   let S be sigma_Field_Subset of X;
   consider M being Function of S,ExtREAL such that
A1:for A being Element of S holds M.A = 0. by Th57;
   take M;
     for A being Element of S holds 0.<=' M.A by A1;
   hence A2: M is nonnegative by Def4;
     {} is Element of S by Th45;
   hence M.{} = 0. by A1;
   let F be Sep_Sequence of S;
     union rng F is Element of S by Th53;
then A3:M.(union rng F) = 0.by A1;
A4:M*F is nonnegative by A2,Th54;
A5:for r being Nat st 0 <= r holds (M*F).r = 0.
   proof
      let r be Nat;
        dom (M*F) = NAT by FUNCT_2:def 1;
      then (M*F).r = M.(F.r) by FUNCT_1:22;
      hence thesis by A1;
   end;
   then A6:SUM(M*F) = Ser(M*F).0 by A4,SUPINF_2:67;
     Ser(M*F).0 = (M*F).0 by SUPINF_2:63;
   hence thesis by A3,A5,A6;
end;

definition
   let X be set;
   let S be sigma_Field_Subset of X;
 canceled;

   mode sigma_Measure of S -> Function of S,ExtREAL means
:Def11:it is nonnegative &
         it.{} = 0.&
         for F being Sep_Sequence of S holds
         SUM(it*F) = it.(union rng F);
existence by Th58;
end;

definition let X be set;
   cluster sigma_Field_Subset-like compl-closed -> cup-closed
     (non empty Subset-Family of X);
   coherence by Lm1;
end;

canceled;

theorem Th60:
   for S being sigma_Field_Subset of X,
       M being sigma_Measure of S holds M is Measure of S
proof
   let S be sigma_Field_Subset of X,
       M be sigma_Measure of S;
A1:M is nonnegative & M.{} = 0.by Def11;
     for A,B being Element of S holds
   (A misses B implies M.(A \/ B) = M.A + M.B)
   proof
      let A,B be Element of S;
      assume
   A2:A misses B;
   A3:A in S & B in S & {} in S by Th45;
A4:   {} is Subset of X by XBOOLE_1:2;
      reconsider A,B as Subset of X;
      consider F being Function of NAT,bool X such that
   A5:rng F = {A,B,{}} &
      F.0 = A & F.1 = B & for n being Nat st 1 < n holds F.n = {} by A4,Th38;
        F is Function of NAT,S
      proof
           {A,B,{}} c= S
         proof
              for x being set holds x in {A,B,{}} implies x in S
              by A3,ENUMSET1:13;
            hence thesis by TARSKI:def 3;
         end;
         hence thesis by A5,FUNCT_2:8;
      end; then reconsider F as Function of NAT,S;
A6:    for n,m being Nat st n <> m holds F.n misses F.m
      proof
         let n,m be Nat;
         assume
      A7: n <> m;
A8:      (n = 0 or n = 1 or 1 < n) & (m = 0 or m = 1 or 1 < m) by CQC_THE1:2;
      A9:n = 0 & 1 < m implies F.n misses F.m
         proof
            assume n = 0 & 1 < m;
            then F.n = A & F.m = {} by A5;
            then F.n /\ F.m = {};
            hence thesis by XBOOLE_0:def 7;
         end;
      A10:n = 1 & 1 < m implies F.n misses F.m
         proof
            assume n = 1 & 1 < m;
            then F.n = B & F.m = {} by A5;
            then F.n /\ F.m = {};
            hence thesis by XBOOLE_0:def 7;
         end;
     A11:(1 < n & m = 0) implies F.n misses F.m
         proof
            assume 1 < n & m = 0;
            then F.n = {} & F.m = A by A5;
            then F.n /\ F.m = {};
            hence thesis by XBOOLE_0:def 7;
         end;
     A12:1 < n & m = 1 implies F.n misses F.m
         proof
            assume 1 < n & m = 1;
            then F.n = {} & F.m = B by A5;
            then F.n /\ F.m = {};
            hence thesis by XBOOLE_0:def 7;
         end;
           1 < n & 1 < m implies F.n misses F.m
         proof
            assume 1 < n & 1 < m;
            then F.n = {} & F.m = {} by A5;
            then F.n /\ F.m = {};
            hence thesis by XBOOLE_0:def 7;
         end;
         hence thesis by A2,A5,A7,A8,A9,A10,A11,A12;
      end;
        for m,n being set st m <> n holds F.m misses F.n
      proof
        let m,n be set; assume A13: m <> n;
        per cases;
        suppose m in NAT & n in NAT;
        hence thesis by A6,A13;
        suppose not m in NAT or not n in NAT;
        then not m in dom F or not n in dom F by FUNCT_2:def 1;
        then F.m = {} or F.n = {} by FUNCT_1:def 4;
        hence thesis by XBOOLE_1:65;
      end;
      then reconsider F as disjoint_valued Function of NAT, S by PROB_2:def 3;
   A14:union rng F = A \/ B
      proof
           union {A,B,{}} = union {A,B} by Th1;
         hence thesis by A5,ZFMISC_1:93;
      end;
   A15:for r being Nat holds (M*F).0 = M.A & (M*F).1 = M.B &
      (1+1 <= r implies (M*F).r = 0.)
      proof
         let r be Nat;
     A16:for r being Nat holds (M*F).r = M.(F.r)
         proof
            let r be Nat;
              dom (M*F) = NAT by FUNCT_2:def 1;
            hence thesis by FUNCT_1:22;
         end;
           1 + 1 <= r implies (M*F).r = 0.
         proof
            assume 1 + 1 <= r;
            then 1 < r by NAT_1:38; then F.r = {} by A5;
            hence thesis by A1,A16;
         end;
         hence thesis by A5,A16;
      end;
      set H = M*F;
  A17:H is nonnegative by A1,Th54;
A18:  0 + 1 = 0 + 1;
      set y1 = Ser H.1;
      set n2 = 1 + 1;
      reconsider A,B as Element of S;
        Ser H.n2 = y1 + H.n2 by SUPINF_2:63;
      then Ser H.n2 = y1 + 0.by A15
                  .= Ser H.1 by SUPINF_2:18
                  .= Ser H.0 + H.1 by A18,SUPINF_2:63
                  .= M.A + M.B by A15,SUPINF_2:63;
      then SUM(M*F) = M.A + M.B by A15,A17,SUPINF_2:67;
      hence thesis by A14,Def11;
   end;
   hence thesis by A1,Def5;
end;

theorem
     for S being sigma_Field_Subset of X,
       M being sigma_Measure of S,
       A,B being Element of S st
   A misses B holds M.(A \/ B) = M.A + M.B
proof
   let S be sigma_Field_Subset of X,
       M be sigma_Measure of S,
       A,B be Element of S;
   assume
A1:A misses B;
     M is Measure of S by Th60;
   hence thesis by A1,Def5;
end;

theorem Th62:
   for S being sigma_Field_Subset of X,
       M being sigma_Measure of S,
       A,B being Element of S st A c= B holds M.A <=' M.B
proof
   let S be sigma_Field_Subset of X,
       M be sigma_Measure of S,
       A,B be Element of S;
   assume
A1:A c= B;
     M is Measure of S by Th60;
   hence thesis by A1,Th25;
end;

theorem
     for S being sigma_Field_Subset of X,
       M being sigma_Measure of S,
       A,B being Element of S st
   A c= B & M.A <' +infty holds M.(B \ A) = M.B - M.A
proof
   let S be sigma_Field_Subset of X,
       M be sigma_Measure of S,
       A,B be Element of S;
   assume
A1:A c= B & M.A <' +infty;
     M is Measure of S by Th60;
   hence thesis by A1,Th26;
end;

theorem Th64:
   for S being sigma_Field_Subset of X,
       M being sigma_Measure of S,
       A,B being Element of S holds
   M.(A \/ B) <=' M.A + M.B
proof
   let S be sigma_Field_Subset of X,
       M be sigma_Measure of S,
       A,B be Element of S;
     M is Measure of S by Th60;
   hence thesis by Th27;
end;

definition
   let X be set,
       S be sigma_Field_Subset of X,
       M be sigma_Measure of S,
       A be set;
   pred A is_measurable M means
:Def12:A in S;
end;

canceled;

theorem
     for S being sigma_Field_Subset of X,
       M being sigma_Measure of S holds
   {} is_measurable M & X is_measurable M &
   for A,B being set st A is_measurable M & B is_measurable M holds
   X \ A is_measurable M & A \/ B is_measurable M &
   A /\ B is_measurable M
proof
   let S be sigma_Field_Subset of X,
       M be sigma_Measure of S;
A1:{} in S & X in S by Th45;
     for A,B being set st
   A is_measurable M & B is_measurable M holds
   X \ A is_measurable M & A \/ B is_measurable M &
   A /\ B is_measurable M
   proof
      let A,B be set;
      assume A is_measurable M & B is_measurable M;
      then A in S & B in S by Def12;
      then X \ A in S & A \/ B in S & A /\ B in S & A \ B in S by Def3,Th46,
Th47;
      hence thesis by Def12;
   end;
   hence thesis by A1,Def12;
end;

theorem
     for S being sigma_Field_Subset of X,
       M being sigma_Measure of S,
       T being N_Sub_set_fam of X st
   (for A being set st A in T holds A is_measurable M) holds
   union T is_measurable M & meet T is_measurable M
proof
   let S be sigma_Field_Subset of X,
       M be sigma_Measure of S,
       T be N_Sub_set_fam of X;
   assume
A1:for A being set st A in T holds A is_measurable M;
A2:T c= S
   proof
     let x be set;
     assume x in T;
     then x is_measurable M by A1;
     hence thesis by Def12;
   end;
then A3:union T in S by Def9;
     meet T in S by A2,Th49;
   hence thesis by A3,Def12;
end;

definition
   let X be set,
       S be sigma_Field_Subset of X,
       M be sigma_Measure of S;
   mode measure_zero of M -> Element of S means
:Def13:M.it = 0.;
existence
proof
     {} is Element of S by Th45;
   then consider A being Element of S such that
A1:A = {};
   take A;
   thus thesis by A1,Def11;
end;
end;

canceled;

theorem
     for S being sigma_Field_Subset of X,
       M being sigma_Measure of S,
       A being Element of S,
       B being measure_zero of M st
   A c= B holds A is measure_zero of M
proof
   let S be sigma_Field_Subset of X,
       M be sigma_Measure of S,
       A be Element of S,
       B be measure_zero of M;
   assume
A1:A c= B;
     M is nonnegative by Def11;
   then 0.<=' M.A & M.A <=' M.B by A1,Def4,Th62;
   then 0.<=' M.A & M.A <=' 0.by Def13;
   then M.A = 0.by SUPINF_1:22;
   hence thesis by Def13;
end;

theorem
     for S being sigma_Field_Subset of X,
       M being sigma_Measure of S,
       A,B being measure_zero of M holds
   A \/ B is measure_zero of M & A /\ B is measure_zero of M &
   A \ B is measure_zero of M
proof
   let S be sigma_Field_Subset of X,
       M be sigma_Measure of S,
       A,B be measure_zero of M;
A1:M.A = 0.& M.B = 0.by Def13;
     M is nonnegative by Def11;
   then A2: 0.<=' M.(A \/ B) & 0.<=' M.(A /\ B) & 0.<=' M.(A \ B) by Def4;
     M.(A \/ B) <=' 0.+ 0.by A1,Th64;
   then M.(A \/ B) <=' 0.by SUPINF_2:18;
then A3:   M.(A \/ B) = 0.by A2,SUPINF_1:22;
     A /\ B c= A by XBOOLE_1:17;
   then M.(A /\ B) <=' 0.by A1,Th62;
then A4:   M.(A /\ B) = 0.by A2,SUPINF_1:22;
     A \ B c= A by XBOOLE_1:36;
   then M.(A \ B) <=' 0.by A1,Th62;
   then M.(A \ B) = 0.by A2,SUPINF_1:22;
   hence thesis by A3,A4,Def13;
end;

theorem
     for S being sigma_Field_Subset of X,
       M being sigma_Measure of S,
       A being Element of S,
       B being measure_zero of M holds
   M.(A \/ B) = M.A & M.(A /\ B) = 0.& M.(A \ B) = M.A
proof
   let S be sigma_Field_Subset of X,
       M be sigma_Measure of S,
       A be Element of S,
       B be measure_zero of M;
A1:M.B = 0.by Def13;
     M is nonnegative by Def11;
   then A2: 0.<=' M.(A \/ B) & 0.<=' M.(A /\ B) & 0.<=' M.(A \ B) by Def4;
     M.(A \/ B) <=' M.A + 0.by A1,Th64;
then A3:M.(A \/ B) <=' M.A by SUPINF_2:18;
     A c= A \/ B by XBOOLE_1:7; then A4: M.A <=' M.(A \/ B) by Th62;
     A /\ B c= B by XBOOLE_1:17;
then A5:M.(A /\ B) <=' 0.by A1,Th62;
then A6:M.(A /\ B) = 0.by A2,SUPINF_1:22;
     A \ B c= A by XBOOLE_1:36;
then A7:M.(A \ B) <=' M.A by Th62;
     M.A = M.((A /\ B) \/ (A \ B)) by XBOOLE_1:51;
   then M.A <=' 0.+ M.(A \ B) by A6,Th64;
   then M.A <=' M.(A \ B) by SUPINF_2:18;
   hence thesis by A2,A3,A4,A5,A7,SUPINF_1:22;
end;

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