lpspace2.miz



    begin

    reserve X for non empty set,

x for Element of X,

S for SigmaField of X,

M for sigma_Measure of S,

f,g,f1,g1 for PartFunc of X, REAL ,

l,m,n,n1,n2 for Nat,

a,b,c for Real;

    theorem :: LPSPACE2:1

    

     Th1: for m,n be positive Real st ((1 / m) + (1 / n)) = 1 holds m > 1

    proof

      let m,n be positive Real;

      assume ((1 / m) + (1 / n)) = 1;

      then

       A1: (1 / n) = (1 - (1 / m));

      assume m <= 1;

      then 1 <= (1 / m) by XREAL_1: 181;

      hence contradiction by A1, XREAL_1: 47;

    end;

    theorem :: LPSPACE2:2

    

     Th2: for X be non empty set, S be SigmaField of X, M be sigma_Measure of S, A be Element of S, f be PartFunc of X, ExtREAL st A = ( dom f) & f is A -measurable & f is nonnegative holds ( Integral (M,f)) in REAL iff f is_integrable_on M

    proof

      let X be non empty set, S be SigmaField of X, M be sigma_Measure of S, A be Element of S, f be PartFunc of X, ExtREAL ;

      assume

       A1: A = ( dom f) & f is A -measurable & f is nonnegative;

       A2:

      now

        assume f is_integrable_on M;

        then -infty < ( Integral (M,f)) & ( Integral (M,f)) < +infty by MESFUNC5: 96;

        hence ( Integral (M,f)) in REAL by XXREAL_0: 14;

      end;

      now

        assume

         A3: ( Integral (M,f)) in REAL ;

        

         A4: ( dom ( max- f)) = A & ( max- f) is A -measurable by A1, MESFUNC2: 26, MESFUNC2:def 3;

        

         A5: ( dom ( max+ f)) = A & ( max+ f) is A -measurable by A1, MESFUNC2: 25, MESFUNC2:def 2;

        for x be Element of X holds 0 <= (( max+ f) . x) by MESFUNC2: 12;

        then ( max+ f) is nonnegative by SUPINF_2: 39;

        then

         A6: ( Integral (M,( max+ f))) = ( integral+ (M,( max+ f))) by A5, MESFUNC5: 88;

        

         A7: for x be Element of X st x in ( dom f) holds (( max+ f) . x) = (f . x)

        proof

          let x be Element of X;

          

           A8: (f . x) >= 0 by A1, SUPINF_2: 39;

          assume x in ( dom f);

          then (( max+ f) . x) = ( max ((f . x), 0 )) by A1, A5, MESFUNC2:def 2;

          hence thesis by A8, XXREAL_0:def 10;

        end;

        then ( max+ f) = f by A1, A5, PARTFUN1: 5;

        then

         A9: ( Integral (M,( max+ f))) < +infty by A3, XXREAL_0: 9;

        for x be Element of X holds 0 <= (( max- f) . x) by MESFUNC2: 13;

        then ( max- f) is nonnegative by SUPINF_2: 39;

        then

         A10: ( Integral (M,( max- f))) = ( integral+ (M,( max- f))) by A4, MESFUNC5: 88;

        for x be Element of X st x in ( dom ( max- f)) holds 0 = (( max- f) . x)

        proof

          let x be Element of X;

          assume x in ( dom ( max- f));

          (( max+ f) . x) = (f . x) by A1, A5, A7, PARTFUN1: 5;

          hence 0 = (( max- f) . x) by MESFUNC2: 19;

        end;

        then ( Integral (M,( max- f))) = 0 by A4, LPSPACE1: 22;

        hence f is_integrable_on M by A1, A6, A9, A10;

      end;

      hence thesis by A2;

    end;

    definition

      let r be Real;

      :: LPSPACE2:def1

      attr r is geq_than_1 means

      : Def1: 1 <= r;

    end

    registration

      cluster geq_than_1 -> positive for Real;

      coherence ;

    end

    reconsider jj = 1 as Real;

    registration

      cluster geq_than_1 for Real;

      existence by Def1;

    end

    registration

      cluster geq_than_1 for Real;

      existence by Def1;

    end

    reserve k for positive Real;

    theorem :: LPSPACE2:3

    

     Th3: for a,b,p be Real st 0 < p & 0 <= a & a < b holds (a to_power p) < (b to_power p)

    proof

      let a,b,p be Real;

      assume

       A1: 0 < p & 0 <= a & a < b;

      now

        assume a = 0 ;

        then (a to_power p) = 0 by A1, POWER:def 2;

        hence (a to_power p) < (b to_power p) by A1, POWER: 34;

      end;

      hence thesis by A1, POWER: 37;

    end;

    theorem :: LPSPACE2:4

    

     Th4: a >= 0 & b > 0 implies (a to_power b) >= 0

    proof

      assume

       A1: a >= 0 ;

      assume b > 0 ;

      then a = 0 implies (a to_power b) >= 0 by POWER:def 2;

      hence thesis by A1, POWER: 34;

    end;

    theorem :: LPSPACE2:5

    

     Th5: a >= 0 & b >= 0 & c > 0 implies ((a * b) to_power c) = ((a to_power c) * (b to_power c))

    proof

      assume that

       A1: a >= 0 & b >= 0 and

       A2: c > 0 ;

      now

        assume

         A3: a = 0 or b = 0 ;

        then ((a * b) to_power c) = 0 by A2, POWER:def 2;

        hence ((a * b) to_power c) = ((a to_power c) * (b to_power c)) by A3;

      end;

      hence thesis by A1, POWER: 30;

    end;

    theorem :: LPSPACE2:6

    

     Th6: for a,b be Real, f st f is nonnegative & a > 0 & b > 0 holds ((f to_power a) to_power b) = (f to_power (a * b))

    proof

      let a,b be Real;

      let f;

      assume

       A1: f is nonnegative & a > 0 & b > 0 ;

      

       A2: ( dom (f to_power a)) = ( dom f) & ( dom ((f to_power a) to_power b)) = ( dom (f to_power a)) & ( dom (f to_power (a * b))) = ( dom f) by MESFUN6C:def 4;

      for x be object st x in ( dom ((f to_power a) to_power b)) holds (((f to_power a) to_power b) . x) = ((f to_power (a * b)) . x)

      proof

        let x be object;

        assume

         A3: x in ( dom ((f to_power a) to_power b));

        

        then

         A4: (((f to_power a) to_power b) . x) = (((f to_power a) . x) to_power b) by MESFUN6C:def 4

        .= (((f . x) to_power a) to_power b) by A2, A3, MESFUN6C:def 4;

        

         A5: ((f to_power (a * b)) . x) = ((f . x) to_power (a * b)) by A2, A3, MESFUN6C:def 4;

        then

         A6: (f . x) > 0 implies (((f to_power a) to_power b) . x) = ((f to_power (a * b)) . x) by A4, POWER: 33;

        now

          assume

           A7: (f . x) = 0 ;

          then (((f to_power a) to_power b) . x) = ( 0 to_power b) by A1, A4, POWER:def 2;

          then (((f to_power a) to_power b) . x) = 0 by A1, POWER:def 2;

          hence (((f to_power a) to_power b) . x) = ((f to_power (a * b)) . x) by A1, A7, A5, POWER:def 2;

        end;

        hence thesis by A6, A1, MESFUNC6: 51;

      end;

      hence thesis by A2, FUNCT_1: 2;

    end;

    theorem :: LPSPACE2:7

    

     Th7: for a,b be Real, f st f is nonnegative & a > 0 & b > 0 holds ((f to_power a) (#) (f to_power b)) = (f to_power (a + b))

    proof

      let a,b be Real;

      let f;

      assume

       A1: f is nonnegative & a > 0 & b > 0 ;

      

       A2: ( dom (f to_power a)) = ( dom f) & ( dom (f to_power b)) = ( dom f) by MESFUN6C:def 4;

      

       A3: ( dom ((f to_power a) (#) (f to_power b))) = (( dom (f to_power a)) /\ ( dom (f to_power b))) by VALUED_1:def 4;

      then

       A4: ( dom ((f to_power a) (#) (f to_power b))) = ( dom (f to_power (a + b))) by A2, MESFUN6C:def 4;

      for x be object st x in ( dom ((f to_power a) (#) (f to_power b))) holds (((f to_power a) (#) (f to_power b)) . x) = ((f to_power (a + b)) . x)

      proof

        let x be object;

        assume

         A5: x in ( dom ((f to_power a) (#) (f to_power b)));

        then ((f to_power a) . x) = ((f . x) to_power a) & ((f to_power b) . x) = ((f . x) to_power b) by A2, A3, MESFUN6C:def 4;

        then

         A6: (((f to_power a) (#) (f to_power b)) . x) = (((f . x) to_power a) * ((f . x) to_power b)) by A5, VALUED_1:def 4;

        

         A7: ((f to_power (a + b)) . x) = ((f . x) to_power (a + b)) by A4, A5, MESFUN6C:def 4;

        then

         A8: (f . x) > 0 implies (((f to_power a) (#) (f to_power b)) . x) = ((f to_power (a + b)) . x) by A6, POWER: 27;

        now

          assume

           A9: (f . x) = 0 ;

          then (((f to_power a) (#) (f to_power b)) . x) = ( 0 * ( 0 to_power b)) by A1, A6, POWER:def 2;

          hence (((f to_power a) (#) (f to_power b)) . x) = ((f to_power (a + b)) . x) by A7, A1, A9, POWER:def 2;

        end;

        hence thesis by A1, A8, MESFUNC6: 51;

      end;

      hence thesis by A4, FUNCT_1: 2;

    end;

    theorem :: LPSPACE2:8

    

     Th8: (f to_power 1) = f

    proof

      

       A1: ( dom (f to_power 1)) = ( dom f) by MESFUN6C:def 4;

      for x be object st x in ( dom (f to_power 1)) holds ((f to_power 1) . x) = (f . x)

      proof

        let x be object;

        assume x in ( dom (f to_power 1));

        then ((f to_power 1) . x) = ((f . x) to_power 1) by MESFUN6C:def 4;

        hence thesis by POWER: 25;

      end;

      hence thesis by A1, FUNCT_1: 2;

    end;

    theorem :: LPSPACE2:9

    

     Th9: for seq1,seq2 be Real_Sequence, k be positive Real st for n be Nat holds (seq1 . n) = ((seq2 . n) to_power k) & (seq2 . n) >= 0 holds (seq1 is convergent iff seq2 is convergent)

    proof

      let seq1,seq2 be Real_Sequence, k be positive Real;

      assume

       A1: for n be Nat holds (seq1 . n) = ((seq2 . n) to_power k) & (seq2 . n) >= 0 ;

      

       A2: for n holds (seq1 . n) >= 0

      proof

        let n;

        ((seq2 . n) to_power k) >= 0 by A1, Th4;

        hence thesis by A1;

      end;

      thus seq1 is convergent implies seq2 is convergent

      proof

        assume

         A3: seq1 is convergent;

        for n be Nat holds (seq2 . n) = ((seq1 . n) to_power (1 / k))

        proof

          let n be Nat;

          ((seq1 . n) to_power (1 / k)) = (((seq2 . n) to_power k) to_power (1 / k)) by A1

          .= ((seq2 . n) to_power (k * (1 / k))) by A1, HOLDER_1: 2

          .= ((seq2 . n) to_power 1) by XCMPLX_1: 106;

          hence thesis by POWER: 25;

        end;

        hence thesis by A2, A3, HOLDER_1: 10;

      end;

      assume seq2 is convergent;

      hence thesis by A1, HOLDER_1: 10;

    end;

    theorem :: LPSPACE2:10

    

     Th10: for seq be Real_Sequence, n,m be Nat st m <= n holds |.((( Partial_Sums seq) . n) - (( Partial_Sums seq) . m)).| <= ((( Partial_Sums ( abs seq)) . n) - (( Partial_Sums ( abs seq)) . m)) & |.((( Partial_Sums seq) . n) - (( Partial_Sums seq) . m)).| <= (( Partial_Sums ( abs seq)) . n)

    proof

      let seq be Real_Sequence;

      let n,m be Nat;

      assume

       A1: m <= n;

      

       A2: for n holds (( abs seq) . n) >= 0

      proof

        let n;

         |.(seq . n).| = (( abs seq) . n) by SEQ_1: 12;

        hence thesis by COMPLEX1: 46;

      end;

      then

       A3: |.((( Partial_Sums ( abs seq)) . n) - (( Partial_Sums ( abs seq)) . m)).| = ((( Partial_Sums ( abs seq)) . n) - (( Partial_Sums ( abs seq)) . m)) by A1, COMSEQ_3: 9;

      (( Partial_Sums ( abs seq)) . m) >= 0 by A2, SERIES_3: 34;

      then |.((( Partial_Sums seq) . n) - (( Partial_Sums seq) . m)).| <= (((( Partial_Sums ( abs seq)) . n) - (( Partial_Sums ( abs seq)) . m)) + (( Partial_Sums ( abs seq)) . m)) by A3, A1, SERIES_1: 34, XREAL_1: 38;

      hence thesis by A3, A1, SERIES_1: 34;

    end;

    theorem :: LPSPACE2:11

    

     Th11: for seq,seq2 be Real_Sequence, k be positive Real st seq is convergent & for n be Nat holds (seq2 . n) = ( |.(( lim seq) - (seq . n)) qua Complex.| to_power k) holds seq2 is convergent & ( lim seq2) = 0

    proof

      let seq,seq2 be Real_Sequence, k be positive Real;

      set r = ( lim seq);

      assume

       A1: seq is convergent & for n be Nat holds (seq2 . n) = ( |.(r - (seq . n)) qua Complex.| to_power k);

      deffunc U( Nat) = |.(r - (seq . $1)) qua Complex.|;

      consider seq1 be Real_Sequence such that

       A2: for n holds (seq1 . n) = U(n) from SEQ_1:sch 1;

      deffunc U( Nat) = r;

      consider seq0 be Real_Sequence such that

       A3: for n holds (seq0 . n) = U(n) from SEQ_1:sch 1;

      reconsider r as Element of REAL by XREAL_0:def 1;

      for n be Nat holds (seq0 . n) = r by A3;

      then

       A4: seq0 is constant by VALUED_0:def 18;

      then

       A5: (seq0 - seq) is convergent by A1;

      

       A6: ( dom seq0) = NAT & ( dom seq) = NAT & ( dom (seq0 - seq)) = NAT & ( dom seq1) = NAT by FUNCT_2:def 1;

      

       A7: ( dom ( abs (seq0 - seq))) = ( dom (seq0 - seq)) by VALUED_1:def 11;

      for n be Element of NAT holds (( abs (seq0 - seq)) . n) = (seq1 . n)

      proof

        let n be Element of NAT ;

        (seq1 . n) = |.(r - (seq . n)).| by A2;

        then (seq1 . n) = |.((seq0 . n) - (seq . n)).| by A3;

        then (seq1 . n) = |.((seq0 - seq) . n).| by A6, VALUED_1: 13;

        hence thesis by A6, A7, VALUED_1:def 11;

      end;

      then

       A8: ( abs (seq0 - seq)) = seq1 by FUNCT_2: 63;

      then

       A9: seq1 is convergent by A5;

      ( lim (seq0 - seq)) = ((seq0 . 0 ) - ( lim seq)) by A4, A1, SEQ_4: 42;

      then ( lim (seq0 - seq)) = (r - ( lim seq)) by A3;

      then

       A10: ( lim seq1) = 0 by A5, A8, COMPLEX1: 44, SEQ_4: 14;

      for n holds (seq2 . n) = ((seq1 . n) to_power k) & (seq1 . n) >= 0

      proof

        let n;

         |.(r - (seq . n)).| = (seq1 . n) by A2;

        hence thesis by A1, COMPLEX1: 46;

      end;

      then seq2 is convergent & ( lim seq2) = (( lim seq1) to_power k) by A9, HOLDER_1: 10;

      hence thesis by A10, POWER:def 2;

    end;

    

     Lm1: for seq be Real_Sequence, n be Nat holds |.(( Partial_Sums seq) . n) qua Complex.| <= (( Partial_Sums ( abs seq)) . n) by NAGATA_2: 13;

    begin

    theorem :: LPSPACE2:12

    

     Th12: for k be positive Real, X be non empty set holds ((X --> 0 ) to_power k) = (X --> 0 )

    proof

      let k be positive Real, X be non empty set;

      

       A1: ( dom ((X --> 0 ) to_power k)) = ( dom (X --> 0 )) by MESFUN6C:def 4;

      now

        let x be Element of X;

        assume x in ( dom ((X --> 0 ) to_power k));

        then (((X --> 0 ) to_power k) . x) = (((X --> 0 ) . x) to_power k) by MESFUN6C:def 4;

        then (((X --> 0 ) to_power k) . x) = ( 0 to_power k) by FUNCOP_1: 7;

        then (((X --> 0 ) to_power k) . x) = 0 by POWER:def 2;

        hence (((X --> 0 ) to_power k) . x) = ((X --> 0 ) . x) by FUNCOP_1: 7;

      end;

      hence thesis by A1, PARTFUN1: 5;

    end;

    theorem :: LPSPACE2:13

    

     Th13: for f be PartFunc of X, REAL , D be set holds ( abs (f | D)) = (( abs f) | D)

    proof

      let f be PartFunc of X, REAL ;

      let D be set;

      

       A1: ( dom ( abs (f | D))) = ( dom (f | D)) by VALUED_1:def 11;

      then ( dom ( abs (f | D))) = (( dom f) /\ D) by RELAT_1: 61;

      then ( dom ( abs (f | D))) = (( dom ( abs f)) /\ D) by VALUED_1:def 11;

      then

       A2: ( dom ( abs (f | D))) = ( dom (( abs f) | D)) by RELAT_1: 61;

      for x be Element of X st x in ( dom ( abs (f | D))) holds (( abs (f | D)) . x) = ((( abs f) | D) . x)

      proof

        let x be Element of X;

        assume

         A3: x in ( dom ( abs (f | D)));

        then x in ( dom f) by A1, RELAT_1: 57;

        then

         A4: x in ( dom ( abs f)) by VALUED_1:def 11;

        (( abs (f | D)) . x) = |.((f | D) . x).| by A3, VALUED_1:def 11;

        then (( abs (f | D)) . x) = |.(f . x).| by A3, A1, FUNCT_1: 47;

        then (( abs (f | D)) . x) = (( abs f) . x) by A4, VALUED_1:def 11;

        hence (( abs (f | D)) . x) = ((( abs f) | D) . x) by A3, A2, FUNCT_1: 47;

      end;

      hence thesis by A2, PARTFUN1: 5;

    end;

    registration

      let X;

      let f be PartFunc of X, REAL ;

      cluster ( abs f) -> nonnegative;

      coherence

      proof

        now

          let x be object;

          assume x in ( dom ( abs f));

          then (( abs f) . x) = |.(f . x).| by VALUED_1:def 11;

          hence 0 <= (( abs f) . x) by COMPLEX1: 46;

        end;

        hence thesis by MESFUNC6: 52;

      end;

    end

    theorem :: LPSPACE2:14

    

     Th14: for f be PartFunc of X, REAL st f is nonnegative holds ( abs f) = f

    proof

      let f be PartFunc of X, REAL ;

      

       A1: ( dom f) = ( dom ( abs f)) by VALUED_1:def 11;

      assume

       A2: f is nonnegative;

      now

        let x be Element of X;

        

         A3: (f . x) >= 0 by A2, MESFUNC6: 51;

        assume x in ( dom f);

        then x in ( dom ( abs f)) by VALUED_1:def 11;

        then (( abs f) . x) = |.(f . x).| by VALUED_1:def 11;

        hence (( abs f) . x) = (f . x) by A3, ABSVALUE:def 1;

      end;

      hence thesis by A1, PARTFUN1: 5;

    end;

    theorem :: LPSPACE2:15

    

     Th15: (X = ( dom f) & for x st x in ( dom f) holds 0 = (f . x)) implies f is_integrable_on M & ( Integral (M,f)) = 0

    proof

      assume

       A1: X = ( dom f) & for x st x in ( dom f) holds 0 = (f . x);

      X is Element of S by MEASURE1: 7;

      then ( R_EAL f) is_integrable_on M & ( Integral (M,( R_EAL f))) = 0 by A1, LPSPACE1: 22;

      hence thesis;

    end;

    definition

      let X be non empty set, S be SigmaField of X, M be sigma_Measure of S, k be positive Real;

      :: LPSPACE2:def2

      func Lp_Functions (M,k) -> non empty Subset of ( RLSp_PFunct X) equals { f where f be PartFunc of X, REAL : ex Ef be Element of S st (M . (Ef ` )) = 0 & ( dom f) = Ef & f is Ef -measurable & (( abs f) to_power k) is_integrable_on M };

      correctness

      proof

        set V = { f where f be PartFunc of X, REAL : ex Ef be Element of S st (M . (Ef ` )) = 0 & ( dom f) = Ef & f is Ef -measurable & (( abs f) to_power k) is_integrable_on M };

        

         A1: V c= ( PFuncs (X, REAL ))

        proof

          let x be object;

          assume x in V;

          then ex f be PartFunc of X, REAL st x = f & ex Ef be Element of S st (M . (Ef ` )) = 0 & ( dom f) = Ef & f is Ef -measurable & (( abs f) to_power k) is_integrable_on M;

          hence x in ( PFuncs (X, REAL )) by PARTFUN1: 45;

        end;

        reconsider g = (X --> ( In ( 0 , REAL ))) as Function of X, REAL by FUNCOP_1: 46;

        reconsider Ef = X as Element of S by MEASURE1: 34;

        set h = (( abs g) to_power k);

        

         A2: ( dom g) = X by FUNCOP_1: 13;

        for x be set st x in ( dom g) holds (g . x) = 0 by FUNCOP_1: 7;

        then

         A3: g is Ef -measurable by A2, LPSPACE1: 52;

        (Ef ` ) = {} by XBOOLE_1: 37;

        then

         A4: (M . (Ef ` )) = 0 by VALUED_0:def 19;

        for x be object st x in ( dom (X --> 0 )) holds 0 <= ((X --> 0 ) . x);

        then ( abs g) = (X --> 0 ) by Th14, MESFUNC6: 52;

        then

         A5: h = g by Th12;

        then for x be Element of X st x in ( dom h) holds (h . x) = 0 by FUNCOP_1: 7;

        then h is_integrable_on M by Th15, A5, A2;

        then g in V by A3, A4, A2;

        hence thesis by A1;

      end;

    end

    theorem :: LPSPACE2:16

    

     Th16: for a,b,k be Real st k > 0 holds ( |.(a + b) qua Complex.| to_power k) <= (( |.a qua Complex.| + |.b qua Complex.|) to_power k) & (( |.a qua Complex.| + |.b qua Complex.|) to_power k) <= ((2 * ( max ( |.a qua Complex.|, |.b qua Complex.|))) to_power k) & ( |.(a + b) qua Complex.| to_power k) <= ((2 * ( max ( |.a qua Complex.|, |.b qua Complex.|))) to_power k)

    proof

      let a,b,k be Real;

      assume

       A1: k > 0 ;

      

       A2: |.(a + b) qua Complex.| <= ( |.a qua Complex.| + |.b qua Complex.|) by ABSVALUE: 9;

       |.a.| <= ( max ( |.a.|, |.b.|)) & |.b.| <= ( max ( |.a.|, |.b.|)) by XXREAL_0: 25;

      then

       A3: ( |.a qua Complex.| + |.b qua Complex.|) <= (( max ( |.a qua Complex.|, |.b qua Complex.|)) + ( max ( |.a qua Complex.|, |.b qua Complex.|))) by XREAL_1: 7;

      then

       A4: |.(a + b) qua Complex.| <= (2 * ( max ( |.a qua Complex.|, |.b qua Complex.|))) by A2, XXREAL_0: 2;

       0 <= |.(a + b) qua Complex.| by COMPLEX1: 46;

      hence thesis by A1, A2, A3, A4, HOLDER_1: 3;

    end;

    theorem :: LPSPACE2:17

    

     Th17: for a,b,k be Real st a >= 0 & b >= 0 & k > 0 holds (( max (a,b)) to_power k) <= ((a to_power k) + (b to_power k))

    proof

      let a,b,k be Real;

      assume

       A1: a >= 0 & b >= 0 & k > 0 ;

      per cases ;

        suppose a <> 0 & b <> 0 ;

        then

         A2: (a to_power k) >= 0 & (b to_power k) >= 0 by A1, POWER: 34;

        ( max (a,b)) = a or ( max (a,b)) = b by XXREAL_0:def 10;

        hence (( max (a,b)) to_power k) <= ((a to_power k) + (b to_power k)) by A2, XREAL_1: 40;

      end;

        suppose

         A3: a = 0 ;

        then (a to_power k) = 0 by A1, POWER:def 2;

        hence (( max (a,b)) to_power k) <= ((a to_power k) + (b to_power k)) by A1, A3, XXREAL_0:def 10;

      end;

        suppose

         A4: b = 0 ;

        then (b to_power k) = 0 by A1, POWER:def 2;

        hence (( max (a,b)) to_power k) <= ((a to_power k) + (b to_power k)) by A1, A4, XXREAL_0:def 10;

      end;

    end;

    theorem :: LPSPACE2:18

    

     Th18: for f be PartFunc of X, REAL , a,b be Real st b > 0 holds (( |.a qua Complex.| to_power b) (#) (( abs f) to_power b)) = (( abs (a (#) f)) to_power b)

    proof

      let f be PartFunc of X, REAL ;

      let a,b be Real;

      assume

       A1: b > 0 ;

      

       A2: ( dom (( |.a qua Complex.| to_power b) (#) (( abs f) to_power b))) = ( dom (( abs f) to_power b)) & ( dom (a (#) f)) = ( dom f) by VALUED_1:def 5;

      

       A3: ( dom (( abs f) to_power b)) = ( dom ( abs f)) & ( dom ( abs (a (#) f))) = ( dom (( abs (a (#) f)) to_power b)) by MESFUN6C:def 4;

      

       A4: ( dom ( abs f)) = ( dom f) & ( dom ( abs (a (#) f))) = ( dom (a (#) f)) by VALUED_1:def 11;

      for x be Element of X st x in ( dom (( |.a qua Complex.| to_power b) (#) (( abs f) to_power b))) holds ((( |.a qua Complex.| to_power b) (#) (( abs f) to_power b)) . x) = ((( abs (a (#) f)) to_power b) . x)

      proof

        let x be Element of X;

        assume

         A5: x in ( dom (( |.a qua Complex.| to_power b) (#) (( abs f) to_power b)));

        

         A6: |.(f . x).| >= 0 & |.a.| >= 0 by COMPLEX1: 46;

        ((( |.a qua Complex.| to_power b) (#) (( abs f) to_power b)) . x) = (( |.a qua Complex.| to_power b) * ((( abs f) to_power b) . x)) by A5, VALUED_1:def 5

        .= (( |.a qua Complex.| to_power b) * ((( abs f) . x) to_power b)) by A2, A5, MESFUN6C:def 4

        .= (( |.a qua Complex.| to_power b) * ( |.(f . x) qua Complex.| to_power b)) by VALUED_1: 18

        .= (( |.a qua Complex.| * |.(f . x) qua Complex.|) to_power b) by A1, A6, Th5

        .= ( |.(a * (f . x)) qua Complex.| to_power b) by COMPLEX1: 65

        .= ( |.((a (#) f) . x) qua Complex.| to_power b) by VALUED_1: 6

        .= ((( abs (a (#) f)) . x) to_power b) by VALUED_1: 18;

        hence thesis by A2, A3, A4, A5, MESFUN6C:def 4;

      end;

      hence thesis by A2, A3, A4, PARTFUN1: 5;

    end;

    theorem :: LPSPACE2:19

    

     Th19: for f be PartFunc of X, REAL , a,b be Real st a > 0 & b > 0 holds ((a to_power b) (#) (( abs f) to_power b)) = ((a (#) ( abs f)) to_power b)

    proof

      let f be PartFunc of X, REAL ;

      let a,b be Real;

      assume

       A1: a > 0 & b > 0 ;

      then

       A2: |.a.| = a by COMPLEX1: 43;

      then ((a to_power b) (#) (( abs f) to_power b)) = (( abs (a (#) f)) to_power b) by A1, Th18;

      hence thesis by A2, RFUNCT_1: 25;

    end;

    theorem :: LPSPACE2:20

    

     Th20: for f be PartFunc of X, REAL , k be Real, E be set holds ((f | E) to_power k) = ((f to_power k) | E)

    proof

      let f be PartFunc of X, REAL ;

      let k be Real;

      let E be set;

      

       A1: ( dom ((f | E) to_power k)) = ( dom (f | E)) by MESFUN6C:def 4;

      then ( dom ((f | E) to_power k)) = (( dom f) /\ E) by RELAT_1: 61;

      then

       A2: ( dom ((f | E) to_power k)) = (( dom (f to_power k)) /\ E) by MESFUN6C:def 4;

      then

       A3: ( dom ((f | E) to_power k)) = ( dom ((f to_power k) | E)) by RELAT_1: 61;

      now

        let x be Element of X;

        assume

         A4: x in ( dom ((f | E) to_power k));

        then (((f | E) to_power k) . x) = (((f | E) . x) to_power k) by MESFUN6C:def 4;

        then

         A5: (((f | E) to_power k) . x) = ((f . x) to_power k) by A1, A4, FUNCT_1: 47;

        x in ( dom (f to_power k)) by A2, A4, XBOOLE_0:def 4;

        then (((f | E) to_power k) . x) = ((f to_power k) . x) by A5, MESFUN6C:def 4;

        hence (((f | E) to_power k) . x) = (((f to_power k) | E) . x) by A4, A3, FUNCT_1: 47;

      end;

      hence thesis by A3, PARTFUN1: 5;

    end;

    theorem :: LPSPACE2:21

    

     Th21: for a,b,k be Real st k > 0 holds ( |.(a + b) qua Complex.| to_power k) <= ((2 to_power k) * (( |.a qua Complex.| to_power k) + ( |.b qua Complex.| to_power k)))

    proof

      let a,b,k be Real;

      assume

       A1: k > 0 ;

      then

       A2: ( |.(a + b) qua Complex.| to_power k) <= ((2 * ( max ( |.a qua Complex.|, |.b qua Complex.|))) to_power k) by Th16;

      

       A3: |.a.| >= 0 & |.b.| >= 0 by COMPLEX1: 46;

      then

       A4: (( max ( |.a qua Complex.|, |.b qua Complex.|)) to_power k) <= (( |.a qua Complex.| to_power k) + ( |.b qua Complex.| to_power k)) by A1, Th17;

      ( max ( |.a.|, |.b.|)) = |.a.| or ( max ( |.a.|, |.b.|)) = |.b.| by XXREAL_0: 16;

      then

       A5: ((2 * ( max ( |.a qua Complex.|, |.b qua Complex.|))) to_power k) = ((2 to_power k) * (( max ( |.a qua Complex.|, |.b qua Complex.|)) to_power k)) by A1, A3, Th5;

      (2 to_power k) > 0 by POWER: 34;

      then ((2 to_power k) * (( max ( |.a qua Complex.|, |.b qua Complex.|)) to_power k)) <= ((2 to_power k) * (( |.a qua Complex.| to_power k) + ( |.b qua Complex.| to_power k))) by A4, XREAL_1: 64;

      hence thesis by A2, A5, XXREAL_0: 2;

    end;

    theorem :: LPSPACE2:22

    

     Th22: for k be positive Real, f,g be PartFunc of X, REAL st f in ( Lp_Functions (M,k)) & g in ( Lp_Functions (M,k)) holds (( abs f) to_power k) is_integrable_on M & (( abs g) to_power k) is_integrable_on M & ((( abs f) to_power k) + (( abs g) to_power k)) is_integrable_on M

    proof

      let k be positive Real;

      let f,g be PartFunc of X, REAL ;

      assume

       A1: f in ( Lp_Functions (M,k)) & g in ( Lp_Functions (M,k));

      then

       A2: ex f1 be PartFunc of X, REAL st f = f1 & ex Ev be Element of S st (M . (Ev ` )) = 0 & ( dom f1) = Ev & f1 is Ev -measurable & (( abs f1) to_power k) is_integrable_on M;

      ex g1 be PartFunc of X, REAL st g = g1 & ex Eu be Element of S st (M . (Eu ` )) = 0 & ( dom g1) = Eu & g1 is Eu -measurable & (( abs g1) to_power k) is_integrable_on M by A1;

      hence thesis by A2, MESFUNC6: 100;

    end;

    theorem :: LPSPACE2:23

    

     Th23: (X --> 0 ) is PartFunc of X, REAL & (X --> 0 ) in ( Lp_Functions (M,k))

    proof

      reconsider g = (X --> ( In ( 0 , REAL ))) as Function of X, REAL by FUNCOP_1: 46;

      reconsider ND = X as Element of S by MEASURE1: 34;

      (ND ` ) = {} by XBOOLE_1: 37;

      then

       A1: (M . (ND ` )) = 0 by VALUED_0:def 19;

      

       A2: ( dom g) = X by FUNCT_2:def 1;

      for x be Element of X st x in ( dom g) holds (g . x) = 0 by FUNCOP_1: 7;

      then

       A3: g is_integrable_on M by A2, Th15;

      for x be object st x in ( dom g) holds 0 <= (g . x);

      then ( abs g) = g by Th14, MESFUNC6: 52;

      then

       A4: (( abs g) to_power k) = g by Th12;

      for x be set st x in ( dom g) holds (g . x) = 0 by FUNCOP_1: 7;

      then g is ND -measurable by A2, LPSPACE1: 52;

      hence thesis by A1, A2, A3, A4;

    end;

    theorem :: LPSPACE2:24

    

     Th24: for k be Real st k > 0 holds for f,g be PartFunc of X, REAL holds for x be Element of X st x in (( dom f) /\ ( dom g)) holds ((( abs (f + g)) to_power k) . x) <= (((2 to_power k) (#) ((( abs f) to_power k) + (( abs g) to_power k))) . x)

    proof

      let k be Real;

      assume

       A1: k > 0 ;

      let f,g be PartFunc of X, REAL ;

      let x be Element of X;

      assume

       A2: x in (( dom f) /\ ( dom g));

      

       A3: ( dom (f + g)) = (( dom f) /\ ( dom g)) by VALUED_1:def 1;

      then ( dom ( abs (f + g))) = (( dom f) /\ ( dom g)) by VALUED_1:def 11;

      then x in ( dom (( abs (f + g)) to_power k)) by A2, MESFUN6C:def 4;

      

      then

       A4: ((( abs (f + g)) to_power k) . x) = ((( abs (f + g)) . x) to_power k) by MESFUN6C:def 4

      .= ( |.((f + g) . x) qua Complex.| to_power k) by VALUED_1: 18

      .= ( |.((f . x) + (g . x)) qua Complex.| to_power k) by A3, A2, VALUED_1:def 1;

      ( dom ( abs f)) = ( dom f) & ( dom ( abs g)) = ( dom g) by VALUED_1:def 11;

      then x in ( dom ( abs f)) & x in ( dom ( abs g)) by A2, XBOOLE_0:def 4;

      then

       A5: x in ( dom (( abs f) to_power k)) & x in ( dom (( abs g) to_power k)) by MESFUN6C:def 4;

      ( |.(f . x) qua Complex.| to_power k) = ((( abs f) . x) to_power k) & ( |.(g . x) qua Complex.| to_power k) = ((( abs g) . x) to_power k) by VALUED_1: 18;

      then

       A6: ( |.(f . x) qua Complex.| to_power k) = ((( abs f) to_power k) . x) & ( |.(g . x) qua Complex.| to_power k) = ((( abs g) to_power k) . x) by A5, MESFUN6C:def 4;

      ( dom ((( abs f) to_power k) + (( abs g) to_power k))) = (( dom (( abs f) to_power k)) /\ ( dom (( abs g) to_power k))) by VALUED_1:def 1;

      then x in ( dom ((( abs f) to_power k) + (( abs g) to_power k))) by A5, XBOOLE_0:def 4;

      

      then ((2 to_power k) * (( |.(f . x) qua Complex.| to_power k) + ( |.(g . x) qua Complex.| to_power k))) = ((2 to_power k) * (((( abs f) to_power k) + (( abs g) to_power k)) . x)) by A6, VALUED_1:def 1

      .= (((2 to_power k) (#) ((( abs f) to_power k) + (( abs g) to_power k))) . x) by VALUED_1: 6;

      hence thesis by A1, A4, Th21;

    end;

    theorem :: LPSPACE2:25

    

     Th25: f in ( Lp_Functions (M,k)) & g in ( Lp_Functions (M,k)) implies (f + g) in ( Lp_Functions (M,k))

    proof

      set W = ( Lp_Functions (M,k));

      assume

       A1: f in W & g in W;

      then

      consider f1 be PartFunc of X, REAL such that

       A2: f1 = f & ex Ef1 be Element of S st (M . (Ef1 ` )) = 0 & ( dom f1) = Ef1 & f1 is Ef1 -measurable & (( abs f1) to_power k) is_integrable_on M;

      consider Ef be Element of S such that

       A3: (M . (Ef ` )) = 0 & ( dom f1) = Ef & f1 is Ef -measurable & (( abs f1) to_power k) is_integrable_on M by A2;

      consider g1 be PartFunc of X, REAL such that

       A4: g1 = g & ex Eg1 be Element of S st (M . (Eg1 ` )) = 0 & ( dom g1) = Eg1 & g1 is Eg1 -measurable & (( abs g1) to_power k) is_integrable_on M by A1;

      consider Eg be Element of S such that

       A5: (M . (Eg ` )) = 0 & ( dom g1) = Eg & g1 is Eg -measurable & (( abs g1) to_power k) is_integrable_on M by A4;

      

       A6: ( dom (f1 + g1)) = (Ef /\ Eg) by A3, A5, VALUED_1:def 1;

      set Efg = (Ef /\ Eg);

      set s = (( abs (f1 + g1)) to_power k);

      set t = ((2 to_power k) (#) ((( abs f1) to_power k) + (( abs g1) to_power k)));

      

       A7: (Efg ` ) = ((X \ Ef) \/ (X \ Eg)) by XBOOLE_1: 54;

      (Ef ` ) is Element of S & (Eg ` ) is Element of S by MEASURE1: 34;

      then (Ef ` ) is measure_zero of M & (Eg ` ) is measure_zero of M by A3, A5, MEASURE1:def 7;

      then ((Ef ` ) \/ (Eg ` )) is measure_zero of M by MEASURE1: 37;

      then

       A8: (M . (Efg ` )) = 0 by A7, MEASURE1:def 7;

      f1 is Efg -measurable & g1 is Efg -measurable by A3, A5, MESFUNC6: 16, XBOOLE_1: 17;

      then

       A9: (f1 + g1) is Efg -measurable by MESFUNC6: 26;

      then

       A10: ( abs (f1 + g1)) is Efg -measurable by A6, MESFUNC6: 48;

      ((( abs f1) to_power k) + (( abs g1) to_power k)) is_integrable_on M by A1, A2, A4, Th22;

      then

       A11: t is_integrable_on M by MESFUNC6: 102;

      

       A12: ( dom ( abs f1)) = ( dom f1) & ( dom ( abs g1)) = ( dom g1) & ( dom ( abs (f1 + g1))) = ( dom (f1 + g1)) by VALUED_1:def 11;

      then

       A13: s is Efg -measurable by A6, A10, MESFUN6C: 29;

      

       A14: ( abs s) = (( abs (f1 + g1)) to_power k) by Th14;

      

       A15: ( dom s) = Efg by A6, A12, MESFUN6C:def 4;

      

       A16: ( dom t) = ( dom ((( abs f1) to_power k) + (( abs g1) to_power k))) by VALUED_1:def 5

      .= (( dom (( abs f1) to_power k)) /\ ( dom (( abs g1) to_power k))) by VALUED_1:def 1

      .= (( dom ( abs f1)) /\ ( dom (( abs g1) to_power k))) by MESFUN6C:def 4

      .= (( dom ( abs f1)) /\ ( dom ( abs g1))) by MESFUN6C:def 4

      .= ( dom (f1 + g1)) by A12, VALUED_1:def 1

      .= ( dom s) by A12, MESFUN6C:def 4;

      now

        let x be Element of X;

        assume x in ( dom s);

        then (( abs s) . x) <= (t . x) by A14, Th24, A3, A5, A15;

        hence |.(s . x) qua Complex.| <= (t . x) by VALUED_1: 18;

      end;

      then s is_integrable_on M by A13, A15, A16, A11, MESFUNC6: 96;

      hence thesis by A2, A4, A8, A6, A9;

    end;

    theorem :: LPSPACE2:26

    

     Th26: f in ( Lp_Functions (M,k)) implies (a (#) f) in ( Lp_Functions (M,k))

    proof

      assume f in ( Lp_Functions (M,k));

      then

      consider f1 be PartFunc of X, REAL such that

       A1: f1 = f & ex Ef1 be Element of S st (M . (Ef1 ` )) = 0 & ( dom f1) = Ef1 & f1 is Ef1 -measurable & (( abs f1) to_power k) is_integrable_on M;

      consider Ef be Element of S such that

       A2: (M . (Ef ` )) = 0 & ( dom f1) = Ef & f1 is Ef -measurable & (( abs f1) to_power k) is_integrable_on M by A1;

      

       A3: ( dom (a (#) f1)) = Ef & (a (#) f1) is Ef -measurable by A2, MESFUNC6: 21, VALUED_1:def 5;

      (( |.a qua Complex.| to_power k) (#) (( abs f1) to_power k)) is_integrable_on M by A1, MESFUNC6: 102;

      then (( abs (a (#) f1)) to_power k) is_integrable_on M by Th18;

      hence thesis by A1, A2, A3;

    end;

    theorem :: LPSPACE2:27

    

     Th27: f in ( Lp_Functions (M,k)) & g in ( Lp_Functions (M,k)) implies (f - g) in ( Lp_Functions (M,k))

    proof

      assume

       A1: f in ( Lp_Functions (M,k)) & g in ( Lp_Functions (M,k));

      then (( - 1) (#) g) in ( Lp_Functions (M,k)) by Th26;

      hence thesis by Th25, A1;

    end;

    theorem :: LPSPACE2:28

    

     Th28: f in ( Lp_Functions (M,k)) implies ( abs f) in ( Lp_Functions (M,k))

    proof

      set W = ( Lp_Functions (M,k));

      assume f in W;

      then

      consider f1 be PartFunc of X, REAL such that

       A1: f1 = f & ex Ef1 be Element of S st (M . (Ef1 ` )) = 0 & ( dom f1) = Ef1 & f1 is Ef1 -measurable & (( abs f1) to_power k) is_integrable_on M;

      consider Ef be Element of S such that

       A2: (M . (Ef ` )) = 0 & ( dom f1) = Ef & f1 is Ef -measurable & (( abs f1) to_power k) is_integrable_on M by A1;

      ( dom ( abs f1)) = Ef by A2, VALUED_1:def 11;

      then

       Z1: (M . (Ef ` )) = 0 & ( dom ( abs f1)) = Ef & ( abs f1) is Ef -measurable & (( abs ( abs f1)) to_power k) is_integrable_on M by A2, MESFUNC6: 48;

      thus thesis by A1, Z1;

    end;

    

     Lm2: ( Lp_Functions (M,k)) is add-closed & ( Lp_Functions (M,k)) is multi-closed

    proof

      set W = ( Lp_Functions (M,k));

      now

        let v,u be Element of the carrier of ( RLSp_PFunct X);

        assume

         A1: v in W & u in W;

        then

        consider v1 be PartFunc of X, REAL such that

         A2: v1 = v & ex ND be Element of S st (M . (ND ` )) = 0 & ( dom v1) = ND & v1 is ND -measurable & (( abs v1) to_power k) is_integrable_on M;

        consider u1 be PartFunc of X, REAL such that

         A3: u1 = u & ex ND be Element of S st (M . (ND ` )) = 0 & ( dom u1) = ND & u1 is ND -measurable & (( abs u1) to_power k) is_integrable_on M by A1;

        reconsider h = (v + u) as Element of ( PFuncs (X, REAL ));

        ( dom h) = (( dom v1) /\ ( dom u1)) & for x be object st x in ( dom h) holds (h . x) = ((v1 . x) + (u1 . x)) by A2, A3, LPSPACE1: 6;

        then (v + u) = (v1 + u1) by VALUED_1:def 1;

        hence (v + u) in ( Lp_Functions (M,k)) by A1, A2, A3, Th25;

      end;

      hence ( Lp_Functions (M,k)) is add-closed by IDEAL_1:def 1;

      now

        let a be Real, u be VECTOR of ( RLSp_PFunct X);

        assume

         A4: u in W;

        then

        consider u1 be PartFunc of X, REAL such that

         A5: u1 = u & ex ND be Element of S st (M . (ND ` )) = 0 & ( dom u1) = ND & u1 is ND -measurable & (( abs u1) to_power k) is_integrable_on M;

        reconsider h = (a * u) as Element of ( PFuncs (X, REAL ));

        

         A6: ( dom h) = ( dom u1) & for x be Element of X st x in ( dom u1) holds (h . x) = (a * (u1 . x)) by A5, LPSPACE1: 9;

        then for x be object st x in ( dom h) holds (h . x) = (a * (u1 . x));

        then (a * u) = (a (#) u1) by A6, VALUED_1:def 5;

        hence (a * u) in ( Lp_Functions (M,k)) by Th26, A4, A5;

      end;

      hence ( Lp_Functions (M,k)) is multi-closed;

    end;

    registration

      let X be non empty set, S be SigmaField of X, M be sigma_Measure of S, k be positive Real;

      cluster ( Lp_Functions (M,k)) -> multi-closed add-closed;

      coherence by Lm2;

    end

    

     Lm3: RLSStruct (# ( Lp_Functions (M,k)), ( In (( 0. ( RLSp_PFunct X)),( Lp_Functions (M,k)))), ( add| (( Lp_Functions (M,k)),( RLSp_PFunct X))), ( Mult_ ( Lp_Functions (M,k))) #) is Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital

    proof

      ( 0. ( RLSp_PFunct X)) in ( Lp_Functions (M,k)) by Th23;

      hence thesis by LPSPACE1: 3;

    end;

    registration

      let X be non empty set, S be SigmaField of X, M be sigma_Measure of S, k be positive Real;

      cluster RLSStruct (# ( Lp_Functions (M,k)), ( In (( 0. ( RLSp_PFunct X)),( Lp_Functions (M,k)))), ( add| (( Lp_Functions (M,k)),( RLSp_PFunct X))), ( Mult_ ( Lp_Functions (M,k))) #) -> Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital;

      coherence by Lm3;

    end

    definition

      let X be non empty set, S be SigmaField of X, M be sigma_Measure of S, k be positive Real;

      :: LPSPACE2:def3

      func RLSp_LpFunct (M,k) -> strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital non empty RLSStruct equals RLSStruct (# ( Lp_Functions (M,k)), ( In (( 0. ( RLSp_PFunct X)),( Lp_Functions (M,k)))), ( add| (( Lp_Functions (M,k)),( RLSp_PFunct X))), ( Mult_ ( Lp_Functions (M,k))) #);

      coherence ;

    end

    begin

    reserve v,u for VECTOR of ( RLSp_LpFunct (M,k));

    theorem :: LPSPACE2:29

    

     Th29: f = v & g = u implies (f + g) = (v + u)

    proof

      reconsider v2 = v, u2 = u as VECTOR of ( RLSp_PFunct X) by TARSKI:def 3;

      reconsider h = (v2 + u2) as Element of ( PFuncs (X, REAL ));

      reconsider v2, u2 as Element of ( PFuncs (X, REAL ));

      assume

       A1: f = v & g = u;

      

       A2: ( dom h) = (( dom v2) /\ ( dom u2)) & for x be Element of X st x in ( dom h) holds (h . x) = ((v2 . x) + (u2 . x)) by LPSPACE1: 6;

      for x be object st x in ( dom h) holds (h . x) = ((f . x) + (g . x)) by A1, LPSPACE1: 6;

      then h = (f + g) by A1, A2, VALUED_1:def 1;

      hence thesis by LPSPACE1: 4;

    end;

    theorem :: LPSPACE2:30

    

     Th30: f = u implies (a (#) f) = (a * u)

    proof

      reconsider u2 = u as VECTOR of ( RLSp_PFunct X) by TARSKI:def 3;

      reconsider h = (a * u2) as Element of ( PFuncs (X, REAL ));

      assume

       A1: f = u;

      then

       A2: ( dom h) = ( dom f) by LPSPACE1: 9;

      then for x be object st x in ( dom h) holds (h . x) = (a * (f . x)) by A1, LPSPACE1: 9;

      then h = (a (#) f) by A2, VALUED_1:def 5;

      hence thesis by LPSPACE1: 5;

    end;

    theorem :: LPSPACE2:31

    

     Th31: f = u implies (u + (( - 1) * u)) = ((X --> 0 ) | ( dom f)) & ex v,g be PartFunc of X, REAL st v in ( Lp_Functions (M,k)) & g in ( Lp_Functions (M,k)) & v = (u + (( - 1) * u)) & g = (X --> 0 ) & v a.e.= (g,M)

    proof

      reconsider u2 = u as VECTOR of ( RLSp_PFunct X) by TARSKI:def 3;

      assume

       A1: f = u;

      (( - 1) * u) = (( - 1) * u2) by LPSPACE1: 5;

      then

       A2: (u + (( - 1) * u)) = (u2 + (( - 1) * u2)) by LPSPACE1: 4;

      hence (u + (( - 1) * u)) = ((X --> 0 ) | ( dom f)) by A1, LPSPACE1: 16;

      (u + (( - 1) * u)) in ( Lp_Functions (M,k));

      then

      consider v be PartFunc of X, REAL such that

       A3: v = (u + (( - 1) * u)) & ex ND be Element of S st (M . (ND ` )) = 0 & ( dom v) = ND & v is ND -measurable & (( abs v) to_power k) is_integrable_on M;

      u in ( Lp_Functions (M,k));

      then ex uu1 be PartFunc of X, REAL st uu1 = u & ex ND be Element of S st (M . (ND ` )) = 0 & ( dom uu1) = ND & uu1 is ND -measurable & (( abs uu1) to_power k) is_integrable_on M;

      then

      consider ND be Element of S such that

       A4: (M . (ND ` )) = 0 & ( dom f) = ND & f is ND -measurable & (( abs f) to_power k) is_integrable_on M by A1;

      set g = (X --> 0 );

      

       A5: (ND ` ) is Element of S & ((ND ` ) ` ) = ND by MEASURE1: 34;

      

       A6: g in ( Lp_Functions (M,k)) by Th23;

      (v | ND) = ((g | ND) | ND) by A2, A3, A4, A1, LPSPACE1: 16;

      then (v | ND) = (g | ND) by FUNCT_1: 51;

      then v a.e.= (g,M) by A4, A5;

      hence thesis by A3, A6;

    end;

    definition

      let X be non empty set, S be SigmaField of X, M be sigma_Measure of S, k be positive Real;

      :: LPSPACE2:def4

      func AlmostZeroLpFunctions (M,k) -> non empty Subset of ( RLSp_LpFunct (M,k)) equals { f where f be PartFunc of X, REAL : f in ( Lp_Functions (M,k)) & f a.e.= ((X --> 0 ),M) };

      coherence

      proof

         A1:

        now

          let x be object;

          assume x in { f where f be PartFunc of X, REAL : f in ( Lp_Functions (M,k)) & f a.e.= ((X --> 0 ),M) };

          then ex f be PartFunc of X, REAL st x = f & f in ( Lp_Functions (M,k)) & f a.e.= ((X --> 0 ),M);

          hence x in the carrier of ( RLSp_LpFunct (M,k));

        end;

        

         A2: (X --> 0 ) a.e.= ((X --> 0 ),M) by LPSPACE1: 28;

        (X --> 0 ) in ( Lp_Functions (M,k)) by Th23;

        then (X --> 0 ) in { f where f be PartFunc of X, REAL : f in ( Lp_Functions (M,k)) & f a.e.= ((X --> 0 ),M) } by A2;

        hence thesis by A1, TARSKI:def 3;

      end;

    end

    

     Lm4: ( AlmostZeroLpFunctions (M,k)) is add-closed & ( AlmostZeroLpFunctions (M,k)) is multi-closed

    proof

      set Z = ( AlmostZeroLpFunctions (M,k));

      set V = ( RLSp_LpFunct (M,k));

      now

        let v,u be VECTOR of V;

        assume

         A1: v in Z & u in Z;

        then

        consider v1 be PartFunc of X, REAL such that

         A2: v1 = v & v1 in ( Lp_Functions (M,k)) & v1 a.e.= ((X --> 0 ),M);

        consider u1 be PartFunc of X, REAL such that

         A3: u1 = u & u1 in ( Lp_Functions (M,k)) & u1 a.e.= ((X --> 0 ),M) by A1;

        

         A4: (v + u) = (v1 + u1) by Th29, A2, A3;

        ((X --> 0 ) + (X --> 0 )) = (X --> 0 ) by LPSPACE1: 33;

        then (v1 + u1) in ( Lp_Functions (M,k)) & (v1 + u1) a.e.= ((X --> 0 ),M) by A4, A2, A3, LPSPACE1: 31;

        hence (v + u) in Z by A4;

      end;

      hence Z is add-closed by IDEAL_1:def 1;

      now

        let a be Real, u be VECTOR of V;

        assume u in Z;

        then

        consider u1 be PartFunc of X, REAL such that

         A5: u1 = u & u1 in ( Lp_Functions (M,k)) & u1 a.e.= ((X --> 0 ),M);

        

         A6: (a * u) = (a (#) u1) by Th30, A5;

        (a (#) (X --> 0 )) = (X --> 0 ) by LPSPACE1: 33;

        then (a (#) u1) in ( Lp_Functions (M,k)) & (a (#) u1) a.e.= ((X --> 0 ),M) by A6, A5, LPSPACE1: 32;

        hence (a * u) in Z by A6;

      end;

      hence Z is multi-closed;

    end;

    registration

      let X be non empty set, S be SigmaField of X, M be sigma_Measure of S, k be positive Real;

      cluster ( AlmostZeroLpFunctions (M,k)) -> add-closed multi-closed;

      coherence by Lm4;

    end

    theorem :: LPSPACE2:32

    ( 0. ( RLSp_LpFunct (M,k))) = (X --> 0 ) & ( 0. ( RLSp_LpFunct (M,k))) in ( AlmostZeroLpFunctions (M,k))

    proof

      thus ( 0. ( RLSp_LpFunct (M,k))) = (X --> 0 ) by Th23, SUBSET_1:def 8;

      

       A1: (X --> 0 ) a.e.= ((X --> 0 ),M) & (X --> 0 ) in ( Lp_Functions (M,k)) by Th23, LPSPACE1: 28;

      ( 0. ( RLSp_LpFunct (M,k))) = ( 0. ( RLSp_PFunct X)) by Th23, SUBSET_1:def 8;

      hence ( 0. ( RLSp_LpFunct (M,k))) in ( AlmostZeroLpFunctions (M,k)) by A1;

    end;

    definition

      let X be non empty set, S be SigmaField of X, M be sigma_Measure of S, k be positive Real;

      :: LPSPACE2:def5

      func RLSp_AlmostZeroLpFunct (M,k) -> non empty RLSStruct equals RLSStruct (# ( AlmostZeroLpFunctions (M,k)), ( In (( 0. ( RLSp_LpFunct (M,k))),( AlmostZeroLpFunctions (M,k)))), ( add| (( AlmostZeroLpFunctions (M,k)),( RLSp_LpFunct (M,k)))), ( Mult_ ( AlmostZeroLpFunctions (M,k))) #);

      coherence ;

    end

    registration

      let X be non empty set, S be SigmaField of X, M be sigma_Measure of S, k be positive Real;

      cluster ( RLSp_LpFunct (M,k)) -> strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital;

      coherence ;

    end

    reserve v,u for VECTOR of ( RLSp_AlmostZeroLpFunct (M,k));

    theorem :: LPSPACE2:33

    f = v & g = u implies (f + g) = (v + u)

    proof

      reconsider v2 = v, u2 = u as VECTOR of ( RLSp_LpFunct (M,k)) by TARSKI:def 3;

      assume

       A1: f = v & g = u;

      (v + u) = (v2 + u2) by LPSPACE1: 4;

      hence (v + u) = (f + g) by Th29, A1;

    end;

    theorem :: LPSPACE2:34

    f = u implies (a (#) f) = (a * u)

    proof

      reconsider u2 = u as VECTOR of ( RLSp_LpFunct (M,k)) by TARSKI:def 3;

      assume

       A1: f = u;

      (a * u) = (a * u2) by LPSPACE1: 5;

      hence (a * u) = (a (#) f) by Th30, A1;

    end;

    definition

      let X be non empty set, S be SigmaField of X, M be sigma_Measure of S, f be PartFunc of X, REAL , k be positive Real;

      :: LPSPACE2:def6

      func a.e-eq-class_Lp (f,M,k) -> Subset of ( Lp_Functions (M,k)) equals { h where h be PartFunc of X, REAL : h in ( Lp_Functions (M,k)) & f a.e.= (h,M) };

      correctness

      proof

        now

          let x be object;

          assume x in { g where g be PartFunc of X, REAL : g in ( Lp_Functions (M,k)) & f a.e.= (g,M) };

          then ex g be PartFunc of X, REAL st x = g & g in ( Lp_Functions (M,k)) & f a.e.= (g,M);

          hence x in ( Lp_Functions (M,k));

        end;

        hence thesis by TARSKI:def 3;

      end;

    end

    theorem :: LPSPACE2:35

    

     Th35: f in ( Lp_Functions (M,k)) implies ex E be Element of S st (M . (E ` )) = 0 & ( dom f) = E & f is E -measurable

    proof

      assume f in ( Lp_Functions (M,k));

      then ex f1 be PartFunc of X, REAL st f = f1 & (ex E be Element of S st (M . (E ` )) = 0 & ( dom f1) = E & f1 is E -measurable & (( abs f1) to_power k) is_integrable_on M);

      hence thesis;

    end;

    theorem :: LPSPACE2:36

    

     Th36: g in ( Lp_Functions (M,k)) & g a.e.= (f,M) implies g in ( a.e-eq-class_Lp (f,M,k))

    proof

      assume that

       A1: g in ( Lp_Functions (M,k)) and

       A2: g a.e.= (f,M);

      f a.e.= (g,M) by A2;

      hence g in ( a.e-eq-class_Lp (f,M,k)) by A1;

    end;

    theorem :: LPSPACE2:37

    

     Th37: (ex E be Element of S st (M . (E ` )) = 0 & E = ( dom f) & f is E -measurable) & g in ( a.e-eq-class_Lp (f,M,k)) implies g a.e.= (f,M) & f in ( Lp_Functions (M,k))

    proof

      assume that

       A1: ex E be Element of S st (M . (E ` )) = 0 & E = ( dom f) & f is E -measurable and

       A2: g in ( a.e-eq-class_Lp (f,M,k));

      

       A3: ex r be PartFunc of X, REAL st g = r & r in ( Lp_Functions (M,k)) & f a.e.= (r,M) by A2;

      hence g a.e.= (f,M);

      g in ( Lp_Functions (M,k)) by A2;

      then

      consider g1 be PartFunc of X, REAL such that

       A4: g = g1 & ex E be Element of S st (M . (E ` )) = 0 & ( dom g1) = E & g1 is E -measurable & (( abs g1) to_power k) is_integrable_on M;

      consider Eh be Element of S such that

       A5: (M . (Eh ` )) = 0 & ( dom g) = Eh & g is Eh -measurable & (( abs g) to_power k) is_integrable_on M by A4;

      reconsider ND = (Eh ` ) as Element of S by MEASURE1: 34;

      ex E be Element of S st (M . (E ` )) = 0 & ( dom f) = E & f is E -measurable & (( abs f) to_power k) is_integrable_on M

      proof

        set AFK = (( abs f) to_power k);

        set AGK = (( abs g) to_power k);

        consider Ef be Element of S such that

         A6: (M . (Ef ` )) = 0 & Ef = ( dom f) & f is Ef -measurable by A1;

        take Ef;

        consider EE be Element of S such that

         A7: (M . EE) = 0 & (g | (EE ` )) = (f | (EE ` )) by A3;

        reconsider E1 = (ND \/ EE) as Element of S;

        EE c= E1 by XBOOLE_1: 7;

        then (E1 ` ) c= (EE ` ) by SUBSET_1: 12;

        then

         A8: (f | (E1 ` )) = ((f | (EE ` )) | (E1 ` )) & (g | (E1 ` )) = ((g | (EE ` )) | (E1 ` )) by FUNCT_1: 51;

        

         A9: ( dom ( abs f)) = Ef by A6, VALUED_1:def 11;

        then ( dom AFK) = Ef by MESFUN6C:def 4;

        then

         A10: ( dom ( max+ ( R_EAL AFK))) = Ef & ( dom ( max- ( R_EAL AFK))) = Ef by MESFUNC2:def 2, MESFUNC2:def 3;

        ( abs f) is Ef -measurable by A6, MESFUNC6: 48;

        then AFK is Ef -measurable by A9, MESFUN6C: 29;

        then

         A11: Ef = ( dom ( R_EAL AFK)) & ( R_EAL AFK) is Ef -measurable by A9, MESFUN6C:def 4;

        then

         A12: ( max+ ( R_EAL AFK)) is Ef -measurable & ( max- ( R_EAL AFK)) is Ef -measurable by MESFUNC2: 25, MESFUNC2: 26;

        (for x be Element of X holds 0. <= (( max+ ( R_EAL AFK)) . x)) & (for x be Element of X holds 0. <= (( max- ( R_EAL AFK)) . x)) by MESFUNC2: 12, MESFUNC2: 13;

        then

         A13: ( max+ ( R_EAL AFK)) is nonnegative & ( max- ( R_EAL AFK)) is nonnegative by SUPINF_2: 39;

        

         A14: Ef = ((Ef /\ E1) \/ (Ef \ E1)) by XBOOLE_1: 51;

        reconsider E0 = (Ef /\ E1) as Element of S;

        reconsider E2 = (Ef \ E1) as Element of S;

        ( max+ ( R_EAL AFK)) = (( max+ ( R_EAL AFK)) | ( dom ( max+ ( R_EAL AFK)))) & ( max- ( R_EAL AFK)) = (( max- ( R_EAL AFK)) | ( dom ( max- ( R_EAL AFK)))) by RELAT_1: 69;

        then

         A15: ( integral+ (M,( max+ ( R_EAL AFK)))) = (( integral+ (M,(( max+ ( R_EAL AFK)) | E0))) + ( integral+ (M,(( max+ ( R_EAL AFK)) | E2)))) & ( integral+ (M,( max- ( R_EAL AFK)))) = (( integral+ (M,(( max- ( R_EAL AFK)) | E0))) + ( integral+ (M,(( max- ( R_EAL AFK)) | E2)))) by A10, A12, A13, A14, MESFUNC5: 81, XBOOLE_1: 89;

        

         A16: ( integral+ (M,(( max+ ( R_EAL AFK)) | E0))) >= 0 & ( integral+ (M,(( max- ( R_EAL AFK)) | E0))) >= 0 by A12, A13, A10, MESFUNC5: 80;

        ND is measure_zero of M & EE is measure_zero of M by A5, A7, MEASURE1:def 7;

        then E1 is measure_zero of M by MEASURE1: 37;

        then (M . E1) = 0 by MEASURE1:def 7;

        then ( integral+ (M,(( max+ ( R_EAL AFK)) | E1))) = 0 & ( integral+ (M,(( max- ( R_EAL AFK)) | E1))) = 0 by A10, A12, A13, MESFUNC5: 82;

        then ( integral+ (M,(( max+ ( R_EAL AFK)) | E0))) = 0 & ( integral+ (M,(( max- ( R_EAL AFK)) | E0))) = 0 by A10, A12, A13, A16, MESFUNC5: 83, XBOOLE_1: 17;

        then

         A17: ( integral+ (M,( max+ ( R_EAL AFK)))) = ( integral+ (M,(( max+ ( R_EAL AFK)) | E2))) & ( integral+ (M,( max- ( R_EAL AFK)))) = ( integral+ (M,(( max- ( R_EAL AFK)) | E2))) by A15, XXREAL_3: 4;

        (Ef \ E1) = (Ef /\ (E1 ` )) by SUBSET_1: 13;

        then

         A18: E2 c= (E1 ` ) by XBOOLE_1: 17;

        then (f | E2) = ((g | (E1 ` )) | E2) by A7, A8, FUNCT_1: 51;

        then

         A19: (f | E2) = (g | E2) by A18, FUNCT_1: 51;

        

         A20: (( abs f) | E2) = ( abs (f | E2)) & (( abs g) | E2) = ( abs (g | E2)) by RFUNCT_1: 46;

        

         A21: ((( abs f) | E2) to_power k) = (AFK | E2) & ((( abs g) | E2) to_power k) = (AGK | E2) by Th20;

        

         A22: (( max+ ( R_EAL AFK)) | E2) = ( max+ (( R_EAL AFK) | E2)) & (( max+ ( R_EAL AGK)) | E2) = ( max+ (( R_EAL AGK) | E2)) & (( max- ( R_EAL AFK)) | E2) = ( max- (( R_EAL AFK) | E2)) & (( max- ( R_EAL AGK)) | E2) = ( max- (( R_EAL AGK) | E2)) by MESFUNC5: 28;

        

         A23: ( R_EAL AGK) is_integrable_on M by A5;

        then

         A24: ( integral+ (M,( max+ ( R_EAL AGK)))) < +infty & ( integral+ (M,( max- ( R_EAL AGK)))) < +infty ;

        ( integral+ (M,( max+ (( R_EAL AGK) | E2)))) <= ( integral+ (M,( max+ ( R_EAL AGK)))) & ( integral+ (M,( max- (( R_EAL AGK) | E2)))) <= ( integral+ (M,( max- ( R_EAL AGK)))) by A23, MESFUNC5: 97;

        then ( integral+ (M,( max+ ( R_EAL AFK)))) < +infty & ( integral+ (M,( max- ( R_EAL AFK)))) < +infty by A17, A19, A20, A21, A22, A24, XXREAL_0: 2;

        then ( R_EAL (( abs f) to_power k)) is_integrable_on M by A11;

        hence thesis by A6;

      end;

      hence f in ( Lp_Functions (M,k));

    end;

    theorem :: LPSPACE2:38

    

     Th38: f in ( Lp_Functions (M,k)) implies f in ( a.e-eq-class_Lp (f,M,k))

    proof

      assume

       A1: f in ( Lp_Functions (M,k));

      f a.e.= (f,M) by LPSPACE1: 28;

      hence thesis by A1;

    end;

    theorem :: LPSPACE2:39

    

     Th39: (ex E be Element of S st (M . (E ` )) = 0 & E = ( dom g) & g is E -measurable) & ( a.e-eq-class_Lp (f,M,k)) <> {} & ( a.e-eq-class_Lp (f,M,k)) = ( a.e-eq-class_Lp (g,M,k)) implies f a.e.= (g,M)

    proof

      assume that

       A1: (ex E be Element of S st (M . (E ` )) = 0 & E = ( dom g) & g is E -measurable) and

       A2: ( a.e-eq-class_Lp (f,M,k)) <> {} and

       A3: ( a.e-eq-class_Lp (f,M,k)) = ( a.e-eq-class_Lp (g,M,k));

      consider x be object such that

       A4: x in ( a.e-eq-class_Lp (f,M,k)) by A2, XBOOLE_0:def 1;

      consider r be PartFunc of X, REAL such that

       A5: x = r & r in ( Lp_Functions (M,k)) & f a.e.= (r,M) by A4;

      r a.e.= (g,M) by A1, A3, A4, A5, Th37;

      hence thesis by A5, LPSPACE1: 30;

    end;

    theorem :: LPSPACE2:40

    f in ( Lp_Functions (M,k)) & (ex E be Element of S st (M . (E ` )) = 0 & E = ( dom g) & g is E -measurable) & ( a.e-eq-class_Lp (f,M,k)) = ( a.e-eq-class_Lp (g,M,k)) implies f a.e.= (g,M)

    proof

      assume that

       A1: f in ( Lp_Functions (M,k)) and

       A2: (ex E be Element of S st (M . (E ` )) = 0 & E = ( dom g) & g is E -measurable) and

       A3: ( a.e-eq-class_Lp (f,M,k)) = ( a.e-eq-class_Lp (g,M,k));

      ( a.e-eq-class_Lp (f,M,k)) is non empty by A1, Th38;

      hence thesis by A2, A3, Th39;

    end;

    theorem :: LPSPACE2:41

    

     Th41: f a.e.= (g,M) implies ( a.e-eq-class_Lp (f,M,k)) = ( a.e-eq-class_Lp (g,M,k))

    proof

      assume

       A1: f a.e.= (g,M);

      now

        let x be object;

        assume x in ( a.e-eq-class_Lp (f,M,k));

        then

        consider r be PartFunc of X, REAL such that

         A2: x = r & r in ( Lp_Functions (M,k)) & f a.e.= (r,M);

        r a.e.= (f,M) by A2;

        then r a.e.= (g,M) by A1, LPSPACE1: 30;

        then g a.e.= (r,M);

        hence x in ( a.e-eq-class_Lp (g,M,k)) by A2;

      end;

      then

       A3: ( a.e-eq-class_Lp (f,M,k)) c= ( a.e-eq-class_Lp (g,M,k));

      now

        let x be object;

        assume x in ( a.e-eq-class_Lp (g,M,k));

        then

        consider r be PartFunc of X, REAL such that

         A4: x = r & r in ( Lp_Functions (M,k)) & g a.e.= (r,M);

        r a.e.= (g,M) & g a.e.= (f,M) by A1, A4;

        then r a.e.= (f,M) by LPSPACE1: 30;

        then f a.e.= (r,M);

        hence x in ( a.e-eq-class_Lp (f,M,k)) by A4;

      end;

      then ( a.e-eq-class_Lp (g,M,k)) c= ( a.e-eq-class_Lp (f,M,k));

      hence ( a.e-eq-class_Lp (f,M,k)) = ( a.e-eq-class_Lp (g,M,k)) by A3;

    end;

    theorem :: LPSPACE2:42

    

     Th42: f a.e.= (g,M) implies ( a.e-eq-class_Lp (f,M,k)) = ( a.e-eq-class_Lp (g,M,k)) by Th41;

    theorem :: LPSPACE2:43

    f in ( Lp_Functions (M,k)) & g in ( a.e-eq-class_Lp (f,M,k)) implies ( a.e-eq-class_Lp (f,M,k)) = ( a.e-eq-class_Lp (g,M,k))

    proof

      assume that

       A1: f in ( Lp_Functions (M,k)) and

       A2: g in ( a.e-eq-class_Lp (f,M,k));

      ex E be Element of S st (M . (E ` )) = 0 & ( dom f) = E & f is E -measurable by A1, Th35;

      hence thesis by Th41, A2, Th37;

    end;

    theorem :: LPSPACE2:44

    (ex E be Element of S st (M . (E ` )) = 0 & E = ( dom f) & f is E -measurable) & (ex E be Element of S st (M . (E ` )) = 0 & E = ( dom f1) & f1 is E -measurable) & (ex E be Element of S st (M . (E ` )) = 0 & E = ( dom g) & g is E -measurable) & (ex E be Element of S st (M . (E ` )) = 0 & E = ( dom g1) & g1 is E -measurable) & ( a.e-eq-class_Lp (f,M,k)) is non empty & ( a.e-eq-class_Lp (g,M,k)) is non empty & ( a.e-eq-class_Lp (f,M,k)) = ( a.e-eq-class_Lp (f1,M,k)) & ( a.e-eq-class_Lp (g,M,k)) = ( a.e-eq-class_Lp (g1,M,k)) implies ( a.e-eq-class_Lp ((f + g),M,k)) = ( a.e-eq-class_Lp ((f1 + g1),M,k))

    proof

      assume (ex E be Element of S st (M . (E ` )) = 0 & E = ( dom f) & f is E -measurable) & (ex E be Element of S st (M . (E ` )) = 0 & E = ( dom f1) & f1 is E -measurable) & (ex E be Element of S st (M . (E ` )) = 0 & E = ( dom g) & g is E -measurable) & (ex E be Element of S st (M . (E ` )) = 0 & E = ( dom g1) & g1 is E -measurable) & ( a.e-eq-class_Lp (f,M,k)) is non empty & ( a.e-eq-class_Lp (g,M,k)) is non empty & ( a.e-eq-class_Lp (f,M,k)) = ( a.e-eq-class_Lp (f1,M,k)) & ( a.e-eq-class_Lp (g,M,k)) = ( a.e-eq-class_Lp (g1,M,k));

      then f a.e.= (f1,M) & g a.e.= (g1,M) by Th39;

      hence thesis by Th41, LPSPACE1: 31;

    end;

    theorem :: LPSPACE2:45

    

     Th45: f in ( Lp_Functions (M,k)) & f1 in ( Lp_Functions (M,k)) & g in ( Lp_Functions (M,k)) & g1 in ( Lp_Functions (M,k)) & ( a.e-eq-class_Lp (f,M,k)) = ( a.e-eq-class_Lp (f1,M,k)) & ( a.e-eq-class_Lp (g,M,k)) = ( a.e-eq-class_Lp (g1,M,k)) implies ( a.e-eq-class_Lp ((f + g),M,k)) = ( a.e-eq-class_Lp ((f1 + g1),M,k))

    proof

      assume that

       A1: f in ( Lp_Functions (M,k)) and

       A2: f1 in ( Lp_Functions (M,k)) and

       A3: g in ( Lp_Functions (M,k)) and

       A4: g1 in ( Lp_Functions (M,k)) and

       A5: ( a.e-eq-class_Lp (f,M,k)) = ( a.e-eq-class_Lp (f1,M,k)) & ( a.e-eq-class_Lp (g,M,k)) = ( a.e-eq-class_Lp (g1,M,k));

      

       A6: (ex E be Element of S st (M . (E ` )) = 0 & ( dom f1) = E & f1 is E -measurable) & (ex E be Element of S st (M . (E ` )) = 0 & ( dom g1) = E & g1 is E -measurable) by A2, A4, Th35;

      f in ( a.e-eq-class_Lp (f,M,k)) & g in ( a.e-eq-class_Lp (g,M,k)) by A1, A3, Th38;

      then f a.e.= (f1,M) & g a.e.= (g1,M) by A5, A6, Th37;

      hence thesis by Th41, LPSPACE1: 31;

    end;

    theorem :: LPSPACE2:46

    (ex E be Element of S st (M . (E ` )) = 0 & ( dom f) = E & f is E -measurable) & (ex E be Element of S st (M . (E ` )) = 0 & ( dom g) = E & g is E -measurable) & ( a.e-eq-class_Lp (f,M,k)) is non empty & ( a.e-eq-class_Lp (f,M,k)) = ( a.e-eq-class_Lp (g,M,k)) implies ( a.e-eq-class_Lp ((a (#) f),M,k)) = ( a.e-eq-class_Lp ((a (#) g),M,k))

    proof

      assume (ex E be Element of S st (M . (E ` )) = 0 & ( dom f) = E & f is E -measurable) & (ex E be Element of S st (M . (E ` )) = 0 & ( dom g) = E & g is E -measurable) & ( a.e-eq-class_Lp (f,M,k)) is non empty & ( a.e-eq-class_Lp (f,M,k)) = ( a.e-eq-class_Lp (g,M,k));

      then (a (#) f) a.e.= ((a (#) g),M) by Th39, LPSPACE1: 32;

      hence thesis by Th41;

    end;

    theorem :: LPSPACE2:47

    

     Th47: f in ( Lp_Functions (M,k)) & g in ( Lp_Functions (M,k)) & ( a.e-eq-class_Lp (f,M,k)) = ( a.e-eq-class_Lp (g,M,k)) implies ( a.e-eq-class_Lp ((a (#) f),M,k)) = ( a.e-eq-class_Lp ((a (#) g),M,k))

    proof

      assume

       A1: f in ( Lp_Functions (M,k)) & g in ( Lp_Functions (M,k)) & ( a.e-eq-class_Lp (f,M,k)) = ( a.e-eq-class_Lp (g,M,k));

      then

       A2: (ex E be Element of S st (M . (E ` )) = 0 & ( dom f) = E & f is E -measurable) & (ex E be Element of S st (M . (E ` )) = 0 & ( dom g) = E & g is E -measurable) by Th35;

      f in ( a.e-eq-class_Lp (g,M,k)) by A1, Th38;

      then f a.e.= (g,M) & (a (#) f) in ( Lp_Functions (M,k)) & (a (#) g) in ( Lp_Functions (M,k)) by A2, Th37, Th26;

      hence thesis by Th42, LPSPACE1: 32;

    end;

    definition

      let X be non empty set, S be SigmaField of X, M be sigma_Measure of S, k be positive Real;

      :: LPSPACE2:def7

      func CosetSet (M,k) -> non empty Subset-Family of ( Lp_Functions (M,k)) equals { ( a.e-eq-class_Lp (f,M,k)) where f be PartFunc of X, REAL : f in ( Lp_Functions (M,k)) };

      correctness

      proof

        set C = { ( a.e-eq-class_Lp (f,M,k)) where f be PartFunc of X, REAL : f in ( Lp_Functions (M,k)) };

        

         A1: C c= ( bool ( Lp_Functions (M,k)))

        proof

          let x be object;

          assume x in C;

          then ex f be PartFunc of X, REAL st ( a.e-eq-class_Lp (f,M,k)) = x & f in ( Lp_Functions (M,k));

          hence x in ( bool ( Lp_Functions (M,k)));

        end;

        (X --> 0 ) in ( Lp_Functions (M,k)) by Th23;

        then ( a.e-eq-class_Lp ((X --> 0 ),M,k)) in C;

        hence thesis by A1;

      end;

    end

    definition

      let X be non empty set, S be SigmaField of X, M be sigma_Measure of S, k be positive Real;

      :: LPSPACE2:def8

      func addCoset (M,k) -> BinOp of ( CosetSet (M,k)) means

      : Def8: for A,B be Element of ( CosetSet (M,k)), a,b be PartFunc of X, REAL st a in A & b in B holds (it . (A,B)) = ( a.e-eq-class_Lp ((a + b),M,k));

      existence

      proof

        set C = ( CosetSet (M,k));

        defpred P[ set, set, set] means for a,b be PartFunc of X, REAL st a in $1 & b in $2 holds $3 = ( a.e-eq-class_Lp ((a + b),M,k));

         A1:

        now

          let A,B be Element of C;

          A in C;

          then

          consider a be PartFunc of X, REAL such that

           A2: A = ( a.e-eq-class_Lp (a,M,k)) & a in ( Lp_Functions (M,k));

          

           A3: ex E be Element of S st (M . (E ` )) = 0 & ( dom a) = E & a is E -measurable by A2, Th35;

          B in C;

          then

          consider b be PartFunc of X, REAL such that

           A4: B = ( a.e-eq-class_Lp (b,M,k)) & b in ( Lp_Functions (M,k));

          

           A5: ex E be Element of S st (M . (E ` )) = 0 & ( dom b) = E & b is E -measurable by A4, Th35;

          set z = ( a.e-eq-class_Lp ((a + b),M,k));

          (a + b) in ( Lp_Functions (M,k)) by Th25, A2, A4;

          then z in C;

          then

          reconsider z as Element of C;

          take z;

          now

            let a1,b1 be PartFunc of X, REAL ;

            assume a1 in A & b1 in B;

            then a1 a.e.= (a,M) & b1 a.e.= (b,M) by A2, A3, A4, A5, Th37;

            hence z = ( a.e-eq-class_Lp ((a1 + b1),M,k)) by Th42, LPSPACE1: 31;

          end;

          hence P[A, B, z];

        end;

        consider f be Function of [:C, C:], C such that

         A6: for A,B be Element of C holds P[A, B, (f . (A,B))] from BINOP_1:sch 3( A1);

        reconsider f as BinOp of C;

        take f;

        let A,B be Element of C;

        let a,b be PartFunc of X, REAL ;

        assume a in A & b in B;

        hence (f . (A,B)) = ( a.e-eq-class_Lp ((a + b),M,k)) by A6;

      end;

      uniqueness

      proof

        let f1,f2 be BinOp of ( CosetSet (M,k)) such that

         A7: for A,B be Element of ( CosetSet (M,k)), a,b be PartFunc of X, REAL st a in A & b in B holds (f1 . (A,B)) = ( a.e-eq-class_Lp ((a + b),M,k)) and

         A8: for A,B be Element of ( CosetSet (M,k)), a,b be PartFunc of X, REAL st a in A & b in B holds (f2 . (A,B)) = ( a.e-eq-class_Lp ((a + b),M,k));

        now

          let A,B be Element of ( CosetSet (M,k));

          A in ( CosetSet (M,k));

          then

          consider a1 be PartFunc of X, REAL such that

           A9: A = ( a.e-eq-class_Lp (a1,M,k)) & a1 in ( Lp_Functions (M,k));

          B in ( CosetSet (M,k));

          then

          consider b1 be PartFunc of X, REAL such that

           A10: B = ( a.e-eq-class_Lp (b1,M,k)) & b1 in ( Lp_Functions (M,k));

          

           A11: a1 in A & b1 in B by A9, A10, Th38;

          then (f1 . (A,B)) = ( a.e-eq-class_Lp ((a1 + b1),M,k)) by A7;

          hence (f1 . (A,B)) = (f2 . (A,B)) by A8, A11;

        end;

        hence thesis by BINOP_1: 2;

      end;

    end

    definition

      let X be non empty set, S be SigmaField of X, M be sigma_Measure of S, k be positive Real;

      :: LPSPACE2:def9

      func zeroCoset (M,k) -> Element of ( CosetSet (M,k)) equals ( a.e-eq-class_Lp ((X --> 0 ),M,k));

      correctness

      proof

        (X --> 0 ) in ( Lp_Functions (M,k)) by Th23;

        then ( a.e-eq-class_Lp ((X --> 0 ),M,k)) in ( CosetSet (M,k));

        hence thesis;

      end;

    end

    definition

      let X be non empty set, S be SigmaField of X, M be sigma_Measure of S, k be positive Real;

      :: LPSPACE2:def10

      func lmultCoset (M,k) -> Function of [: REAL , ( CosetSet (M,k)):], ( CosetSet (M,k)) means

      : Def10: for z be Real, A be Element of ( CosetSet (M,k)), f be PartFunc of X, REAL st f in A holds (it . (z,A)) = ( a.e-eq-class_Lp ((z (#) f),M,k));

      existence

      proof

        set C = ( CosetSet (M,k));

        defpred P[ Real, set, set] means for f be PartFunc of X, REAL st f in $2 holds $3 = ( a.e-eq-class_Lp (($1 (#) f),M,k));

         A1:

        now

          let z be Element of REAL , A be Element of C;

          A in C;

          then

          consider a be PartFunc of X, REAL such that

           A2: A = ( a.e-eq-class_Lp (a,M,k)) & a in ( Lp_Functions (M,k));

          

           A3: ex E be Element of S st (M . (E ` )) = 0 & E = ( dom a) & a is E -measurable by A2, Th35;

          set c = ( a.e-eq-class_Lp ((z (#) a),M,k));

          (z (#) a) in ( Lp_Functions (M,k)) by Th26, A2;

          then c in C;

          then

          reconsider c as Element of C;

          take c;

          now

            let a1 be PartFunc of X, REAL ;

            assume a1 in A;

            then (z (#) a1) a.e.= ((z (#) a),M) by A2, A3, Th37, LPSPACE1: 32;

            hence c = ( a.e-eq-class_Lp ((z (#) a1),M,k)) by Th42;

          end;

          hence P[z, A, c];

        end;

        consider f be Function of [: REAL , C:], C such that

         A4: for z be Element of REAL , A be Element of C holds P[z, A, (f . (z,A))] from BINOP_1:sch 3( A1);

        

         A5: for z be Real, A be Element of C holds P[z, A, (f . (z,A))]

        proof

          let z be Real, A be Element of C;

          reconsider z as Element of REAL by XREAL_0:def 1;

           P[z, A, (f . (z,A))] by A4;

          hence thesis;

        end;

        take f;

        let z be Real, A be Element of C, a be PartFunc of X, REAL ;

        assume a in A;

        hence (f . (z,A)) = ( a.e-eq-class_Lp ((z (#) a),M,k)) by A5;

      end;

      uniqueness

      proof

        set C = ( CosetSet (M,k));

        let f1,f2 be Function of [: REAL , C:], C such that

         A6: for z be Real, A be Element of ( CosetSet (M,k)), a be PartFunc of X, REAL st a in A holds (f1 . (z,A)) = ( a.e-eq-class_Lp ((z (#) a),M,k)) and

         A7: for z be Real, A be Element of ( CosetSet (M,k)), a be PartFunc of X, REAL st a in A holds (f2 . (z,A)) = ( a.e-eq-class_Lp ((z (#) a),M,k));

        now

          let z be Element of REAL , A be Element of ( CosetSet (M,k));

          A in C;

          then

          consider a1 be PartFunc of X, REAL such that

           A8: A = ( a.e-eq-class_Lp (a1,M,k)) & a1 in ( Lp_Functions (M,k));

          

          thus (f1 . (z,A)) = ( a.e-eq-class_Lp ((z (#) a1),M,k)) by A6, A8, Th38

          .= (f2 . (z,A)) by A7, A8, Th38;

        end;

        hence thesis by BINOP_1: 2;

      end;

    end

    definition

      let X be non empty set, S be SigmaField of X, M be sigma_Measure of S, k be positive Real;

      :: LPSPACE2:def11

      func Pre-Lp-Space (M,k) -> strict RLSStruct means

      : Def11: the carrier of it = ( CosetSet (M,k)) & the addF of it = ( addCoset (M,k)) & ( 0. it ) = ( zeroCoset (M,k)) & the Mult of it = ( lmultCoset (M,k));

      existence

      proof

        take RLSStruct (# ( CosetSet (M,k)), ( zeroCoset (M,k)), ( addCoset (M,k)), ( lmultCoset (M,k)) #);

        thus thesis;

      end;

      uniqueness ;

    end

    registration

      let X be non empty set, S be SigmaField of X, M be sigma_Measure of S, k be positive Real;

      cluster ( Pre-Lp-Space (M,k)) -> non empty;

      coherence

      proof

        the carrier of ( Pre-Lp-Space (M,k)) = ( CosetSet (M,k)) by Def11;

        hence thesis;

      end;

    end

    registration

      let X be non empty set, S be SigmaField of X, M be sigma_Measure of S, k be positive Real;

      cluster ( Pre-Lp-Space (M,k)) -> Abelian add-associative right_zeroed right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital;

      coherence

      proof

        set C = ( CosetSet (M,k)), aC = ( addCoset (M,k)), lC = ( lmultCoset (M,k));

        set A = ( Pre-Lp-Space (M,k));

        

         A1: the carrier of A = ( CosetSet (M,k)) & the addF of A = ( addCoset (M,k)) & ( 0. A) = ( zeroCoset (M,k)) & the Mult of A = ( lmultCoset (M,k)) by Def11;

        thus A is Abelian

        proof

          let A1,A2 be Element of A;

          A1 in C by A1;

          then

          consider a be PartFunc of X, REAL such that

           A2: A1 = ( a.e-eq-class_Lp (a,M,k)) & a in ( Lp_Functions (M,k));

          A2 in C by A1;

          then

          consider b be PartFunc of X, REAL such that

           A3: A2 = ( a.e-eq-class_Lp (b,M,k)) & b in ( Lp_Functions (M,k));

          

           A4: a in A1 & b in A2 by A2, A3, Th38;

          then (A1 + A2) = ( a.e-eq-class_Lp ((a + b),M,k)) by A1, Def8;

          hence (A1 + A2) = (A2 + A1) by A1, A4, Def8;

        end;

        thus A is add-associative

        proof

          let A1,A2,A3 be Element of A;

          A1 in C by A1;

          then

          consider a be PartFunc of X, REAL such that

           A5: A1 = ( a.e-eq-class_Lp (a,M,k)) & a in ( Lp_Functions (M,k));

          A2 in C by A1;

          then

          consider b be PartFunc of X, REAL such that

           A6: A2 = ( a.e-eq-class_Lp (b,M,k)) & b in ( Lp_Functions (M,k));

          A3 in C by A1;

          then

          consider c be PartFunc of X, REAL such that

           A7: A3 = ( a.e-eq-class_Lp (c,M,k)) & c in ( Lp_Functions (M,k));

          

           A8: a in A1 & b in A2 & c in A3 by A5, A6, A7, Th38;

          then (aC . (A1,A2)) = ( a.e-eq-class_Lp ((a + b),M,k)) & (aC . (A2,A3)) = ( a.e-eq-class_Lp ((b + c),M,k)) by A1, Def8;

          then

           A9: (a + b) in (A1 + A2) & (b + c) in (A2 + A3) by A1, Th38, Th25, A5, A6, A7;

          reconsider a1 = a, b1 = b, c1 = c as VECTOR of ( RLSp_LpFunct (M,k)) by A5, A6, A7;

          

           A10: (a + b) = (a1 + b1) & (b + c) = (b1 + c1) by Th29;

          then (a + (b + c)) = (a1 + (b1 + c1)) by Th29;

          then (a + (b + c)) = ((a1 + b1) + c1) by RLVECT_1:def 3;

          then (a + (b + c)) = ((a + b) + c) by A10, Th29;

          then ((A1 + A2) + A3) = ( a.e-eq-class_Lp ((a + (b + c)),M,k)) by A8, A9, Def8, A1;

          hence ((A1 + A2) + A3) = (A1 + (A2 + A3)) by A8, A9, Def8, A1;

        end;

        thus A is right_zeroed

        proof

          let A1 be Element of A;

          A1 in C by A1;

          then

          consider a be PartFunc of X, REAL such that

           A11: A1 = ( a.e-eq-class_Lp (a,M,k)) & a in ( Lp_Functions (M,k));

          

           A12: a in A1 by A11, Th38;

          set z = (X --> 0 );

          

           A13: z in ( 0. A) by A1, Th38, Th23;

          reconsider a1 = a, z1 = z as VECTOR of ( RLSp_LpFunct (M,k)) by A11, Th23;

          (a + z) = (a1 + z1) by Th29

          .= (a1 + ( 0. ( RLSp_LpFunct (M,k))))

          .= a by RLVECT_1:def 4;

          hence (A1 + ( 0. A)) = A1 by A1, A11, A12, A13, Def8;

        end;

        thus A is right_complementable

        proof

          let A1 be Element of A;

          A1 in C by A1;

          then

          consider a be PartFunc of X, REAL such that

           A14: A1 = ( a.e-eq-class_Lp (a,M,k)) & a in ( Lp_Functions (M,k));

          

           A15: a in A1 by A14, Th38;

          reconsider a1 = a as VECTOR of ( RLSp_LpFunct (M,k)) by A14;

          

           A16: (( - 1) (#) a) in ( Lp_Functions (M,k)) by A14, Th26;

          set A2 = ( a.e-eq-class_Lp ((( - 1) (#) a),M,k));

          A2 in C by A16;

          then

          reconsider A2 as Element of A by A1;

          take A2;

          

           A17: (( - 1) (#) a) in A2 by Th38, A14, Th26;

          consider v,g be PartFunc of X, REAL such that

           A18: v in ( Lp_Functions (M,k)) & g in ( Lp_Functions (M,k)) & v = (a1 + (( - 1) * a1)) & g = (X --> 0 ) & v a.e.= (g,M) by Th31;

          (( - 1) (#) a) = (( - 1) * a1) by Th30;

          then (a + (( - 1) (#) a)) a.e.= (g,M) by Th29, A18;

          then ( 0. A) = ( a.e-eq-class_Lp ((a + (( - 1) (#) a)),M,k)) by Th42, A18, A1;

          hence (A1 + A2) = ( 0. A) by A15, A17, Def8, A1;

        end;

        now

          let x0,y0 be Real, A1,A2 be Element of A;

          reconsider x = x0, y = y0 as Real;

          A1 in C by A1;

          then

          consider a be PartFunc of X, REAL such that

           A19: A1 = ( a.e-eq-class_Lp (a,M,k)) & a in ( Lp_Functions (M,k));

          A2 in C by A1;

          then

          consider b be PartFunc of X, REAL such that

           A20: A2 = ( a.e-eq-class_Lp (b,M,k)) & b in ( Lp_Functions (M,k));

          

           A21: a in A1 & b in A2 by A19, A20, Th38;

          then (aC . (A1,A2)) = ( a.e-eq-class_Lp ((a + b),M,k)) by A1, Def8;

          then

           A22: (a + b) in (A1 + A2) by Th38, Th25, A19, A20, A1;

          reconsider a1 = a, b1 = b as VECTOR of ( RLSp_LpFunct (M,k)) by A19, A20;

          

           A23: (y (#) a) = (y * a1) & (x (#) a) = (x * a1) & (x (#) b) = (x * b1) & ((x + y) (#) a) = ((x + y) * a1) & (1 (#) a) = (1 * a1) by Th30;

          (a + b) = (a1 + b1) by Th29;

          then (x (#) (a + b)) = (x * (a1 + b1)) by Th30;

          then (x (#) (a + b)) = ((x * a1) + (x * b1)) by RLVECT_1:def 5;

          then

           A24: (x (#) (a + b)) = ((x (#) a) + (x (#) b)) by A23, Th29;

          ((x + y) (#) a) = ((x * a1) + (y * a1)) by A23, RLVECT_1:def 6;

          then

           A25: ((x + y) (#) a) = ((x (#) a) + (y (#) a)) by A23, Th29;

          (x (#) (y (#) a)) = (x * (y * a1)) by A23, Th30

          .= ((x * y) * a1) by RLVECT_1:def 7;

          then

           A26: (x (#) (y (#) a)) = ((x * y) (#) a) by Th30;

          (lC . (x,A1)) = ( a.e-eq-class_Lp ((x (#) a),M,k)) & (lC . (x,A2)) = ( a.e-eq-class_Lp ((x (#) b),M,k)) & (lC . (y,A1)) = ( a.e-eq-class_Lp ((y (#) a),M,k)) by A1, A21, Def10;

          then

           A27: (x (#) a) in (x * A1) & (x (#) b) in (x * A2) & (y (#) a) in (y * A1) by A1, Th38, Th26, A19, A20;

          (x * (A1 + A2)) = ( a.e-eq-class_Lp (((x (#) a) + (x (#) b)),M,k)) by A1, A24, A22, Def10;

          hence (x0 * (A1 + A2)) = ((x0 * A1) + (x0 * A2)) by A1, A27, Def8;

          ((x + y) * A1) = ( a.e-eq-class_Lp (((x (#) a) + (y (#) a)),M,k)) by A1, A25, A21, Def10;

          hence ((x0 + y0) * A1) = ((x0 * A1) + (y0 * A1)) by A27, Def8, A1;

          ((x0 * y0) * A1) = ( a.e-eq-class_Lp ((x (#) (y (#) a)),M,k)) by A1, A26, A21, Def10;

          hence ((x0 * y0) * A1) = (x0 * (y0 * A1)) by A27, Def10, A1;

          (1 (#) a) = a by A23, RLVECT_1:def 8;

          hence (1 * A1) = A1 by A19, A21, Def10, A1;

        end;

        hence thesis;

      end;

    end

    begin

    theorem :: LPSPACE2:48

    

     Th48: f in ( Lp_Functions (M,k)) & g in ( Lp_Functions (M,k)) & f a.e.= (g,M) implies ( Integral (M,(( abs f) to_power k))) = ( Integral (M,(( abs g) to_power k)))

    proof

      set t = (( abs f) to_power k);

      set s = (( abs g) to_power k);

      assume

       A1: f in ( Lp_Functions (M,k)) & g in ( Lp_Functions (M,k)) & f a.e.= (g,M);

      then ex f1 be PartFunc of X, REAL st f = f1 & ex E be Element of S st (M . (E ` )) = 0 & ( dom f1) = E & f1 is E -measurable & (( abs f1) to_power k) is_integrable_on M;

      then

      consider Df be Element of S such that

       A2: (M . (Df ` )) = 0 & ( dom f) = Df & f is Df -measurable & t is_integrable_on M;

      ex g1 be PartFunc of X, REAL st g = g1 & ex E be Element of S st (M . (E ` )) = 0 & ( dom g1) = E & g1 is E -measurable & (( abs g1) to_power k) is_integrable_on M by A1;

      then

      consider Dg be Element of S such that

       A3: (M . (Dg ` )) = 0 & ( dom g) = Dg & g is Dg -measurable & s is_integrable_on M;

      

       A4: ( dom ( abs f)) = ( dom f) & ( dom ( abs g)) = ( dom g) by VALUED_1:def 11;

      consider E1 be Element of S such that

       A5: (M . E1) = 0 & (f | (E1 ` )) = (g | (E1 ` )) by A1;

      reconsider NDf = (Df ` ), NDg = (Dg ` ) as Element of S by MEASURE1: 34;

      set Ef = (Df \ (NDg \/ E1));

      set Eg = (Dg \ (NDf \/ E1));

      set E2 = ((NDf \/ NDg) \/ E1);

      NDf is measure_zero of M & NDg is measure_zero of M & E1 is measure_zero of M by A2, A3, A5, MEASURE1:def 7;

      then (NDf \/ E1) is measure_zero of M & (NDg \/ E1) is measure_zero of M by MEASURE1: 37;

      then

       A6: (M . (NDf \/ E1)) = 0 & (M . (NDg \/ E1)) = 0 by MEASURE1:def 7;

      (X \ NDf) = (X /\ Df) & (X \ NDg) = (X /\ Dg) by XBOOLE_1: 48;

      then

       A7: (X \ NDf) = Df & (X \ NDg) = Dg by XBOOLE_1: 28;

      Ef = ((Df \ NDg) \ E1) & Eg = ((Dg \ NDf) \ E1) by XBOOLE_1: 41;

      then

       A8: Ef = ((X \ (NDf \/ NDg)) \ E1) & Eg = ((X \ (NDf \/ NDg)) \ E1) by A7, XBOOLE_1: 41;

      then

       A9: Ef = (X \ E2) & Eg = (X \ E2) by XBOOLE_1: 41;

      ( abs f) is Df -measurable & ( abs g) is Dg -measurable by A2, A3, MESFUNC6: 48;

      then

       A10: t is Df -measurable & s is Dg -measurable by A2, A3, A4, MESFUN6C: 29;

      

       A11: ( dom t) = Df & ( dom s) = Dg by A2, A3, A4, MESFUN6C:def 4;

      then

       A12: ( Integral (M,(t | Ef))) = ( Integral (M,t)) & ( Integral (M,(s | Eg))) = ( Integral (M,s)) by A6, A10, MESFUNC6: 89;

      ( dom (t | Ef)) = (( dom t) /\ Ef) & ( dom (s | Ef)) = (( dom s) /\ Ef) by RELAT_1: 61;

      then

       A13: ( dom (t | Ef)) = ((Df /\ Df) \ (NDg \/ E1)) & ( dom (s | Ef)) = ((Dg /\ Dg) \ (NDf \/ E1)) by A11, A8, XBOOLE_1: 49;

      now

        let x be Element of X;

        assume

         A14: x in ( dom (t | Ef));

        

         A15: ( dom (t | Ef)) c= ( dom t) & ( dom (s | Ef)) c= ( dom s) by RELAT_1: 60;

        (E2 ` ) c= (E1 ` ) by XBOOLE_1: 7, XBOOLE_1: 34;

        then

         A16: (f . x) = ((f | (E1 ` )) . x) & (g . x) = ((g | (E1 ` )) . x) by A14, A13, A9, FUNCT_1: 49;

        ((t | Ef) . x) = (t . x) & ((s | Ef) . x) = (s . x) by A14, A13, FUNCT_1: 49;

        then ((t | Ef) . x) = ((( abs f) . x) to_power k) & ((s | Ef) . x) = ((( abs g) . x) to_power k) by A8, A13, A14, A15, MESFUN6C:def 4;

        then ((t | Ef) . x) = ( |.(f . x) qua Complex.| to_power k) & ((s | Ef) . x) = ( |.(g . x) qua Complex.| to_power k) by VALUED_1: 18;

        hence ((t | Ef) . x) = ((s | Ef) . x) by A5, A16;

      end;

      hence thesis by A12, A13, A8, PARTFUN1: 5;

    end;

    theorem :: LPSPACE2:49

    

     Th49: f in ( Lp_Functions (M,k)) implies ( Integral (M,(( abs f) to_power k))) in REAL & 0 <= ( Integral (M,(( abs f) to_power k)))

    proof

      assume f in ( Lp_Functions (M,k));

      then

       A1: ex f1 be PartFunc of X, REAL st f = f1 & ex ND be Element of S st (M . (ND ` )) = 0 & ( dom f1) = ND & f1 is ND -measurable & (( abs f1) to_power k) is_integrable_on M;

      then -infty < ( Integral (M,(( abs f) to_power k))) & ( Integral (M,(( abs f) to_power k))) < +infty by MESFUNC6: 90;

      hence ( Integral (M,(( abs f) to_power k))) in REAL by XXREAL_0: 14;

      ( R_EAL (( abs f) to_power k)) is_integrable_on M by A1;

      then

      consider A be Element of S such that

       A2: A = ( dom ( R_EAL (( abs f) to_power k))) & ( R_EAL (( abs f) to_power k)) is A -measurable;

      A = ( dom (( abs f) to_power k)) & (( abs f) to_power k) is A -measurable by A2;

      hence thesis by MESFUNC6: 84;

    end;

    theorem :: LPSPACE2:50

    

     Th50: (ex x be VECTOR of ( Pre-Lp-Space (M,k)) st f in x & g in x) implies f a.e.= (g,M) & f in ( Lp_Functions (M,k)) & g in ( Lp_Functions (M,k))

    proof

      assume ex x be VECTOR of ( Pre-Lp-Space (M,k)) st f in x & g in x;

      then

      consider x be VECTOR of ( Pre-Lp-Space (M,k)) such that

       A1: f in x & g in x;

      x in the carrier of ( Pre-Lp-Space (M,k));

      then x in ( CosetSet (M,k)) by Def11;

      then

      consider h be PartFunc of X, REAL such that

       A2: x = ( a.e-eq-class_Lp (h,M,k)) & h in ( Lp_Functions (M,k));

      (ex i be PartFunc of X, REAL st f = i & i in ( Lp_Functions (M,k)) & h a.e.= (i,M)) & (ex j be PartFunc of X, REAL st g = j & j in ( Lp_Functions (M,k)) & h a.e.= (j,M)) by A1, A2;

      then f a.e.= (h,M) & h a.e.= (g,M);

      hence thesis by A1, A2, LPSPACE1: 30;

    end;

    reserve x for Point of ( Pre-Lp-Space (M,k));

    theorem :: LPSPACE2:51

    

     Th51: f in x implies (( abs f) to_power k) is_integrable_on M & f in ( Lp_Functions (M,k))

    proof

      assume

       A1: f in x;

      x in the carrier of ( Pre-Lp-Space (M,k));

      then x in ( CosetSet (M,k)) by Def11;

      then

      consider h be PartFunc of X, REAL such that

       A2: x = ( a.e-eq-class_Lp (h,M,k)) & h in ( Lp_Functions (M,k));

      ex g be PartFunc of X, REAL st f = g & g in ( Lp_Functions (M,k)) & h a.e.= (g,M) by A1, A2;

      then ex f0 be PartFunc of X, REAL st f = f0 & ex ND be Element of S st (M . (ND ` )) = 0 & ( dom f0) = ND & f0 is ND -measurable & (( abs f0) to_power k) is_integrable_on M;

      hence thesis;

    end;

    theorem :: LPSPACE2:52

    

     Th52: f in x & g in x implies f a.e.= (g,M) & ( Integral (M,(( abs f) to_power k))) = ( Integral (M,(( abs g) to_power k)))

    proof

      assume f in x & g in x;

      then f a.e.= (g,M) & f in ( Lp_Functions (M,k)) & g in ( Lp_Functions (M,k)) by Th50;

      hence thesis by Th48;

    end;

    definition

      let X be non empty set, S be SigmaField of X, M be sigma_Measure of S, k be positive Real;

      :: LPSPACE2:def12

      func Lp-Norm (M,k) -> Function of the carrier of ( Pre-Lp-Space (M,k)), REAL means

      : Def12: for x be Point of ( Pre-Lp-Space (M,k)) holds ex f be PartFunc of X, REAL st f in x & ex r be Real st r = ( Integral (M,(( abs f) to_power k))) & (it . x) = (r to_power (1 / k));

      existence

      proof

        defpred P[ set, set] means ex f be PartFunc of X, REAL st f in $1 & ex r be Real st r = ( Integral (M,(( abs f) to_power k))) & $2 = (r to_power (1 / k));

        

         A1: for x be Point of ( Pre-Lp-Space (M,k)) holds ex y be Element of REAL st P[x, y]

        proof

          let x be Point of ( Pre-Lp-Space (M,k));

          x in the carrier of ( Pre-Lp-Space (M,k));

          then x in ( CosetSet (M,k)) by Def11;

          then

          consider f be PartFunc of X, REAL such that

           A2: x = ( a.e-eq-class_Lp (f,M,k)) & f in ( Lp_Functions (M,k));

          reconsider r1 = ( Integral (M,(( abs f) to_power k))) as Element of REAL by A2, Th49;

          (r1 to_power (1 / k)) in REAL by XREAL_0:def 1;

          hence thesis by A2, Th38;

        end;

        consider F be Function of the carrier of ( Pre-Lp-Space (M,k)), REAL such that

         A3: for x be Point of ( Pre-Lp-Space (M,k)) holds P[x, (F . x)] from FUNCT_2:sch 3( A1);

        take F;

        thus thesis by A3;

      end;

      uniqueness

      proof

        let N1,N2 be Function of the carrier of ( Pre-Lp-Space (M,k)), REAL ;

        assume

         A4: (for x be Point of ( Pre-Lp-Space (M,k)) holds ex f be PartFunc of X, REAL st f in x & ex r1 be Real st r1 = ( Integral (M,(( abs f) to_power k))) & (N1 . x) = (r1 to_power (1 / k))) & (for x be Point of ( Pre-Lp-Space (M,k)) holds ex g be PartFunc of X, REAL st g in x & ex r2 be Real st r2 = ( Integral (M,(( abs g) to_power k))) & (N2 . x) = (r2 to_power (1 / k)));

        now

          let x be Point of ( Pre-Lp-Space (M,k));

          (ex f be PartFunc of X, REAL st f in x & ex r1 be Real st r1 = ( Integral (M,(( abs f) to_power k))) & (N1 . x) = (r1 to_power (1 / k))) & (ex g be PartFunc of X, REAL st g in x & ex r2 be Real st r2 = ( Integral (M,(( abs g) to_power k))) & (N2 . x) = (r2 to_power (1 / k))) by A4;

          hence (N1 . x) = (N2 . x) by Th52;

        end;

        hence N1 = N2 by FUNCT_2: 63;

      end;

    end

    definition

      let X be non empty set, S be SigmaField of X, M be sigma_Measure of S, k be positive Real;

      :: LPSPACE2:def13

      func Lp-Space (M,k) -> non empty NORMSTR equals NORMSTR (# the carrier of ( Pre-Lp-Space (M,k)), the ZeroF of ( Pre-Lp-Space (M,k)), the addF of ( Pre-Lp-Space (M,k)), the Mult of ( Pre-Lp-Space (M,k)), ( Lp-Norm (M,k)) #);

      coherence ;

    end

    reserve x,y for Point of ( Lp-Space (M,k));

    theorem :: LPSPACE2:53

    

     Th53: (ex f be PartFunc of X, REAL st f in ( Lp_Functions (M,k)) & x = ( a.e-eq-class_Lp (f,M,k))) & for f be PartFunc of X, REAL st f in x holds ex r be Real st 0 <= r & r = ( Integral (M,(( abs f) to_power k))) & ||.x.|| = (r to_power (1 / k))

    proof

      x in the carrier of ( Pre-Lp-Space (M,k));

      then x in ( CosetSet (M,k)) by Def11;

      then ex g be PartFunc of X, REAL st x = ( a.e-eq-class_Lp (g,M,k)) & g in ( Lp_Functions (M,k));

      hence ex f be PartFunc of X, REAL st f in ( Lp_Functions (M,k)) & x = ( a.e-eq-class_Lp (f,M,k));

      consider f be PartFunc of X, REAL such that

       A1: f in x & ex r be Real st r = ( Integral (M,(( abs f) to_power k))) & (( Lp-Norm (M,k)) . x) = (r to_power (1 / k)) by Def12;

      hereby

        let g be PartFunc of X, REAL ;

        assume

         A2: g in x;

        then

         A3: g in ( Lp_Functions (M,k)) by Th50;

        ( Integral (M,(( abs g) to_power k))) = ( Integral (M,(( abs f) to_power k))) by A1, Th52, A2;

        hence ex r be Real st 0 <= r & r = ( Integral (M,(( abs g) to_power k))) & ||.x.|| = (r to_power (1 / k)) by A1, A3, Th49;

      end;

    end;

    theorem :: LPSPACE2:54

    

     Th54: (f in x & g in y implies (f + g) in (x + y)) & (f in x implies (a (#) f) in (a * x))

    proof

      set C = ( CosetSet (M,k));

      hereby

        assume

         A1: f in x & g in y;

        x in the carrier of ( Pre-Lp-Space (M,k));

        then

         A2: x in C by Def11;

        then

        consider a be PartFunc of X, REAL such that

         A3: x = ( a.e-eq-class_Lp (a,M,k)) & a in ( Lp_Functions (M,k));

        

         A4: a in x by A3, Th38;

        y in the carrier of ( Pre-Lp-Space (M,k));

        then

         A5: y in C by Def11;

        then

        consider b be PartFunc of X, REAL such that

         A6: y = ( a.e-eq-class_Lp (b,M,k)) & b in ( Lp_Functions (M,k));

        b in y by A6, Th38;

        then (( addCoset (M,k)) . (x,y)) = ( a.e-eq-class_Lp ((a + b),M,k)) by A2, A5, A4, Def8;

        then

         A7: (x + y) = ( a.e-eq-class_Lp ((a + b),M,k)) by Def11;

        ex r be PartFunc of X, REAL st f = r & r in ( Lp_Functions (M,k)) & a a.e.= (r,M) by A1, A3;

        then

         A8: ( a.e-eq-class_Lp (a,M,k)) = ( a.e-eq-class_Lp (f,M,k)) by Th42;

        ex r be PartFunc of X, REAL st g = r & r in ( Lp_Functions (M,k)) & b a.e.= (r,M) by A1, A6;

        then ( a.e-eq-class_Lp (b,M,k)) = ( a.e-eq-class_Lp (g,M,k)) by Th42;

        then ( a.e-eq-class_Lp ((a + b),M,k)) = ( a.e-eq-class_Lp ((f + g),M,k)) by A1, A3, A6, A8, Th45;

        hence (f + g) in (x + y) by Th38, A7, Th25, A3, A1, A6;

      end;

      hereby

        assume

         A9: f in x;

        x in the carrier of ( Pre-Lp-Space (M,k));

        then

         A10: x in C by Def11;

        then

        consider f1 be PartFunc of X, REAL such that

         A11: x = ( a.e-eq-class_Lp (f1,M,k)) & f1 in ( Lp_Functions (M,k));

        f1 in x by A11, Th38;

        then (( lmultCoset (M,k)) . (a,x)) = ( a.e-eq-class_Lp ((a (#) f1),M,k)) by A10, Def10;

        then

         A12: (a * x) = ( a.e-eq-class_Lp ((a (#) f1),M,k)) by Def11;

        ex r be PartFunc of X, REAL st f = r & r in ( Lp_Functions (M,k)) & f1 a.e.= (r,M) by A9, A11;

        then ( a.e-eq-class_Lp (f1,M,k)) = ( a.e-eq-class_Lp (f,M,k)) by Th42;

        then ( a.e-eq-class_Lp ((a (#) f1),M,k)) = ( a.e-eq-class_Lp ((a (#) f),M,k)) by A11, A9, Th47;

        hence (a (#) f) in (a * x) by A12, Th26, A9, A11, Th38;

      end;

    end;

    theorem :: LPSPACE2:55

    

     Th55: f in x implies x = ( a.e-eq-class_Lp (f,M,k)) & (ex r be Real st 0 <= r & r = ( Integral (M,(( abs f) to_power k))) & ||.x.|| = (r to_power (1 / k)))

    proof

      assume

       A1: f in x;

      x in the carrier of ( Pre-Lp-Space (M,k));

      then x in ( CosetSet (M,k)) by Def11;

      then

      consider g be PartFunc of X, REAL such that

       A2: x = ( a.e-eq-class_Lp (g,M,k)) & g in ( Lp_Functions (M,k));

      g in x by A2, Th38;

      then f a.e.= (g,M) & f in ( Lp_Functions (M,k)) & g in ( Lp_Functions (M,k)) by A1, Th50;

      hence thesis by Th53, A1, A2, Th42;

    end;

    theorem :: LPSPACE2:56

    

     Th56: (X --> 0 ) in ( L1_Functions M)

    proof

      reconsider ND = {} as Element of S by MEASURE1: 34;

      

       A1: (M . ND) = 0 by VALUED_0:def 19;

      (X --> ( In ( 0 , REAL ))) is Function of X, REAL by FUNCOP_1: 46;

      then

       A2: ( dom (X --> 0 )) = (ND ` ) by FUNCT_2:def 1;

      for x be Element of X st x in ( dom (X --> 0 )) holds ((X --> 0 ) . x) = 0 by FUNCOP_1: 7;

      then (X --> 0 ) is_integrable_on M by A2, Th15;

      hence thesis by A1, A2;

    end;

    theorem :: LPSPACE2:57

    

     Th57: f in ( Lp_Functions (M,k)) & ( Integral (M,(( abs f) to_power k))) = 0 implies f a.e.= ((X --> 0 ),M)

    proof

      assume that

       A1: f in ( Lp_Functions (M,k)) and

       A2: ( Integral (M,(( abs f) to_power k))) = 0 ;

      ex h be PartFunc of X, REAL st f = h & ex ND be Element of S st (M . (ND ` )) = 0 & ( dom h) = ND & h is ND -measurable & (( abs h) to_power k) is_integrable_on M by A1;

      then

      consider NDf be Element of S such that

       A3: (M . (NDf ` )) = 0 & ( dom f) = NDf & f is NDf -measurable & (( abs f) to_power k) is_integrable_on M;

      reconsider t = (( abs f) to_power k) as PartFunc of X, REAL ;

      reconsider ND = (NDf ` ) as Element of S by MEASURE1: 34;

      

       A4: ( dom t) = ( dom ( abs f)) by MESFUN6C:def 4;

      then

       A5: ( dom t) = NDf by A3, VALUED_1:def 11;

      ( dom t) = (ND ` ) by A4, A3, VALUED_1:def 11;

      then

       A6: t in ( L1_Functions M) by A3;

      ( abs t) = t by Th14;

      then t a.e.= ((X --> 0 ),M) by A2, A6, LPSPACE1: 53;

      then

      consider ND1 be Element of S such that

       A7: (M . ND1) = 0 & ((( abs f) to_power k) | (ND1 ` )) = ((X --> 0 ) | (ND1 ` ));

      set ND2 = (ND \/ ND1);

      ND is measure_zero of M & ND1 is measure_zero of M by A3, A7, MEASURE1:def 7;

      then ND2 is measure_zero of M by MEASURE1: 37;

      then

       A8: (M . ND2) = 0 by MEASURE1:def 7;

      

       A9: (ND2 ` ) c= (ND ` ) & (ND2 ` ) c= (ND1 ` ) by XBOOLE_1: 7, XBOOLE_1: 34;

      ( dom (X --> 0 )) = X by FUNCOP_1: 13;

      then

       A10: ( dom ((X --> 0 ) | (ND2 ` ))) = (ND2 ` ) by RELAT_1: 62;

      

       A11: ( dom (f | (ND2 ` ))) = (ND2 ` ) by A3, A9, RELAT_1: 62;

      for x be object st x in ( dom (f | (ND2 ` ))) holds ((f | (ND2 ` )) . x) = (((X --> 0 ) | (ND2 ` )) . x)

      proof

        let x be object;

        assume

         A12: x in ( dom (f | (ND2 ` )));

         A13:

        now

          assume (f . x) <> 0 ;

          then |.(f . x).| > 0 by COMPLEX1: 47;

          then ( |.(f . x) qua Complex.| to_power k) <> 0 by POWER: 34;

          then ((( abs f) . x) to_power k) <> 0 by VALUED_1: 18;

          then

           A14: ((( abs f) to_power k) . x) <> 0 by A5, A9, A12, A11, MESFUN6C:def 4;

          (((X --> 0 ) | (ND1 ` )) . x) = ((X --> 0 ) . x) by A9, A12, A11, FUNCT_1: 49;

          then (((X --> 0 ) | (ND1 ` )) . x) = 0 by A12, FUNCOP_1: 7;

          hence contradiction by A14, A7, A9, A12, A11, FUNCT_1: 49;

        end;

        (((X --> 0 ) | (ND2 ` )) . x) = ((X --> 0 ) . x) by A11, A12, FUNCT_1: 49;

        then (((X --> 0 ) | (ND2 ` )) . x) = 0 by A12, FUNCOP_1: 7;

        hence thesis by A11, A12, A13, FUNCT_1: 49;

      end;

      then (f | (ND2 ` )) = ((X --> 0 ) | (ND2 ` )) by A10, A11, FUNCT_1:def 11;

      hence thesis by A8;

    end;

    theorem :: LPSPACE2:58

    

     Th58: ( Integral (M,(( abs (X --> 0 )) to_power k))) = 0

    proof

      

       A1: for x be object st x in ( dom (X --> 0 )) holds 0 <= ((X --> 0 ) . x);

      

      then ( Integral (M,(( abs (X --> 0 )) to_power k))) = ( Integral (M,((X --> 0 ) to_power k))) by Th14, MESFUNC6: 52

      .= ( Integral (M,(X --> 0 ))) by Th12

      .= ( Integral (M,( abs (X --> 0 )))) by A1, Th14, MESFUNC6: 52;

      hence thesis by LPSPACE1: 54;

    end;

    theorem :: LPSPACE2:59

    

     Th59: for m,n be positive Real st ((1 / m) + (1 / n)) = 1 & f in ( Lp_Functions (M,m)) & g in ( Lp_Functions (M,n)) holds (f (#) g) in ( L1_Functions M) & (f (#) g) is_integrable_on M

    proof

      let m,n be positive Real;

      assume that

       A1: ((1 / m) + (1 / n)) = 1 and

       A2: f in ( Lp_Functions (M,m)) & g in ( Lp_Functions (M,n));

      

       A3: m > 1 & n > 1 by A1, Th1;

      consider f1 be PartFunc of X, REAL such that

       A4: f = f1 & ex NDf be Element of S st (M . (NDf ` )) = 0 & ( dom f1) = NDf & f1 is NDf -measurable & (( abs f1) to_power m) is_integrable_on M by A2;

      consider EDf be Element of S such that

       A5: (M . (EDf ` )) = 0 & ( dom f1) = EDf & f1 is EDf -measurable by A4;

      consider g1 be PartFunc of X, REAL such that

       A6: g = g1 & ex NDg be Element of S st (M . (NDg ` )) = 0 & ( dom g1) = NDg & g1 is NDg -measurable & (( abs g1) to_power n) is_integrable_on M by A2;

      consider EDg be Element of S such that

       A7: (M . (EDg ` )) = 0 & ( dom g1) = EDg & g1 is EDg -measurable by A6;

      set u = (( abs f1) to_power m);

      set v = (( abs g1) to_power n);

      set w = (f1 (#) g1);

      set z = (((1 / m) (#) u) + ((1 / n) (#) v));

      

       A8: ( dom f1) = ( dom ( abs f1)) & ( dom g1) = ( dom ( abs g1)) by VALUED_1:def 11;

      then

       A9: ( dom u) = ( dom f1) & ( dom v) = ( dom g1) by MESFUN6C:def 4;

      then

       A10: ( dom w) = (( dom u) /\ ( dom v)) by VALUED_1:def 4;

      set Nf = (EDf ` );

      set Ng = (EDg ` );

      set E = (EDf /\ EDg);

      reconsider Nf, Ng as Element of S by MEASURE1: 34;

      ( dom u) = (Nf ` ) & ( dom v) = (Ng ` ) by A5, A7, A8, MESFUN6C:def 4;

      then u in ( L1_Functions M) & v in ( L1_Functions M) by A4, A5, A6, A7;

      then ((1 / m) (#) u) in ( L1_Functions M) & ((1 / n) (#) v) in ( L1_Functions M) by LPSPACE1: 24;

      then z in ( L1_Functions M) by LPSPACE1: 23;

      then

       A11: ex h be PartFunc of X, REAL st z = h & ex ND be Element of S st (M . ND) = 0 & ( dom h) = (ND ` ) & h is_integrable_on M;

      ( dom ((1 / m) (#) u)) = ( dom u) & ( dom ((1 / n) (#) v)) = ( dom v) by VALUED_1:def 5;

      then

       A12: ( dom z) = (( dom u) /\ ( dom v)) by VALUED_1:def 1;

      

       A13: (E ` ) = ((EDf ` ) \/ (EDg ` )) by XBOOLE_1: 54;

      Nf is measure_zero of M & Ng is measure_zero of M by A5, A7, MEASURE1:def 7;

      then (Nf \/ Ng) is measure_zero of M by MEASURE1: 37;

      then

       A14: (M . (E ` )) = 0 by A13, MEASURE1:def 7;

      f1 is E -measurable & g1 is E -measurable by A5, A7, MESFUNC6: 16, XBOOLE_1: 17;

      then

       A15: w is E -measurable by A5, A7, MESFUN7C: 31;

      for x be Element of X st x in ( dom w) holds |.(w . x) qua Complex.| <= (z . x)

      proof

        let x be Element of X;

        assume

         A16: x in ( dom w);

        ( abs (f1 (#) g1)) = (( abs f1) (#) ( abs g1)) by RFUNCT_1: 24;

        then (( abs (f1 (#) g1)) . x) = ((( abs f1) . x) * (( abs g1) . x)) by VALUED_1: 5;

        then

         A17: |.((f1 (#) g1) . x).| = ((( abs f1) . x) * (( abs g1) . x)) by VALUED_1: 18;

        

         A18: (( abs f1) . x) >= 0 & (( abs g1) . x) >= 0 by MESFUNC6: 51;

        x in ( dom u) & x in ( dom v) by A16, A10, XBOOLE_0:def 4;

        then (((( abs f1) . x) to_power m) / m) = ((1 / m) * ((( abs f1) to_power m) . x)) & (((( abs g1) . x) to_power n) / n) = ((1 / n) * ((( abs g1) to_power n) . x)) by MESFUN6C:def 4;

        then (((( abs f1) . x) to_power m) / m) = (((1 / m) (#) (( abs f1) to_power m)) . x) & (((( abs g1) . x) to_power n) / n) = (((1 / n) (#) (( abs g1) to_power n)) . x) by VALUED_1: 6;

        then |.(w . x).| <= ((((1 / m) (#) u) . x) + (((1 / n) (#) v) . x)) by A1, A3, A17, A18, HOLDER_1: 5;

        hence thesis by A16, A10, A12, VALUED_1:def 1;

      end;

      then

       A19: w is_integrable_on M by A5, A7, A9, A10, A11, A15, A12, MESFUNC6: 96;

      set ND = (E ` );

      reconsider ND as Element of S by MEASURE1: 34;

      ( dom w) = (ND ` ) by A5, A7, VALUED_1:def 4;

      hence thesis by A4, A6, A14, A19;

    end;

    theorem :: LPSPACE2:60

    

     Th60: for m,n be positive Real st ((1 / m) + (1 / n)) = 1 & f in ( Lp_Functions (M,m)) & g in ( Lp_Functions (M,n)) holds ex r1 be Real st r1 = ( Integral (M,(( abs f) to_power m))) & ex r2 be Real st r2 = ( Integral (M,(( abs g) to_power n))) & ( Integral (M,( abs (f (#) g)))) <= ((r1 to_power (1 / m)) * (r2 to_power (1 / n)))

    proof

      let m,n be positive Real;

      assume

       A1: ((1 / m) + (1 / n)) = 1 & f in ( Lp_Functions (M,m)) & g in ( Lp_Functions (M,n));

      then

       A2: m > 1 & n > 1 by Th1;

      consider f1 be PartFunc of X, REAL such that

       A3: f = f1 & ex NDf be Element of S st (M . (NDf ` )) = 0 & ( dom f1) = NDf & f1 is NDf -measurable & (( abs f1) to_power m) is_integrable_on M by A1;

      consider EDf be Element of S such that

       A4: (M . (EDf ` )) = 0 & ( dom f1) = EDf & f1 is EDf -measurable by A3;

      consider g1 be PartFunc of X, REAL such that

       A5: g = g1 & ex NDg be Element of S st (M . (NDg ` )) = 0 & ( dom g1) = NDg & g1 is NDg -measurable & (( abs g1) to_power n) is_integrable_on M by A1;

      consider EDg be Element of S such that

       A6: (M . (EDg ` )) = 0 & ( dom g1) = EDg & g1 is EDg -measurable by A5;

      set u = (( abs f1) to_power m);

      set v = (( abs g1) to_power n);

      

       A7: 0 <= ( Integral (M,u)) & 0 <= ( Integral (M,v)) by A3, A5, A1, Th49;

      reconsider s1 = ( Integral (M,u)), s2 = ( Integral (M,v)) as Element of REAL by A3, A5, A1, Th49;

      

       A8: ( dom f1) = ( dom ( abs f1)) & ( dom g1) = ( dom ( abs g1)) by VALUED_1:def 11;

      reconsider Nf = (EDf ` ), Ng = (EDg ` ) as Element of S by MEASURE1: 34;

      set t1 = (s1 to_power (1 / m));

      set t2 = (s2 to_power (1 / n));

      set E = (EDf /\ EDg);

      

       A9: (E ` ) = ((EDf ` ) \/ (EDg ` )) by XBOOLE_1: 54;

      Nf is measure_zero of M & Ng is measure_zero of M by A4, A6, MEASURE1:def 7;

      then

       A10: (E ` ) is measure_zero of M by A9, MEASURE1: 37;

      

       A11: ( dom (f1 (#) g1)) = (EDf /\ EDg) by A4, A6, VALUED_1:def 4;

      f1 is E -measurable & g1 is E -measurable by A4, A6, MESFUNC6: 16, XBOOLE_1: 17;

      then

       A12: (f1 (#) g1) is E -measurable by A4, A6, MESFUN7C: 31;

      

       A13: (f1 (#) g1) in ( L1_Functions M) by A1, A3, A5, Th59;

      then

       A14: ex fg1 be PartFunc of X, REAL st fg1 = (f1 (#) g1) & ex ND be Element of S st (M . ND) = 0 & ( dom fg1) = (ND ` ) & fg1 is_integrable_on M;

      then

       A15: ( Integral (M,( abs (f1 (#) g1)))) in REAL & ( abs (f1 (#) g1)) is_integrable_on M by LPSPACE1: 44;

      per cases by A3, A5, A1, Th49;

        suppose

         A16: s1 = 0 & s2 >= 0 ;

        f1 in ( Lp_Functions (M,m)) by A3;

        then f1 a.e.= ((X --> 0 ),M) by A16, Th57;

        then

        consider Nf1 be Element of S such that

         A17: (M . Nf1) = 0 & (f1 | (Nf1 ` )) = ((X --> 0 ) | (Nf1 ` ));

        reconsider Z = ((E \ Nf1) ` ) as Element of S by MEASURE1: 34;

        

         A18: ((E \ Nf1) ` ) = ((E ` ) \/ Nf1) by SUBSET_1: 14;

        Nf1 is measure_zero of M by A17, MEASURE1:def 7;

        then Z is measure_zero of M by A10, A18, MEASURE1: 37;

        then

         A19: (M . Z) = 0 by MEASURE1:def 7;

        ( dom (X --> 0 )) = X by FUNCOP_1: 13;

        then

         A20: ( dom ((X --> 0 ) | (Z ` ))) = (Z ` ) by RELAT_1: 62;

        

         A21: ( dom ((f1 (#) g1) | (Z ` ))) = (Z ` ) by A11, RELAT_1: 62, XBOOLE_1: 36;

        for x be object st x in ( dom ((f1 (#) g1) | (Z ` ))) holds (((f1 (#) g1) | (Z ` )) . x) = (((X --> 0 ) | (Z ` )) . x)

        proof

          let x be object;

          assume

           A22: x in ( dom ((f1 (#) g1) | (Z ` )));

          then x in X & not x in Nf1 by A21, XBOOLE_0:def 5;

          then x in (Nf1 ` ) by XBOOLE_0:def 5;

          then (f1 . x) = ((f1 | (Nf1 ` )) . x) & ((X --> 0 ) . x) = (((X --> 0 ) | (Nf1 ` )) . x) by FUNCT_1: 49;

          then

           A23: (f1 . x) = 0 by A17, A22, FUNCOP_1: 7;

          

           A24: ( dom ((f1 (#) g1) | (Z ` ))) c= ( dom (f1 (#) g1)) by RELAT_1: 60;

          (((f1 (#) g1) | (Z ` )) . x) = ((f1 (#) g1) . x) by A22, FUNCT_1: 47

          .= ((f1 . x) * (g1 . x)) by A22, A24, VALUED_1:def 4

          .= (((Z ` ) --> 0 ) . x) by A22, A21, A23, FUNCOP_1: 7

          .= (((X /\ (Z ` )) --> 0 ) . x) by XBOOLE_1: 28;

          hence thesis by FUNCOP_1: 12;

        end;

        then ((f1 (#) g1) | (Z ` )) = ((X --> 0 ) | (Z ` )) by A20, A21, FUNCT_1:def 11;

        then

         A25: (f1 (#) g1) a.e.= ((X --> 0 ),M) by A19;

        (X --> 0 ) in ( L1_Functions M) by Th56;

        then ( Integral (M,( abs (f1 (#) g1)))) = ( Integral (M,( abs (X --> 0 )))) by A13, A25, LPSPACE1: 45;

        then

         A26: ( Integral (M,( abs (f1 (#) g1)))) = 0 by LPSPACE1: 54;

        (t1 * t2) = ( 0 * t2) by A16, POWER:def 2;

        hence thesis by A3, A5, A26;

      end;

        suppose

         A27: s1 > 0 & s2 = 0 ;

        g1 in ( Lp_Functions (M,n)) by A5;

        then g1 a.e.= ((X --> 0 ),M) by A27, Th57;

        then

        consider Ng1 be Element of S such that

         A28: (M . Ng1) = 0 & (g1 | (Ng1 ` )) = ((X --> 0 ) | (Ng1 ` ));

        reconsider Z = ((E \ Ng1) ` ) as Element of S by MEASURE1: 34;

        

         A29: ((E \ Ng1) ` ) = ((E ` ) \/ Ng1) by SUBSET_1: 14;

        Ng1 is measure_zero of M by A28, MEASURE1:def 7;

        then Z is measure_zero of M by A10, A29, MEASURE1: 37;

        then

         A30: (M . Z) = 0 by MEASURE1:def 7;

        ( dom (X --> 0 )) = X by FUNCOP_1: 13;

        then

         A31: ( dom ((X --> 0 ) | (Z ` ))) = (Z ` ) by RELAT_1: 62;

        

         A32: ( dom ((f1 (#) g1) | (Z ` ))) = (Z ` ) by A11, RELAT_1: 62, XBOOLE_1: 36;

        for x be object st x in ( dom ((f1 (#) g1) | (Z ` ))) holds (((f1 (#) g1) | (Z ` )) . x) = (((X --> 0 ) | (Z ` )) . x)

        proof

          let x be object;

          assume

           A33: x in ( dom ((f1 (#) g1) | (Z ` )));

          then x in X & not x in Ng1 by A32, XBOOLE_0:def 5;

          then x in (Ng1 ` ) by XBOOLE_0:def 5;

          then (g1 . x) = ((g1 | (Ng1 ` )) . x) & ((X --> 0 ) . x) = (((X --> 0 ) | (Ng1 ` )) . x) by FUNCT_1: 49;

          then

           A34: (g1 . x) = 0 by A28, A33, FUNCOP_1: 7;

          

           A35: ( dom ((f1 (#) g1) | (Z ` ))) c= ( dom (f1 (#) g1)) by RELAT_1: 60;

          (((f1 (#) g1) | (Z ` )) . x) = ((f1 (#) g1) . x) by A33, FUNCT_1: 47

          .= ((f1 . x) * (g1 . x)) by A33, A35, VALUED_1:def 4

          .= (((Z ` ) --> 0 ) . x) by A33, A32, A34, FUNCOP_1: 7

          .= (((X /\ (Z ` )) --> 0 ) . x) by XBOOLE_1: 28;

          hence thesis by FUNCOP_1: 12;

        end;

        then ((f1 (#) g1) | (Z ` )) = ((X --> 0 ) | (Z ` )) by A31, A32, FUNCT_1:def 11;

        then

         A36: (f1 (#) g1) a.e.= ((X --> 0 ),M) by A30;

        (X --> 0 ) in ( L1_Functions M) by Th56;

        then ( Integral (M,( abs (f1 (#) g1)))) = ( Integral (M,( abs (X --> 0 )))) by A13, A36, LPSPACE1: 45;

        then

         A37: ( Integral (M,( abs (f1 (#) g1)))) = 0 by LPSPACE1: 54;

        (t1 * t2) = (t1 * 0 ) by A27, POWER:def 2;

        hence thesis by A3, A5, A37;

      end;

        suppose

         A38: s1 <> 0 & s2 <> 0 ;

        then

         A39: t1 > 0 & t2 > 0 by A7, POWER: 34;

        then

         A40: |.(1 / (t1 * t2)).| = (1 / (t1 * t2)) by ABSVALUE:def 1;

        set w = ((1 / (t1 * t2)) (#) (f1 (#) g1));

        set F = ((1 / m) (#) (((1 / t1) (#) ( abs f1)) to_power m));

        set G = ((1 / n) (#) (((1 / t2) (#) ( abs g1)) to_power n));

        set z = (F + G);

        

         A41: ( dom ((1 / t1) (#) ( abs f1))) = ( dom ( abs f1)) & ( dom ((1 / t2) (#) ( abs g1))) = ( dom ( abs g1)) by VALUED_1:def 5;

        ( dom F) = ( dom (((1 / t1) (#) ( abs f1)) to_power m)) & ( dom G) = ( dom (((1 / t2) (#) ( abs g1)) to_power n)) by VALUED_1:def 5;

        then

         A42: ( dom F) = ( dom ( abs f1)) & ( dom G) = ( dom ( abs g1)) by A41, MESFUN6C:def 4;

        then

         A43: ( dom z) = (( dom ( abs f1)) /\ ( dom ( abs g1))) by VALUED_1:def 1;

        (((1 / t1) (#) ( abs f1)) to_power m) = (((1 / t1) to_power m) (#) u) & (((1 / t2) (#) ( abs g1)) to_power n) = (((1 / t2) to_power n) (#) v) by A39, Th19;

        then

         A44: (((1 / t1) (#) ( abs f1)) to_power m) is_integrable_on M & (((1 / t2) (#) ( abs g1)) to_power n) is_integrable_on M by A3, A5, MESFUNC6: 102;

        then

         A45: F is_integrable_on M & G is_integrable_on M by MESFUNC6: 102;

        then

         A46: z is_integrable_on M by MESFUNC6: 100;

        

         A47: ( dom w) = ( dom (f1 (#) g1)) by VALUED_1:def 5;

        then

         A48: ( dom w) = (( dom f1) /\ ( dom g1)) by VALUED_1:def 4;

        ( dom ((1 / (t1 * t2)) (#) ( abs (f1 (#) g1)))) = ( dom ( abs (f1 (#) g1))) by VALUED_1:def 5;

        then

         A49: ( dom ((1 / (t1 * t2)) (#) ( abs (f1 (#) g1)))) = ( dom (f1 (#) g1)) by VALUED_1:def 11;

        

         A50: w is E -measurable by A11, A12, MESFUNC6: 21;

        for x be Element of X st x in ( dom w) holds |.(w . x) qua Complex.| <= (z . x)

        proof

          let x be Element of X;

          assume

           A51: x in ( dom w);

          (( abs f1) . x) >= 0 & (( abs g1) . x) >= 0 by MESFUNC6: 51;

          then

           A52: (((1 / t1) * (( abs f1) . x)) * ((1 / t2) * (( abs g1) . x))) <= (((((1 / t1) * (( abs f1) . x)) to_power m) / m) + ((((1 / t2) * (( abs g1) . x)) to_power n) / n)) by A1, A2, A39, HOLDER_1: 5;

          ( dom (( abs f1) (#) ( abs g1))) = (( dom ( abs f1)) /\ ( dom ( abs g1))) by VALUED_1:def 4;

          then

           A53: ((( abs f1) (#) ( abs g1)) . x) = ((( abs f1) . x) * (( abs g1) . x)) by A8, A48, A51, VALUED_1:def 4;

          

           A54: (((1 / t1) * (( abs f1) . x)) * ((1 / t2) * (( abs g1) . x))) = ((((1 / t1) * (1 / t2)) * (( abs f1) . x)) * (( abs g1) . x))

          .= (((1 / (t1 * t2)) * (( abs f1) . x)) * (( abs g1) . x)) by XCMPLX_1: 102

          .= ((1 / (t1 * t2)) * ((( abs f1) (#) ( abs g1)) . x)) by A53

          .= ((1 / (t1 * t2)) * (( abs (f1 (#) g1)) . x)) by RFUNCT_1: 24

          .= (((1 / (t1 * t2)) (#) ( abs (f1 (#) g1))) . x) by A47, A51, A49, VALUED_1:def 5

          .= (( abs w) . x) by A40, RFUNCT_1: 25;

          

           A55: ((1 / t1) * (( abs f1) . x)) = (((1 / t1) (#) ( abs f1)) . x) & ((1 / t2) * (( abs g1) . x)) = (((1 / t2) (#) ( abs g1)) . x) by VALUED_1: 6;

          ( dom (((1 / t1) (#) ( abs f1)) to_power m)) = ( dom f1) & ( dom (((1 / t2) (#) ( abs g1)) to_power n)) = ( dom g1) by A8, A41, MESFUN6C:def 4;

          then x in ( dom (((1 / t1) (#) ( abs f1)) to_power m)) & x in ( dom (((1 / t2) (#) ( abs g1)) to_power n)) by A48, A51, XBOOLE_0:def 4;

          then ((((1 / t1) (#) ( abs f1)) . x) to_power m) = ((((1 / t1) (#) ( abs f1)) to_power m) . x) & ((((1 / t2) (#) ( abs g1)) . x) to_power n) = ((((1 / t2) (#) ( abs g1)) to_power n) . x) by MESFUN6C:def 4;

          then (((((1 / t1) (#) ( abs f1)) . x) to_power m) / m) = (F . x) & (((((1 / t2) (#) ( abs g1)) . x) to_power n) / n) = (G . x) by VALUED_1: 6;

          then (((((1 / t1) * (( abs f1) . x)) to_power m) / m) + ((((1 / t2) * (( abs g1) . x)) to_power n) / n)) = (z . x) by A8, A48, A51, A43, A55, VALUED_1:def 1;

          hence thesis by A52, A54, VALUED_1: 18;

        end;

        then

         A56: ( Integral (M,( abs w))) <= ( Integral (M,z)) by A4, A6, A46, A8, A48, A43, A50, MESFUNC6: 96;

        consider E1 be Element of S such that

         A57: E1 = (( dom F) /\ ( dom G)) & ( Integral (M,(F + G))) = (( Integral (M,(F | E1))) + ( Integral (M,(G | E1)))) by A45, MESFUNC6: 101;

        EDf = (X /\ EDf) & EDg = (X /\ EDg) by XBOOLE_1: 28;

        then

         A58: EDf = (X \ Nf) & EDg = (X \ Ng) by XBOOLE_1: 48;

        

         A59: (EDf \ E) = (EDf \ EDg) by XBOOLE_1: 47

        .= (((X \ Nf) \ X) \/ ((X \ Nf) /\ Ng)) by A58, XBOOLE_1: 52

        .= ((X \ (Nf \/ X)) \/ ((X \ Nf) /\ Ng)) by XBOOLE_1: 41

        .= ((X \ X) \/ ((X \ Nf) /\ Ng)) by XBOOLE_1: 12

        .= ( {} \/ ((X \ Nf) /\ Ng)) by XBOOLE_1: 37;

        

         A60: (EDg \ E) = (EDg \ EDf) by XBOOLE_1: 47

        .= (((X \ Ng) \ X) \/ ((X \ Ng) /\ Nf)) by A58, XBOOLE_1: 52

        .= ((X \ (Ng \/ X)) \/ ((X \ Ng) /\ Nf)) by XBOOLE_1: 41

        .= ((X \ X) \/ ((X \ Ng) /\ Nf)) by XBOOLE_1: 12

        .= ( {} \/ ((X \ Ng) /\ Nf)) by XBOOLE_1: 37;

        set NF = (EDf /\ Ng);

        set NG = (EDg /\ Nf);

        Nf is measure_zero of M & Ng is measure_zero of M by A4, A6, MEASURE1:def 7;

        then NF is measure_zero of M & NG is measure_zero of M by MEASURE1: 36, XBOOLE_1: 17;

        then

         A61: (M . NF) = 0 & (M . NG) = 0 by MEASURE1:def 7;

        E = (EDf /\ E) & E = (EDg /\ E) by XBOOLE_1: 17, XBOOLE_1: 28;

        then

         A62: E = (EDf \ NF) & E = (EDg \ NG) by A58, A59, A60, XBOOLE_1: 48;

        ( R_EAL F) is_integrable_on M by A45;

        then ex E be Element of S st E = ( dom ( R_EAL F)) & ( R_EAL F) is E -measurable;

        then

         A63: F is EDf -measurable by A42, A8, A4;

        ( R_EAL G) is_integrable_on M by A45;

        then ex E be Element of S st E = ( dom ( R_EAL G)) & ( R_EAL G) is E -measurable;

        then

         A64: G is EDg -measurable by A42, A8, A6;

        ((1 / t1) to_power m) = (t1 to_power ( - m)) by A38, A7, POWER: 32, POWER: 34;

        then ((1 / t1) to_power m) = (s1 to_power ((1 / m) * ( - m))) by A7, A38, POWER: 33;

        then ((1 / t1) to_power m) = (s1 to_power ( - ((1 * (1 / m)) * m)));

        then ((1 / t1) to_power m) = (s1 to_power ( - 1)) by XCMPLX_1: 106;

        then ((1 / t1) to_power m) = ((1 / s1) to_power 1) by A7, A38, POWER: 32;

        then

         A65: ((1 / t1) to_power m) = (1 / s1) by POWER: 25;

        

         A66: ((1 / s1) qua ExtReal * s1) = 1 & ((1 / s2) qua ExtReal * s2) = 1 by A38, XCMPLX_1: 106;

        

         A67: ((1 / t2) to_power n) = (t2 to_power ( - n)) by A38, A7, POWER: 32, POWER: 34

        .= (s2 to_power ((1 / n) * ( - n))) by A7, A38, POWER: 33

        .= (s2 to_power ( - ((1 * (1 / n)) * n)))

        .= (s2 to_power ( - 1)) by XCMPLX_1: 106

        .= ((1 / s2) to_power 1) by A7, A38, POWER: 32

        .= (1 / s2) by POWER: 25;

        

         A68: ( Integral (M,(F | E))) = ( Integral (M,F)) by A4, A8, A42, A62, A61, A63, MESFUNC6: 89

        .= ((1 / m) * ( Integral (M,(((1 / t1) (#) ( abs f1)) to_power m)))) by A44, MESFUNC6: 102

        .= ((1 / m) * ( Integral (M,(((1 / t1) to_power m) (#) (( abs f1) to_power m))))) by A39, Th19

        .= ((1 / m) * (((1 / t1) to_power m) * ( Integral (M,(( abs f1) to_power m))))) by A3, MESFUNC6: 102

        .= (1 / m) by A65, A66, XXREAL_3: 81;

        

         A69: ( Integral (M,(G | E))) = ( Integral (M,G)) by A6, A8, A42, A62, A61, A64, MESFUNC6: 89

        .= ((1 / n) * ( Integral (M,(((1 / t2) (#) ( abs g1)) to_power n)))) by A44, MESFUNC6: 102

        .= ((1 / n) * ( Integral (M,(((1 / t2) to_power n) (#) (( abs g1) to_power n))))) by A39, Th19

        .= ((1 / n) * (((1 / t2) to_power n) * ( Integral (M,(( abs g1) to_power n))))) by A5, MESFUNC6: 102

        .= (1 / n) by A66, A67, XXREAL_3: 81;

        reconsider n1 = (1 / n), m1 = (1 / m) as Real;

        

         A70: ( Integral (M,(F + G))) = (( Integral (M,(F | E))) + ( Integral (M,(G | E)))) by A42, A4, A6, A8, A57

        .= (m1 + n1) by A69, A68, SUPINF_2: 1

        .= jj by A1;

        ( abs w) = ( |.(1 / (t1 * t2)) qua Complex.| (#) ( abs (f1 (#) g1))) by RFUNCT_1: 25;

        then ( abs w) = ((1 / (t1 * t2)) (#) ( abs (f1 (#) g1))) by A39, ABSVALUE:def 1;

        then

         A71: ( Integral (M,( abs w))) = ((1 / (t1 * t2)) * ( Integral (M,( abs (f1 (#) g1))))) by A15, MESFUNC6: 102;

        reconsider c1 = ( Integral (M,( abs (f1 (#) g1)))) as Element of REAL by A14, LPSPACE1: 44;

        ((1 / (t1 * t2)) qua ExtReal * ( Integral (M,( abs (f1 (#) g1))))) = ((1 / (t1 * t2)) qua ExtReal * c1);

        then ((1 / (t1 * t2)) * ( Integral (M,( abs (f1 (#) g1))))) = ((1 / (t1 * t2)) * c1);

        then ((t1 * t2) * ((1 / (t1 * t2)) * c1)) <= ((t1 * t2) * 1) by A39, A56, A71, A70, XREAL_1: 64;

        then

         A72: (((t1 * t2) * (1 / (t1 * t2))) * c1) <= (t1 * t2);

        ((t1 * t2) * (1 / (t1 * t2))) = 1 by A39, XCMPLX_1: 106;

        hence thesis by A3, A5, A72;

      end;

    end;

    

     Lm5: for m,n be positive Real st ((1 / m) + (1 / n)) = 1 & f in ( Lp_Functions (M,m)) & g in ( Lp_Functions (M,m)) holds ex r1,r2,r3 be Real st r1 = ( Integral (M,(( abs f) to_power m))) & r2 = ( Integral (M,(( abs g) to_power m))) & r3 = ( Integral (M,(( abs (f + g)) to_power m))) & (r3 to_power (1 / m)) <= ((r1 to_power (1 / m)) + (r2 to_power (1 / m)))

    proof

      let m,n be positive Real;

      assume

       A1: ((1 / m) + (1 / n)) = 1 & f in ( Lp_Functions (M,m)) & g in ( Lp_Functions (M,m));

      then (((m + n) * ((m * n) " )) * (m * n)) = (1 * (m * n)) by XCMPLX_1: 211;

      then ((m + n) * (((m * n) " ) * (m * n))) = (m * n);

      then ((m + n) * 1) = (m * n) by XCMPLX_0:def 7;

      then

       A2: m = (n * (m - 1));

      

       A3: (1 - 1) < (m - 1) by A1, Th1, XREAL_1: 14;

      then

       A4: (m - 1) > 0 ;

      ex f1 be PartFunc of X, REAL st f = f1 & ex NDf be Element of S st (M . (NDf ` )) = 0 & ( dom f1) = NDf & f1 is NDf -measurable & (( abs f1) to_power m) is_integrable_on M by A1;

      then

      consider EDf be Element of S such that

       A5: (M . (EDf ` )) = 0 & ( dom f) = EDf & f is EDf -measurable;

      ex g1 be PartFunc of X, REAL st g = g1 & ex NDg be Element of S st (M . (NDg ` )) = 0 & ( dom g1) = NDg & g1 is NDg -measurable & (( abs g1) to_power m) is_integrable_on M by A1;

      then

      consider EDg be Element of S such that

       A6: (M . (EDg ` )) = 0 & ( dom g) = EDg & g is EDg -measurable;

      set E = (EDf /\ EDg);

      

       A7: (f + g) in ( Lp_Functions (M,m)) by A1, Th25;

      then

       A8: ex h1 be PartFunc of X, REAL st (f + g) = h1 & ex NDfg be Element of S st (M . (NDfg ` )) = 0 & ( dom h1) = NDfg & h1 is NDfg -measurable & (( abs h1) to_power m) is_integrable_on M;

      

       A9: ( dom (f + g)) = E by A5, A6, VALUED_1:def 1;

      then

       A10: ( abs (f + g)) is E -measurable by A8, MESFUNC6: 48;

      reconsider s1 = ( Integral (M,(( abs f) to_power m))) as Element of REAL by A1, Th49;

      reconsider s2 = ( Integral (M,(( abs g) to_power m))) as Element of REAL by A1, Th49;

      reconsider s3 = ( Integral (M,(( abs (f + g)) to_power m))) as Element of REAL by A7, Th49;

      set t = (( abs (f + g)) to_power (m - 1));

      

       A11: ( dom t) = ( dom ( abs (f + g))) by MESFUN6C:def 4;

      then

       A12: ( dom t) = E by A9, VALUED_1:def 11;

      then

       A13: t is E -measurable by A3, A10, A11, MESFUN6C: 29;

      

       A14: (t to_power n) = (( abs (f + g)) to_power m) by A2, A3, Th6;

      

       A15: ( abs t) = t by Th14, A4;

      then

       A16: t in ( Lp_Functions (M,n)) by A9, A12, A14, A8, A13;

      then

      reconsider s4 = ( Integral (M,(( abs t) to_power n))) as Element of REAL by Th49;

      (t (#) f) is_integrable_on M & (t (#) g) is_integrable_on M by A1, A16, Th59;

      then

      reconsider u1 = ( Integral (M,( abs (t (#) f)))), u2 = ( Integral (M,( abs (t (#) g)))) as Element of REAL by LPSPACE1: 44;

      

       A17: ( dom ( abs f)) = EDf & ( dom ( abs g)) = EDg by A5, A6, VALUED_1:def 11;

      ( dom (t (#) ( abs f))) = (( dom t) /\ ( dom ( abs f))) & ( dom (t (#) ( abs g))) = (( dom t) /\ ( dom ( abs g))) by VALUED_1:def 4;

      then

       A18: ( dom (t (#) ( abs f))) = E & ( dom (t (#) ( abs g))) = E by A12, A17, XBOOLE_1: 17, XBOOLE_1: 28;

      

       A19: ( abs (t (#) f)) = (t (#) ( abs f)) & ( abs (t (#) g)) = (t (#) ( abs g)) & ( abs (t (#) (f + g))) = (t (#) ( abs (f + g))) by A15, RFUNCT_1: 24;

      (t (#) f) is_integrable_on M & (t (#) g) is_integrable_on M & (t (#) (f + g)) is_integrable_on M by A1, A16, A7, Th59;

      then

       A20: (t (#) ( abs f)) is_integrable_on M & (t (#) ( abs g)) is_integrable_on M & (t (#) ( abs (f + g))) is_integrable_on M by A19, LPSPACE1: 44;

      set F = (t (#) ( abs (f + g)));

      set G = ((t (#) ( abs f)) + (t (#) ( abs g)));

      

       A21: ( dom F) = (E /\ E) by A11, A12, VALUED_1:def 4;

      

       A22: ( dom G) = (E /\ E) by A18, VALUED_1:def 1;

      ( R_EAL F) is_integrable_on M by A20;

      then ex E1 be Element of S st E1 = ( dom ( R_EAL F)) & ( R_EAL F) is E1 -measurable;

      then

       A23: F is E -measurable by A21;

      

       A24: G is_integrable_on M by A20, MESFUNC6: 100;

      for x be Element of X st x in ( dom F) holds |.(F . x) qua Complex.| <= (G . x)

      proof

        let x be Element of X;

        assume

         A25: x in ( dom F);

        then |.((f . x) + (g . x)).| = |.((f + g) . x).| by A9, A21, VALUED_1:def 1;

        then

         A26: |.((f . x) + (g . x)).| = (( abs (f + g)) . x) & |.(f . x).| = (( abs f) . x) & |.(g . x).| = (( abs g) . x) by VALUED_1: 18;

        

         A27: |.(f . x).| = |.(f . x).| & |.(g . x).| = |.(g . x).| & |.((f . x) + (g . x)).| = |.((f . x) + (g . x)).|;

        

         A28: (t . x) >= 0 & (( abs (f + g)) . x) >= 0 by A3, MESFUNC6: 51;

        reconsider fx = (f . x), gx = (g . x) as R_eal by XXREAL_0:def 1;

        

         A29: (fx + gx) = ((f . x) + (g . x)) by SUPINF_2: 1;

         |.(fx + gx).| <= ( |.fx.| + |.gx.|) by EXTREAL1: 24;

        then |.((f . x) + (g . x)).| <= ( |.fx.| + |.gx.|) by A29;

        then |.((f . x) + (g . x)).| <= ( |.(f . x) qua Complex.| + |.(g . x) qua Complex.|) by A27, SUPINF_2: 1;

        then

         A30: ((t . x) * (( abs (f + g)) . x)) <= ((t . x) * ((( abs f) . x) + (( abs g) . x))) by A26, A28, XREAL_1: 64;

        ((t . x) * (( abs f) . x)) = ((t (#) ( abs f)) . x) & ((t . x) * (( abs g) . x)) = ((t (#) ( abs g)) . x) by VALUED_1: 5;

        then ((t . x) * ((( abs f) . x) + (( abs g) . x))) = (((t (#) ( abs f)) . x) + ((t (#) ( abs g)) . x));

        then

         A31: ((t . x) * ((( abs f) . x) + (( abs g) . x))) = (G . x) by A21, A22, A25, VALUED_1:def 1;

        ((t . x) * (( abs (f + g)) . x)) = (F . x) by VALUED_1: 5;

        hence thesis by A31, A30, A28, ABSVALUE:def 1;

      end;

      then

       A32: ( Integral (M,( abs F))) <= ( Integral (M,G)) by A21, A22, A23, A24, MESFUNC6: 96;

      

       A33: ex E1 be Element of S st E1 = (E /\ E) & ( Integral (M,G)) = (( Integral (M,((t (#) ( abs f)) | E1))) + ( Integral (M,((t (#) ( abs g)) | E1)))) by A18, A20, MESFUNC6: 101;

      ( Integral (M,((t (#) ( abs f)) | E))) = ( Integral (M,(t (#) ( abs f)))) & ( Integral (M,((t (#) ( abs g)) | E))) = ( Integral (M,(t (#) ( abs g)))) by A18, RELAT_1: 69;

      then

       A34: ( Integral (M,G)) = (u1 + u2) by A19, A33, SUPINF_2: 1;

      set v1 = ((s4 to_power (1 / n)) * (s1 to_power (1 / m)));

      set v2 = ((s4 to_power (1 / n)) * (s2 to_power (1 / m)));

      (ex r4 be Real st r4 = ( Integral (M,(( abs t) to_power n))) & ex r1 be Real st r1 = ( Integral (M,(( abs f) to_power m))) & ( Integral (M,( abs (t (#) f)))) <= ((r4 to_power (1 / n)) * (r1 to_power (1 / m)))) & (ex r4 be Real st r4 = ( Integral (M,(( abs t) to_power n))) & ex r2 be Real st r2 = ( Integral (M,(( abs g) to_power m))) & ( Integral (M,( abs (t (#) g)))) <= ((r4 to_power (1 / n)) * (r2 to_power (1 / m)))) by A1, A16, Th60;

      then

       A35: (u1 + u2) <= (v1 + v2) by XREAL_1: 7;

      F = ((( abs (f + g)) to_power (m - 1)) (#) (( abs (f + g)) to_power 1)) by Th8

      .= (( abs (f + g)) to_power ((m - 1) + 1)) by Th7, A3;

      then ( Integral (M,( abs F))) = s3 by Th14;

      then

       A36: s3 <= (((s3 to_power (1 / n)) * (s1 to_power (1 / m))) + ((s3 to_power (1 / n)) * (s2 to_power (1 / m)))) by A14, A15, A32, A34, A35, XXREAL_0: 2;

      per cases by A7, Th49;

        suppose s3 = 0 ;

        then

         A37: (s3 to_power (1 / m)) = 0 by POWER:def 2;

        (s1 to_power (1 / m)) >= 0 & (s2 to_power (1 / m)) >= 0 by A1, Th49, Th4;

        then (s3 to_power (1 / m)) <= ((s1 to_power (1 / m)) + (s2 to_power (1 / m))) by A37;

        hence thesis;

      end;

        suppose

         A38: s3 > 0 ;

        then

         A39: (s3 to_power (1 / n)) > 0 by POWER: 34;

        set w1 = (s3 to_power (1 / n));

        ((1 / w1) * ((w1 * (s1 to_power (1 / m))) + (w1 * (s2 to_power (1 / m))))) = (((1 / w1) * w1) * ((s1 to_power (1 / m)) + (s2 to_power (1 / m))));

        then

         A40: ((1 / w1) * ((w1 * (s1 to_power (1 / m))) + (w1 * (s2 to_power (1 / m))))) = (1 * ((s1 to_power (1 / m)) + (s2 to_power (1 / m)))) by A39, XCMPLX_1: 106;

        ((1 / w1) * s3) = ((s3 to_power ( - (1 / n))) * s3) by A38, POWER: 28

        .= ((s3 to_power ( - (1 / n))) * (s3 to_power 1)) by POWER: 25

        .= (s3 to_power (( - (1 / n)) + 1)) by A38, POWER: 27

        .= (s3 to_power (1 / m)) by A1;

        hence thesis by A39, A40, A36, XREAL_1: 64;

      end;

    end;

    theorem :: LPSPACE2:61

    

     Th61: for m be positive Real holds for r1,r2,r3 be Real st 1 <= m & f in ( Lp_Functions (M,m)) & g in ( Lp_Functions (M,m)) & r1 = ( Integral (M,(( abs f) to_power m))) & r2 = ( Integral (M,(( abs g) to_power m))) & r3 = ( Integral (M,(( abs (f + g)) to_power m))) holds (r3 to_power (1 / m)) <= ((r1 to_power (1 / m)) + (r2 to_power (1 / m)))

    proof

      let m be positive Real;

      let r1,r2,r3 be Real;

      assume

       A1: 1 <= m & f in ( Lp_Functions (M,m)) & g in ( Lp_Functions (M,m)) & r1 = ( Integral (M,(( abs f) to_power m))) & r2 = ( Integral (M,(( abs g) to_power m))) & r3 = ( Integral (M,(( abs (f + g)) to_power m)));

      per cases ;

        suppose

         A2: m = 1;

        then

         A3: r1 = ( Integral (M,( abs f))) & r2 = ( Integral (M,( abs g))) & r3 = ( Integral (M,( abs (f + g)))) by A1, Th8;

        

         A4: ex f1 be PartFunc of X, REAL st f = f1 & ex ND be Element of S st (M . (ND ` )) = 0 & ( dom f1) = ND & f1 is ND -measurable & (( abs f1) to_power m) is_integrable_on M by A1;

        

         A5: ex g1 be PartFunc of X, REAL st g = g1 & ex ND be Element of S st (M . (ND ` )) = 0 & ( dom g1) = ND & g1 is ND -measurable & (( abs g1) to_power m) is_integrable_on M by A1;

        then ( abs f) is_integrable_on M & ( abs g) is_integrable_on M by A2, A4, Th8;

        then f is_integrable_on M & g is_integrable_on M by A4, A5, MESFUNC6: 94;

        then ( Integral (M,( abs (f + g)))) <= (( Integral (M,( abs f))) + ( Integral (M,( abs g)))) by LPSPACE1: 55;

        then

         A6: r3 <= (r1 + r2) by A3, XXREAL_3:def 2;

        (r1 to_power (1 / m)) = r1 & (r2 to_power (1 / m)) = r2 by A2, POWER: 25;

        hence thesis by A6, A2, POWER: 25;

      end;

        suppose

         A7: m <> 1;

        set n1 = (1 - (1 / m));

        1 < m by A1, A7, XXREAL_0: 1;

        then (1 / m) < 1 by XREAL_1: 189;

        then 0 < n1 by XREAL_1: 50;

        then

        reconsider n = (1 / n1) as positive Real;

        ((1 / m) + (1 / n)) = 1;

        then ex rr1,rr2,rr3 be Real st rr1 = ( Integral (M,(( abs f) to_power m))) & rr2 = ( Integral (M,(( abs g) to_power m))) & rr3 = ( Integral (M,(( abs (f + g)) to_power m))) & (rr3 to_power (1 / m)) <= ((rr1 to_power (1 / m)) + (rr2 to_power (1 / m))) by A1, Lm5;

        hence thesis by A1;

      end;

    end;

    

     Lm6: for k be geq_than_1 Real holds ( Lp-Space (M,k)) is reflexive discerning RealNormSpace-like vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed right_complementable

    proof

      let k be geq_than_1 Real;

      set x = ( 0. ( Lp-Space (M,k)));

      x = ( 0. ( Pre-Lp-Space (M,k)));

      then x = ( zeroCoset (M,k)) by Def11;

      then (X --> 0 ) in x by Th38, Th23;

      then ex r be Real st 0 <= r & r = ( Integral (M,(( abs (X --> 0 )) to_power k))) & ||.x.|| = (r to_power (1 / k)) by Th55;

      then

      consider r0 be Real such that

       A1: r0 = ( Integral (M,(( abs (X --> 0 )) to_power k))) & (( Lp-Norm (M,k)) . x) = (r0 to_power (1 / k));

      r0 = 0 by A1, Th58;

      hence ||.x.|| = 0 by A1, POWER:def 2;

      now

        let x,y be Point of ( Lp-Space (M,k)), a be Real;

        

         A2: 1 <= k by Def1;

        hereby

          assume

           A3: ||.x.|| = 0 ;

          consider f be PartFunc of X, REAL such that

           A4: f in x & ex r be Real st r = ( Integral (M,(( abs f) to_power k))) & ||.x.|| = (r to_power (1 / k)) by Def12;

          

           A5: f in ( Lp_Functions (M,k)) by Th51, A4;

          then

          consider r1 be Real such that

           A6: r1 = ( Integral (M,(( abs f) to_power k))) & r1 >= 0 & (( Lp-Norm (M,k)) . x) = (r1 to_power (1 / k)) by A4, Th49;

          r1 = 0 by A3, A6, POWER: 34;

          then ( zeroCoset (M,k)) = ( a.e-eq-class_Lp (f,M,k)) by A5, A6, Th57, Th42;

          then ( 0. ( Pre-Lp-Space (M,k))) = ( a.e-eq-class_Lp (f,M,k)) by Def11;

          hence x = ( 0. ( Lp-Space (M,k))) by A4, Th55;

        end;

        consider f be PartFunc of X, REAL such that

         A7: f in x & ex r1 be Real st r1 = ( Integral (M,(( abs f) to_power k))) & ||.x.|| = (r1 to_power (1 / k)) by Def12;

        

         A8: (( abs f) to_power k) is_integrable_on M & f in ( Lp_Functions (M,k)) by Th51, A7;

        consider g be PartFunc of X, REAL such that

         A9: g in y & ex r2 be Real st r2 = ( Integral (M,(( abs g) to_power k))) & ||.y.|| = (r2 to_power (1 / k)) by Def12;

        

         A10: (( abs g) to_power k) is_integrable_on M & g in ( Lp_Functions (M,k)) by Th51, A9;

        consider s1 be Real such that

         A11: s1 = ( Integral (M,(( abs f) to_power k))) & ||.x.|| = (s1 to_power (1 / k)) by A7;

        

         A12: s1 = 0 implies (s1 to_power (1 / k)) >= 0 by POWER:def 2;

        s1 > 0 implies (s1 to_power (1 / k)) >= 0 by POWER: 34;

        hence 0 <= ||.x.|| by A12, A8, A11, Th49;

        set t = (f + g);

        set w = (x + y);

        

         A13: s1 >= 0 by A8, A11, Th49;

        consider s2 be Real such that

         A14: s2 = ( Integral (M,(( abs g) to_power k))) & ||.y.|| = (s2 to_power (1 / k)) by A9;

        (f + g) in (x + y) by Th54, A7, A9;

        then ex r be Real st 0 <= r & r = ( Integral (M,(( abs t) to_power k))) & ||.w.|| = (r to_power (1 / k)) by Th53;

        hence ||.(x + y).|| <= ( ||.x.|| + ||.y.||) by Th61, A2, A8, A10, A14, A11;

        set t = (a (#) f);

        set w = (a * x);

        (a (#) f) in (a * x) by Th54, A7;

        then ex r be Real st 0 <= r & r = ( Integral (M,(( abs t) to_power k))) & ||.w.|| = (r to_power (1 / k)) by Th53;

        then

        consider s be Real such that

         A15: s = ( Integral (M,(( abs t) to_power k))) & ||.w.|| = (s to_power (1 / k));

        reconsider r = ( |.a qua Complex.| to_power k) as Real;

        

         A16: s = ( Integral (M,(r (#) (( abs f) to_power k)))) by A15, Th18

        .= (r * ( Integral (M,(( abs f) to_power k)))) by A8, MESFUNC6: 102

        .= (r * s1) by A11, EXTREAL1: 1

        .= (( |.a qua Complex.| to_power k) * s1);

        ( |.a qua Complex.| to_power k) >= 0 by Th4, COMPLEX1: 46;

        

        then ||.(a * x).|| = ((( |.a qua Complex.| to_power k) to_power (1 / k)) * (s1 to_power (1 / k))) by A13, A15, A16, Th5

        .= (( |.a qua Complex.| to_power (k * (1 / k))) * (s1 to_power (1 / k))) by COMPLEX1: 46, HOLDER_1: 2

        .= (( |.a qua Complex.| to_power 1) * (s1 to_power (1 / k))) by XCMPLX_1: 106;

        hence ||.(a * x).|| = ( |.a qua Complex.| * ||.x.||) by A11, POWER: 25;

      end;

      hence thesis by NORMSP_1:def 1, RSSPACE3: 2;

    end;

    registration

      let k be geq_than_1 Real;

      let X be non empty set, S be SigmaField of X, M be sigma_Measure of S;

      cluster ( Lp-Space (M,k)) -> reflexive discerning RealNormSpace-like vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed right_complementable;

      coherence by Lm6;

    end

    begin

    theorem :: LPSPACE2:62

    

     Th62: for Sq be sequence of ( Lp-Space (M,k)) holds ex Fsq be Functional_Sequence of X, REAL st for n be Nat holds (Fsq . n) in ( Lp_Functions (M,k)) & (Fsq . n) in (Sq . n) & (Sq . n) = ( a.e-eq-class_Lp ((Fsq . n),M,k)) & ex r be Real st r = ( Integral (M,(( abs (Fsq . n)) to_power k))) & ||.(Sq . n).|| = (r to_power (1 / k))

    proof

      let Sq be sequence of ( Lp-Space (M,k));

      defpred P[ Nat, set] means ex f be PartFunc of X, REAL st $2 = f & f in ( Lp_Functions (M,k)) & f in (Sq . $1) & (Sq . $1) = ( a.e-eq-class_Lp (f,M,k)) & ex r be Real st r = ( Integral (M,(( abs f) to_power k))) & ||.(Sq . $1).|| = (r to_power (1 / k));

      

       A1: for x be Element of NAT holds ex y be Element of ( PFuncs (X, REAL )) st P[x, y]

      proof

        let x be Element of NAT ;

        consider y be PartFunc of X, REAL such that

         A2: y in ( Lp_Functions (M,k)) & (Sq . x) = ( a.e-eq-class_Lp (y,M,k)) by Th53;

        ex r be Real st 0 <= r & r = ( Integral (M,(( abs y) to_power k))) & ||.(Sq . x).|| = (r to_power (1 / k)) by Th53, A2, Th38;

        hence thesis by A2, Th38;

      end;

      consider G be sequence of ( PFuncs (X, REAL )) such that

       A3: for n be Element of NAT holds P[n, (G . n)] from FUNCT_2:sch 3( A1);

      reconsider G as Functional_Sequence of X, REAL ;

      now

        let n be Nat;

        n in NAT by ORDINAL1:def 12;

        then ex f be PartFunc of X, REAL st (G . n) = f & f in ( Lp_Functions (M,k)) & f in (Sq . n) & (Sq . n) = ( a.e-eq-class_Lp (f,M,k)) & ex r be Real st r = ( Integral (M,(( abs f) to_power k))) & ||.(Sq . n).|| = (r to_power (1 / k)) by A3;

        hence (G . n) in ( Lp_Functions (M,k)) & (G . n) in (Sq . n) & (Sq . n) = ( a.e-eq-class_Lp ((G . n),M,k)) & ex r be Real st r = ( Integral (M,(( abs (G . n)) to_power k))) & ||.(Sq . n).|| = (r to_power (1 / k));

      end;

      hence thesis;

    end;

    theorem :: LPSPACE2:63

    

     Th63: for Sq be sequence of ( Lp-Space (M,k)) holds ex Fsq be with_the_same_dom Functional_Sequence of X, REAL st for n be Nat holds (Fsq . n) in ( Lp_Functions (M,k)) & (Fsq . n) in (Sq . n) & (Sq . n) = ( a.e-eq-class_Lp ((Fsq . n),M,k)) & ex r be Real st 0 <= r & r = ( Integral (M,(( abs (Fsq . n)) to_power k))) & ||.(Sq . n).|| = (r to_power (1 / k))

    proof

      let Sq be sequence of ( Lp-Space (M,k));

      consider Fsq be Functional_Sequence of X, REAL such that

       A1: for n be Nat holds (Fsq . n) in ( Lp_Functions (M,k)) & (Fsq . n) in (Sq . n) & (Sq . n) = ( a.e-eq-class_Lp ((Fsq . n),M,k)) & ex r be Real st r = ( Integral (M,(( abs (Fsq . n)) to_power k))) & ||.(Sq . n).|| = (r to_power (1 / k)) by Th62;

      defpred P[ Nat, set] means ex DMFSQN be Element of S st $2 = DMFSQN & ex FSQN be PartFunc of X, REAL st (Fsq . $1) = FSQN & (M . (DMFSQN ` )) = 0 & ( dom FSQN) = DMFSQN & FSQN is DMFSQN -measurable & (( abs FSQN) to_power k) is_integrable_on M;

      

       A2: for n be Element of NAT holds ex y be Element of S st P[n, y]

      proof

        let n be Element of NAT ;

        (Fsq . n) in ( Lp_Functions (M,k)) by A1;

        then ex FMF be PartFunc of X, REAL st (Fsq . n) = FMF & ex ND be Element of S st (M . (ND ` )) = 0 & ( dom FMF) = ND & FMF is ND -measurable & (( abs FMF) to_power k) is_integrable_on M;

        hence thesis;

      end;

      consider G be sequence of S such that

       A3: for n be Element of NAT holds P[n, (G . n)] from FUNCT_2:sch 3( A2);

      reconsider E0 = ( meet ( rng G)) as Element of S;

      

       A4: for n be Nat holds (M . (X \ (G . n))) = 0 & E0 c= ( dom (Fsq . n))

      proof

        let n be Nat;

        

         A5: n in NAT by ORDINAL1:def 12;

        ex D be Element of S st (G . n) = D & ex F be PartFunc of X, REAL st (Fsq . n) = F & (M . (D ` )) = 0 & ( dom F) = D & F is D -measurable & (( abs F) to_power k) is_integrable_on M by A3, A5;

        hence (M . (X \ (G . n))) = 0 & E0 c= ( dom (Fsq . n)) by FUNCT_2: 4, SETFAM_1: 3, A5;

      end;

      

       A6: (X \ ( rng G)) is N_Sub_set_fam of X by MEASURE1: 21;

      for A be set st A in (X \ ( rng G)) holds A in S & A is measure_zero of M

      proof

        let A be set;

        assume

         A7: A in (X \ ( rng G));

        then

        reconsider A0 = A as Subset of X;

        (A0 ` ) in ( rng G) by A7, SETFAM_1:def 7;

        then

        consider n be object such that

         A8: n in NAT & (A0 ` ) = (G . n) by FUNCT_2: 11;

        reconsider n as Nat by A8;

        

         A9: ((A0 ` ) ` ) = A0;

        then A0 = (X \ (G . n)) by A8;

        hence A in S by MEASURE1: 34;

        

         A10: (M . A0) = 0 by A4, A8, A9;

        A0 = (X \ (G . n)) by A8, A9;

        then A is Element of S by MEASURE1: 34;

        hence A is measure_zero of M by A10, MEASURE1:def 7;

      end;

      then

       A11: (for A be object st A in (X \ ( rng G)) holds A in S) & (for A be set st A in (X \ ( rng G)) holds A is measure_zero of M);

      then (X \ ( rng G)) c= S;

      then (X \ ( rng G)) is N_Measure_fam of S by A6, MEASURE2:def 1;

      then

       A12: ( union (X \ ( rng G))) is measure_zero of M by A11, MEASURE2: 14;

      (E0 ` ) = (X \ (X \ ( union (X \ ( rng G))))) by MEASURE1: 4

      .= (X /\ ( union (X \ ( rng G)))) by XBOOLE_1: 48

      .= ( union (X \ ( rng G))) by XBOOLE_1: 28;

      then

       A13: (M . (E0 ` )) = 0 by A12, MEASURE1:def 7;

      set Fsq2 = (Fsq || E0);

      

       A14: for n be Nat holds ( dom (Fsq2 . n)) = E0

      proof

        let n be Nat;

        ( dom (Fsq2 . n)) = ( dom ((Fsq . n) | E0)) by MESFUN9C:def 1;

        then ( dom (Fsq2 . n)) = (( dom (Fsq . n)) /\ E0) by RELAT_1: 61;

        hence ( dom (Fsq2 . n)) = E0 by A4, XBOOLE_1: 28;

      end;

      now

        let n,m be Nat;

        ( dom (Fsq2 . n)) = E0 & ( dom (Fsq2 . m)) = E0 by A14;

        hence ( dom (Fsq2 . n)) = ( dom (Fsq2 . m));

      end;

      then

      reconsider Fsq2 as with_the_same_dom Functional_Sequence of X, REAL by MESFUNC8:def 2;

      take Fsq2;

      hereby

        let n be Nat;

        (Fsq . n) in ( Lp_Functions (M,k)) by A1;

        then

         A15: ex FMF be PartFunc of X, REAL st (Fsq . n) = FMF & ex ND be Element of S st (M . (ND ` )) = 0 & ( dom FMF) = ND & FMF is ND -measurable & (( abs FMF) to_power k) is_integrable_on M;

        then

        reconsider E2 = ( dom (Fsq . n)) as Element of S;

        

         A16: (E2 /\ E0) = E0 by A4, XBOOLE_1: 28;

        ( R_EAL (Fsq . n)) is E2 -measurable by A15;

        then ( R_EAL (Fsq . n)) is E0 -measurable by A4, MESFUNC1: 30;

        then (Fsq . n) is E0 -measurable;

        then ((Fsq . n) | E0) is E0 -measurable by A16, MESFUNC6: 76;

        then

         A17: (Fsq2 . n) is E0 -measurable by MESFUN9C:def 1;

        

         A18: ( dom (Fsq2 . n)) = E0 by A14;

        ( dom (( abs (Fsq . n)) to_power k)) = ( dom ( abs (Fsq . n))) & ( dom (( abs (Fsq2 . n)) to_power k)) = ( dom ( abs (Fsq2 . n))) by MESFUN6C:def 4;

        then

         A19: ( dom (( abs (Fsq . n)) to_power k)) = ( dom (Fsq . n)) & ( dom (( abs (Fsq2 . n)) to_power k)) = ( dom (Fsq2 . n)) by VALUED_1:def 11;

        for x be object st x in ( dom (( abs (Fsq2 . n)) to_power k)) holds ((( abs (Fsq2 . n)) to_power k) . x) = ((( abs (Fsq . n)) to_power k) . x)

        proof

          let x be object;

          assume

           A20: x in ( dom (( abs (Fsq2 . n)) to_power k));

          then

          reconsider x0 = x as Element of X;

          

           A21: x in ( dom (( abs (Fsq . n)) to_power k)) by A18, A19, A16, A20, XBOOLE_0:def 4;

          

          thus ((( abs (Fsq2 . n)) to_power k) . x) = ((( abs (Fsq2 . n)) . x0) to_power k) by A20, MESFUN6C:def 4

          .= ( |.((Fsq2 . n) . x0) qua Complex.| to_power k) by VALUED_1: 18

          .= ( |.(((Fsq . n) | E0) . x0) qua Complex.| to_power k) by MESFUN9C:def 1

          .= ( |.((Fsq . n) . x0) qua Complex.| to_power k) by A18, A19, A20, FUNCT_1: 49

          .= ((( abs (Fsq . n)) . x0) to_power k) by VALUED_1: 18

          .= ((( abs (Fsq . n)) to_power k) . x) by A21, MESFUN6C:def 4;

        end;

        then ((( abs (Fsq . n)) to_power k) | E0) = (( abs (Fsq2 . n)) to_power k) by A14, A16, A19, FUNCT_1: 46;

        then (( abs (Fsq2 . n)) to_power k) is_integrable_on M by A15, MESFUNC6: 91;

        hence

         A22: (Fsq2 . n) in ( Lp_Functions (M,k)) by A17, A18, A13;

        

         A23: (Fsq . n) in (Sq . n) & (Sq . n) = ( a.e-eq-class_Lp ((Fsq . n),M,k)) by A1;

        reconsider EB = (E0 ` ) as Element of S by MEASURE1: 34;

        ((Fsq2 . n) | (EB ` )) = (Fsq2 . n) by A18, RELAT_1: 68;

        then ((Fsq2 . n) | (EB ` )) = ((Fsq . n) | (EB ` )) by MESFUN9C:def 1;

        then

         A24: (Fsq2 . n) a.e.= ((Fsq . n),M) by A13;

        hence (Fsq2 . n) in (Sq . n) by A23, A22, Th36;

        ( a.e-eq-class_Lp ((Fsq2 . n),M,k)) = ( a.e-eq-class_Lp ((Fsq . n),M,k)) by Th42, A24;

        hence (Sq . n) = ( a.e-eq-class_Lp ((Fsq2 . n),M,k)) by A1;

        hence ex r be Real st 0 <= r & r = ( Integral (M,(( abs (Fsq2 . n)) to_power k))) & ||.(Sq . n).|| = (r to_power (1 / k)) by Th53, Th38, A22;

      end;

    end;

    

     Lm7: for X be RealNormSpace, Sq be sequence of X, Sq0 be Point of X, R1 be Real_Sequence, N be increasing sequence of NAT st Sq is Cauchy_sequence_by_Norm & (for i be Nat holds (R1 . i) = ||.(Sq0 - (Sq . (N . i))).||) & R1 is convergent & ( lim R1) = 0 holds Sq is convergent & ( lim Sq) = Sq0 & ||.(Sq - Sq0).|| is convergent & ( lim ||.(Sq - Sq0).||) = 0

    proof

      let X be RealNormSpace, Sq be sequence of X, Sq0 be Point of X, R1 be Real_Sequence, N be increasing sequence of NAT ;

      assume that

       A1: Sq is Cauchy_sequence_by_Norm and

       A2: for i be Nat holds (R1 . i) = ||.(Sq0 - (Sq . (N . i))).|| and

       A3: R1 is convergent & ( lim R1) = 0 ;

       A4:

      now

        let p be Real;

        assume

         A5: 0 < p;

        then

        consider n2 such that

         A6: for m, n st n2 <= m & n2 <= n holds ||.((Sq . m) - (Sq . n)).|| < (p / 2) by A1, RSSPACE3: 8;

        consider n1 such that

         A7: for l st n1 <= l holds |.((R1 . l) - 0 ) qua Complex.| < (p / 2) by A3, A5, SEQ_2:def 7;

        reconsider n3 = ( max (n1,n2)) as Nat by TARSKI: 1;

        take n3;

        thus for n be Nat st n3 <= n holds ||.((Sq . n) - Sq0).|| < p

        proof

          let n be Nat;

          assume

           A8: n3 <= n;

          n1 <= n3 by XXREAL_0: 25;

          then n1 <= n by A8, XXREAL_0: 2;

          then |.((R1 . n) - 0 ).| < (p / 2) by A7;

          then

           A9: |. ||.(Sq0 - (Sq . (N . n))).||.| < (p / 2) by A2;

          

           A10: ||.(Sq0 - (Sq . (N . n))).|| < (p / 2) by A9, ABSVALUE:def 1;

          n <= (N . n) by SEQM_3: 14;

          then

           A11: n3 <= (N . n) by A8, XXREAL_0: 2;

          n2 <= n3 by XXREAL_0: 25;

          then n2 <= (N . n) & n2 <= n by A8, A11, XXREAL_0: 2;

          then ||.((Sq . (N . n)) - (Sq . n)).|| < (p / 2) by A6;

          then

           A12: ( ||.(Sq0 - (Sq . (N . n))).|| + ||.((Sq . (N . n)) - (Sq . n)).||) < ((p / 2) + (p / 2)) by A10, XREAL_1: 8;

          

           A13: ||.((Sq . n) - Sq0).|| = ||.(Sq0 - (Sq . n)).|| by NORMSP_1: 7

          .= ||.((Sq0 - (Sq . (N . n))) + ((Sq . (N . n)) - (Sq . n))).|| by LOPBAN_3: 3;

           ||.((Sq0 - (Sq . (N . n))) + ((Sq . (N . n)) - (Sq . n))).|| <= ( ||.(Sq0 - (Sq . (N . n))).|| + ||.((Sq . (N . n)) - (Sq . n)).||) by NORMSP_1:def 1;

          hence ||.((Sq . n) - Sq0).|| < p by A13, A12, XXREAL_0: 2;

        end;

      end;

      hence

       A14: Sq is convergent by NORMSP_1:def 6;

      hence ( lim Sq) = Sq0 by A4, NORMSP_1:def 7;

      hence thesis by A14, NORMSP_1: 24;

    end;

    theorem :: LPSPACE2:64

    for X be RealNormSpace, Sq be sequence of X, Sq0 be Point of X st ||.(Sq - Sq0).|| is convergent & ( lim ||.(Sq - Sq0).||) = 0 holds Sq is convergent & ( lim Sq) = Sq0

    proof

      let X be RealNormSpace, Sq be sequence of X, Sq0 be Point of X;

      assume

       A1: ||.(Sq - Sq0).|| is convergent & ( lim ||.(Sq - Sq0).||) = 0 ;

      

       A2: for p be Real st 0 < p holds ex n be Nat st for m be Nat st n <= m holds ||.((Sq . m) - Sq0).|| < p

      proof

        let p be Real;

        assume 0 < p;

        then

        consider n such that

         A3: for m st n <= m holds |.(( ||.(Sq - Sq0).|| . m) - 0 ) qua Complex.| < p by A1, SEQ_2:def 7;

        take n;

        hereby

          let m be Nat;

          assume n <= m;

          then |.(( ||.(Sq - Sq0).|| . m) - 0 ).| < p by A3;

          then |. ||.((Sq - Sq0) . m).||.| < p by NORMSP_0:def 4;

          then |. ||.((Sq . m) - Sq0).||.| < p by NORMSP_1:def 4;

          hence ||.((Sq . m) - Sq0).|| < p by ABSVALUE:def 1;

        end;

      end;

      hence Sq is convergent by NORMSP_1:def 6;

      hence ( lim Sq) = Sq0 by A2, NORMSP_1:def 7;

    end;

    theorem :: LPSPACE2:65

    

     Th65: for X be RealNormSpace, Sq be sequence of X st Sq is Cauchy_sequence_by_Norm holds ex N be increasing sequence of NAT st for i,j be Nat st j >= (N . i) holds ||.((Sq . j) - (Sq . (N . i))).|| < (2 to_power ( - i))

    proof

      let X be RealNormSpace, Sq be sequence of X;

      assume

       A1: Sq is Cauchy_sequence_by_Norm;

      1 = (2 to_power ( - 0 )) by POWER: 24;

      then

      consider N0 be Nat such that

       A2: for j,i be Nat st j >= N0 & i >= N0 holds ||.((Sq . j) - (Sq . i)).|| < (2 to_power ( - 0 )) by A1, RSSPACE3: 8;

      reconsider N0 as Element of NAT by ORDINAL1:def 12;

      defpred P[ set, set, set] means ex n,x,y be Nat st n = $1 & x = $2 & y = $3 & ((for j be Nat st j >= x holds ||.((Sq . j) - (Sq . x)).|| < (2 to_power ( - n))) implies x < y & (for j be Nat st j >= y holds ||.((Sq . j) - (Sq . y)).|| < (2 to_power ( - (n + 1)))));

      

       A3: for n be Nat, x be Element of NAT holds ex y be Element of NAT st P[n, x, y]

      proof

        let n be Nat, x be Element of NAT ;

        now

          assume for j be Nat st j >= x holds ||.((Sq . j) - (Sq . x)).|| < (2 to_power ( - n));

           0 < (2 to_power ( - (n + 1))) by POWER: 34;

          then

          consider N2 be Nat such that

           A4: for j,i be Nat st j >= N2 & i >= N2 holds ||.((Sq . j) - (Sq . i)).|| < (2 to_power ( - (n + 1))) by A1, RSSPACE3: 8;

          set y = (( max (x,N2)) + 1);

          take y;

          x <= ( max (x,N2)) by XXREAL_0: 25;

          hence x < y by NAT_1: 13;

          N2 <= ( max (x,N2)) by XXREAL_0: 25;

          then

           A5: N2 < y by NAT_1: 13;

          thus for j be Nat st j >= y holds ||.((Sq . j) - (Sq . y)).|| < (2 to_power ( - (n + 1)))

          proof

            let j be Nat;

            assume j >= y;

            then j >= N2 & y >= N2 by A5, XXREAL_0: 2;

            hence thesis by A4;

          end;

        end;

        hence thesis;

      end;

      consider f be sequence of NAT such that

       A6: (f . 0 ) = N0 & for n be Nat holds P[n, (f . n), (f . (n + 1))] from RECDEF_1:sch 2( A3);

      defpred Q[ Nat] means for j be Nat st j >= (f . $1) holds ||.((Sq . j) - (Sq . (f . $1))).|| < (2 to_power ( - $1));

      

       A7: Q[ 0 ] by A2, A6;

       A8:

      now

        let i be Nat;

        assume

         A9: Q[i];

        ex n,x,y be Nat st n = i & x = (f . i) & y = (f . (i + 1)) & ((for j be Nat st j >= x holds ||.((Sq . j) - (Sq . x)).|| < (2 to_power ( - n))) implies x < y & (for j be Nat st j >= y holds ||.((Sq . j) - (Sq . y)).|| < (2 to_power ( - (n + 1))))) by A6;

        hence Q[(i + 1)] by A9;

      end;

      

       A10: for i be Nat holds Q[i] from NAT_1:sch 2( A7, A8);

      now

        let i be Nat;

        ex n,x,y be Nat st n = i & x = (f . i) & y = (f . (i + 1)) & ((for j be Nat st j >= x holds ||.((Sq . j) - (Sq . x)).|| < (2 to_power ( - n))) implies x < y & (for j be Nat st j >= y holds ||.((Sq . j) - (Sq . y)).|| < (2 to_power ( - (n + 1))))) by A6;

        hence (f . i) < (f . (i + 1)) by A10;

      end;

      then f is increasing;

      hence thesis by A10;

    end;

    theorem :: LPSPACE2:66

    

     Th66: for F be Functional_Sequence of X, REAL st (for m be Nat holds (F . m) in ( Lp_Functions (M,k))) holds for m be Nat holds (( Partial_Sums F) . m) in ( Lp_Functions (M,k))

    proof

      let F be Functional_Sequence of X, REAL ;

      assume

       A1: for m be Nat holds (F . m) in ( Lp_Functions (M,k));

      defpred P[ Nat] means (( Partial_Sums F) . $1) in ( Lp_Functions (M,k));

      (( Partial_Sums F) . 0 ) = (F . 0 ) by MESFUN9C:def 2;

      then

       A2: P[ 0 ] by A1;

       A3:

      now

        let j be Nat;

        assume P[j];

        then

         A4: (( Partial_Sums F) . j) in ( Lp_Functions (M,k)) & (F . (j + 1)) in ( Lp_Functions (M,k)) by A1;

        (( Partial_Sums F) . (j + 1)) = ((( Partial_Sums F) . j) + (F . (j + 1))) by MESFUN9C:def 2;

        hence P[(j + 1)] by A4, Th25;

      end;

      for j be Nat holds P[j] from NAT_1:sch 2( A2, A3);

      hence thesis;

    end;

    theorem :: LPSPACE2:67

    

     Th67: for F be Functional_Sequence of X, REAL st (for m be Nat holds (F . m) is nonnegative) holds for m be Nat holds (( Partial_Sums F) . m) is nonnegative

    proof

      let F be Functional_Sequence of X, REAL ;

      assume

       A1: for m be Nat holds (F . m) is nonnegative;

      defpred P[ Nat] means (( Partial_Sums F) . $1) is nonnegative;

      (( Partial_Sums F) . 0 ) = (F . 0 ) by MESFUN9C:def 2;

      then

       A2: P[ 0 ] by A1;

       A3:

      now

        let k be Nat;

        assume P[k];

        then

         A4: (( Partial_Sums F) . k) is nonnegative & (F . (k + 1)) is nonnegative by A1;

        (( Partial_Sums F) . (k + 1)) = ((( Partial_Sums F) . k) + (F . (k + 1))) by MESFUN9C:def 2;

        hence P[(k + 1)] by A4, MESFUNC6: 56;

      end;

      for k be Nat holds P[k] from NAT_1:sch 2( A2, A3);

      hence thesis;

    end;

    theorem :: LPSPACE2:68

    

     Th68: for F be Functional_Sequence of X, REAL , x be Element of X, n,m be Nat st F is with_the_same_dom & x in ( dom (F . 0 )) & (for k be Nat holds (F . k) is nonnegative) & n <= m holds ((( Partial_Sums F) . n) . x) <= ((( Partial_Sums F) . m) . x)

    proof

      let F be Functional_Sequence of X, REAL , x be Element of X, n,m be Nat;

      assume

       A1: F is with_the_same_dom;

      assume

       A2: x in ( dom (F . 0 ));

      assume

       A3: for m be Nat holds (F . m) is nonnegative;

      assume

       A4: n <= m;

      set PF = ( Partial_Sums F);

      defpred P[ Nat] means ((PF . n) . x) <= ((PF . $1) . x);

      

       A5: for k be Nat holds ((PF . k) . x) <= ((PF . (k + 1)) . x)

      proof

        let k be Nat;

        

         A6: (PF . (k + 1)) = ((PF . k) + (F . (k + 1))) by MESFUN9C:def 2;

        

         A7: ( dom (PF . (k + 1))) = ( dom (F . 0 )) by A1, MESFUN9C: 11;

        (F . (k + 1)) is nonnegative & (PF . k) is nonnegative by A3, Th67;

        then 0 <= ((F . (k + 1)) . x) & 0 <= ((PF . k) . x) by MESFUNC6: 51;

        then (((PF . k) . x) + 0 ) <= (((PF . k) . x) + ((F . (k + 1)) . x)) by XREAL_1: 7;

        hence thesis by A7, A2, A6, VALUED_1:def 1;

      end;

      

       A8: for k be Nat st k >= n & (for l be Nat st l >= n & l < k holds P[l]) holds P[k]

      proof

        let k be Nat;

        assume

         A9: k >= n & for l be Nat st l >= n & l < k holds P[l];

        now

          assume k > n;

          then k >= (n + 1) by NAT_1: 13;

          then

           A10: k = (n + 1) or k > (n + 1) by XXREAL_0: 1;

          now

            assume

             A11: k > (n + 1);

            then

            reconsider l = (k - 1) as Nat by NAT_1: 20;

            k < (k + 1) by NAT_1: 13;

            then k > l & l >= n by A11, XREAL_1: 19;

            then

             A12: ((PF . n) . x) <= ((PF . l) . x) by A9;

            k = (l + 1);

            then ((PF . l) . x) <= ((PF . k) . x) by A5;

            hence thesis by A12, XXREAL_0: 2;

          end;

          hence thesis by A10, A5;

        end;

        hence thesis by A9, XXREAL_0: 1;

      end;

      for k be Nat st k >= n holds P[k] from NAT_1:sch 9( A8);

      hence thesis by A4;

    end;

    theorem :: LPSPACE2:69

    

     Th69: for F be Functional_Sequence of X, REAL st F is with_the_same_dom holds ( abs F) is with_the_same_dom

    proof

      let F be Functional_Sequence of X, REAL ;

      assume

       A1: F is with_the_same_dom;

      for n,m be Nat holds ( dom (( abs F) . n)) = ( dom (( abs F) . m))

      proof

        let n,m be Nat;

        (( abs F) . n) = ( abs (F . n)) & (( abs F) . m) = ( abs (F . m)) by SEQFUNC:def 4;

        then ( dom (( abs F) . n)) = ( dom (F . n)) & ( dom (( abs F) . m)) = ( dom (F . m)) by VALUED_1:def 11;

        hence ( dom (( abs F) . n)) = ( dom (( abs F) . m)) by A1, MESFUNC8:def 2;

      end;

      hence ( abs F) is with_the_same_dom by MESFUNC8:def 2;

    end;

    theorem :: LPSPACE2:70

    

     Th70: for k be geq_than_1 Real, Sq be sequence of ( Lp-Space (M,k)) st Sq is Cauchy_sequence_by_Norm holds Sq is convergent

    proof

      let k be geq_than_1 Real;

      let Sq be sequence of ( Lp-Space (M,k));

      

       A1: 1 <= k by Def1;

      assume

       A2: Sq is Cauchy_sequence_by_Norm;

      consider Fsq be with_the_same_dom Functional_Sequence of X, REAL such that

       A3: for n be Nat holds (Fsq . n) in ( Lp_Functions (M,k)) & (Fsq . n) in (Sq . n) & (Sq . n) = ( a.e-eq-class_Lp ((Fsq . n),M,k)) & ex r be Real st 0 <= r & r = ( Integral (M,(( abs (Fsq . n)) to_power k))) & ||.(Sq . n).|| = (r to_power (1 / k)) by Th63;

      (Fsq . 0 ) in ( Lp_Functions (M,k)) by A3;

      then

       A4: ex D be Element of S st (M . (D ` )) = 0 & ( dom (Fsq . 0 )) = D & (Fsq . 0 ) is D -measurable by Th35;

      then

      reconsider E = ( dom (Fsq . 0 )) as Element of S;

      consider N be increasing sequence of NAT such that

       A5: for i,j be Nat st j >= (N . i) holds ||.((Sq . j) - (Sq . (N . i))).|| < (2 to_power ( - i)) by Th65, A2;

      deffunc FsqN( Nat) = (Fsq . (N . $1));

      consider F1 be Functional_Sequence of X, REAL such that

       A6: for n be Nat holds (F1 . n) = FsqN(n) from SEQFUNC:sch 1;

      

       A7: for n be Nat holds ( dom (F1 . n)) = E & (F1 . n) in ( Lp_Functions (M,k)) & (F1 . n) is E -measurable & ( abs (F1 . n)) in ( Lp_Functions (M,k))

      proof

        let n be Nat;

        

         A8: (F1 . n) = (Fsq . (N . n)) by A6;

        hence

         A9: ( dom (F1 . n)) = E & (F1 . n) in ( Lp_Functions (M,k)) by A3, MESFUNC8:def 2;

        then

        consider F be PartFunc of X, REAL such that

         Z1: (F1 . n) = F & ex ND be Element of S st (M . (ND ` )) = 0 & ( dom F) = ND & F is ND -measurable & (( abs F) to_power k) is_integrable_on M;

        consider ND be Element of S such that

         Z2: (M . (ND ` )) = 0 & ( dom F) = ND & F is ND -measurable & (( abs F) to_power k) is_integrable_on M by Z1;

        ND = E by Z1, Z2, A8, MESFUNC8:def 2;

        hence (F1 . n) is E -measurable by Z1, Z2;

        thus ( abs (F1 . n)) in ( Lp_Functions (M,k)) by A9, Th28;

      end;

      for n,m be Nat holds ( dom (F1 . n)) = ( dom (F1 . m))

      proof

        let n,m be Nat;

        ( dom (F1 . n)) = E & ( dom (F1 . m)) = E by A7;

        hence thesis;

      end;

      then

      reconsider F1 as with_the_same_dom Functional_Sequence of X, REAL by MESFUNC8:def 2;

      deffunc FF( Nat) = ((F1 . ($1 + 1)) - (F1 . $1));

      consider FMF be Functional_Sequence of X, REAL such that

       A10: for n be Nat holds (FMF . n) = FF(n) from SEQFUNC:sch 1;

      

       A11: for n be Nat holds ( dom (FMF . n)) = E & (FMF . n) in ( Lp_Functions (M,k))

      proof

        let n be Nat;

        

         A12: ( dom (F1 . n)) = E & ( dom (F1 . (n + 1))) = E by A7;

        (FMF . n) = ((F1 . (n + 1)) - (F1 . n)) by A10;

        then ( dom (FMF . n)) = (( dom (F1 . (n + 1))) /\ ( dom (F1 . n))) by VALUED_1: 12;

        hence ( dom (FMF . n)) = E by A12;

        (Fsq . (N . (n + 1))) in ( Lp_Functions (M,k)) & (Fsq . (N . n)) in ( Lp_Functions (M,k)) by A3;

        then (F1 . (n + 1)) in ( Lp_Functions (M,k)) & (F1 . n) in ( Lp_Functions (M,k)) by A6;

        then ((F1 . (n + 1)) - (F1 . n)) in ( Lp_Functions (M,k)) by Th27;

        hence (FMF . n) in ( Lp_Functions (M,k)) by A10;

      end;

      for n,m be Nat holds ( dom (FMF . n)) = ( dom (FMF . m))

      proof

        let n,m be Nat;

        ( dom (FMF . n)) = E & ( dom (FMF . m)) = E by A11;

        hence thesis;

      end;

      then

      reconsider FMF as with_the_same_dom Functional_Sequence of X, REAL by MESFUNC8:def 2;

      set AbsFMF = ( abs FMF);

      

       A13: for n be Nat holds (AbsFMF . n) is nonnegative & ( dom (AbsFMF . n)) = E & ( abs (AbsFMF . n)) = (AbsFMF . n) & (AbsFMF . n) in ( Lp_Functions (M,k)) & (AbsFMF . n) is E -measurable

      proof

        let n be Nat;

        

         A14: (AbsFMF . n) = ( abs (FMF . n)) by SEQFUNC:def 4;

        hence (AbsFMF . n) is nonnegative;

        

         A15: ( dom (FMF . n)) = E & (FMF . n) in ( Lp_Functions (M,k)) by A11;

        hence ( dom (AbsFMF . n)) = E & ( abs (AbsFMF . n)) = (AbsFMF . n) by A14, VALUED_1:def 11;

        thus (AbsFMF . n) in ( Lp_Functions (M,k)) by A11, A14, Th28;

        then

        consider D be Element of S such that

         Z1: (M . (D ` )) = 0 & ( dom (AbsFMF . n)) = D & (AbsFMF . n) is D -measurable by Th35;

        D = E by Z1, A15, A14, VALUED_1:def 11;

        hence (AbsFMF . n) is E -measurable by Z1;

      end;

      reconsider AbsFMF as with_the_same_dom Functional_Sequence of X, REAL by Th69;

      deffunc Gk( Nat) = (( abs (F1 . 0 )) + (( Partial_Sums AbsFMF) . $1));

      consider G be Functional_Sequence of X, REAL such that

       A16: for n be Nat holds (G . n) = Gk(n) from SEQFUNC:sch 1;

      

       A17: for n be Nat holds ( dom (G . n)) = E & (G . n) in ( Lp_Functions (M,k)) & (G . n) is nonnegative & (G . n) is E -measurable & ( abs (G . n)) = (G . n)

      proof

        let n be Nat;

        

         A18: (G . n) = (( abs (F1 . 0 )) + (( Partial_Sums AbsFMF) . n)) by A16;

        

        then

         A19: ( dom (G . n)) = (( dom ( abs (F1 . 0 ))) /\ ( dom (( Partial_Sums AbsFMF) . n))) by VALUED_1:def 1

        .= (( dom (F1 . 0 )) /\ ( dom (( Partial_Sums AbsFMF) . n))) by VALUED_1:def 11

        .= (( dom (F1 . 0 )) /\ ( dom (AbsFMF . 0 ))) by MESFUN9C: 11;

        

         A20: (( Partial_Sums AbsFMF) . n) in ( Lp_Functions (M,k)) & (( Partial_Sums AbsFMF) . n) is nonnegative & (( Partial_Sums AbsFMF) . n) is E -measurable by A13, Th66, Th67, MESFUN9C: 16;

        

         A21: ( dom (AbsFMF . 0 )) = E by A13;

        

         A22: (F1 . 0 ) in ( Lp_Functions (M,k)) & ( dom (F1 . 0 )) = E & (F1 . 0 ) is E -measurable by A7;

        then ( abs (F1 . 0 )) in ( Lp_Functions (M,k)) & ( abs (F1 . 0 )) is nonnegative & ( abs (F1 . 0 )) is E -measurable by Th28, MESFUNC6: 48;

        hence thesis by A19, A22, A21, A18, A20, Th14, Th25, MESFUNC6: 26, MESFUNC6: 56;

      end;

      deffunc Gpk( Nat) = ((G . $1) to_power k);

      consider Gp be Functional_Sequence of X, REAL such that

       A23: for n be Nat holds (Gp . n) = Gpk(n) from SEQFUNC:sch 1;

      

       A24: for n be Nat holds ((G . n) to_power k) is nonnegative & ((G . n) to_power k) is E -measurable

      proof

        let n be Nat;

        

         A25: (G . n) is nonnegative by A17;

        hence ((G . n) to_power k) is nonnegative;

        (G . n) is E -measurable & ( dom (G . n)) = E by A17;

        hence ((G . n) to_power k) is E -measurable by A25, MESFUN6C: 29;

      end;

      reconsider ExtGp = ( R_EAL Gp) as Functional_Sequence of X, ExtREAL ;

      

       A26: for n be Nat holds ( dom (ExtGp . n)) = E & (ExtGp . n) is E -measurable & (ExtGp . n) is nonnegative

      proof

        let n be Nat;

        (ExtGp . n) = ( R_EAL ((G . n) to_power k)) by A23;

        then ( dom (ExtGp . n)) = ( dom (G . n)) by MESFUN6C:def 4;

        hence ( dom (ExtGp . n)) = E by A17;

        ((G . n) to_power k) is E -measurable by A24;

        then ( R_EAL ((G . n) to_power k)) is E -measurable;

        hence (ExtGp . n) is E -measurable by A23;

        ((G . n) to_power k) is nonnegative by A24;

        hence (ExtGp . n) is nonnegative by A23;

      end;

      then

       A27: ( dom (ExtGp . 0 )) = E & (ExtGp . 0 ) is nonnegative;

      for n,m be Nat holds ( dom (ExtGp . n)) = ( dom (ExtGp . m))

      proof

        let n,m be Nat;

        ( dom (ExtGp . n)) = E & ( dom (ExtGp . m)) = E by A26;

        hence thesis;

      end;

      then

      reconsider ExtGp as with_the_same_dom Functional_Sequence of X, ExtREAL by MESFUNC8:def 2;

      

       A28: for n,m be Nat st n <= m holds for x be Element of X st x in E holds ((ExtGp . n) . x) <= ((ExtGp . m) . x)

      proof

        let n,m be Nat;

        assume

         A29: n <= m;

        let x be Element of X;

        assume

         A30: x in E;

        then

         A31: x in ( dom (G . n)) & x in ( dom (G . m)) by A17;

        then x in ( dom ((G . n) to_power k)) & x in ( dom ((G . m) to_power k)) by MESFUN6C:def 4;

        then (((G . n) . x) to_power k) = (((G . n) to_power k) . x) & (((G . m) . x) to_power k) = (((G . m) to_power k) . x) by MESFUN6C:def 4;

        then

         A32: (((G . n) . x) to_power k) = ((ExtGp . n) . x) & (((G . m) . x) to_power k) = ((ExtGp . m) . x) by A23;

        ( dom (AbsFMF . 0 )) = E by A13;

        then ((( Partial_Sums AbsFMF) . n) . x) <= ((( Partial_Sums AbsFMF) . m) . x) by Th68, A29, A30, A13;

        then

         A33: ((( abs (F1 . 0 )) . x) + ((( Partial_Sums AbsFMF) . n) . x)) <= ((( abs (F1 . 0 )) . x) + ((( Partial_Sums AbsFMF) . m) . x)) by XREAL_1: 6;

        (G . m) = (( abs (F1 . 0 )) + (( Partial_Sums AbsFMF) . m)) & (G . n) = (( abs (F1 . 0 )) + (( Partial_Sums AbsFMF) . n)) by A16;

        then

         A34: ((G . m) . x) = ((( abs (F1 . 0 )) . x) + ((( Partial_Sums AbsFMF) . m) . x)) & ((G . n) . x) = ((( abs (F1 . 0 )) . x) + ((( Partial_Sums AbsFMF) . n) . x)) by A31, VALUED_1:def 1;

        (G . n) is nonnegative by A17;

        then 0 <= ((G . n) . x) by MESFUNC6: 51;

        hence thesis by A32, A33, A34, HOLDER_1: 3;

      end;

      

       A35: for x be Element of X st x in E holds (ExtGp # x) is non-decreasing

      proof

        let x be Element of X;

        assume

         A36: x in E;

        for n,m be Nat st m <= n holds ((ExtGp # x) . m) <= ((ExtGp # x) . n)

        proof

          let n,m be Nat;

          assume m <= n;

          then ((ExtGp . m) . x) <= ((ExtGp . n) . x) by A28, A36;

          then ((ExtGp # x) . m) <= ((ExtGp . n) . x) by MESFUNC5:def 13;

          hence thesis by MESFUNC5:def 13;

        end;

        hence (ExtGp # x) is non-decreasing by RINFSUP2: 7;

      end;

      

       A37: for x be Element of X st x in E holds (ExtGp # x) is convergent

      proof

        let x be Element of X;

        assume x in E;

        then (ExtGp # x) is non-decreasing by A35;

        hence thesis by RINFSUP2: 37;

      end;

      then

      consider I be ExtREAL_sequence such that

       A38: (for n be Nat holds (I . n) = ( Integral (M,(ExtGp . n)))) & I is convergent & ( Integral (M,( lim ExtGp))) = ( lim I) by A27, A26, A28, MESFUNC9: 52;

      now

        let y be object;

        assume y in ( rng I);

        then

        consider x be Element of NAT such that

         A39: y = (I . x) by FUNCT_2: 113;

        

         A40: y = ( Integral (M,(Gp . x))) by A39, A38;

        (G . x) = ( abs (G . x)) by A17;

        then

         A41: (Gp . x) = (( abs (G . x)) to_power k) by A23;

        (G . x) in ( Lp_Functions (M,k)) by A17;

        hence y in REAL by A40, A41, Th49;

      end;

      then ( rng I) c= REAL ;

      then

      reconsider Ir = I as sequence of REAL by FUNCT_2: 6;

      deffunc KAbsFMF( Nat) = ( Integral (M,((AbsFMF . $1) to_power k)));

      

       A42: for x be Element of NAT holds KAbsFMF(x) is Element of REAL

      proof

        let x be Element of NAT ;

        (AbsFMF . x) in ( Lp_Functions (M,k)) by A13;

        then ( Integral (M,(( abs (AbsFMF . x)) to_power k))) in REAL by Th49;

        hence thesis by A13;

      end;

      consider KPAbsFMF be sequence of REAL such that

       A43: for x be Element of NAT holds (KPAbsFMF . x) = KAbsFMF(x) from FUNCT_2:sch 9( A42);

      deffunc KKAbsFMF( Nat) = ((KPAbsFMF . $1) to_power (1 / k));

      

       A44: for x be Element of NAT holds KKAbsFMF(x) is Element of REAL by XREAL_0:def 1;

      consider PAbsFMF be sequence of REAL such that

       A45: for x be Element of NAT holds (PAbsFMF . x) = KKAbsFMF(x) from FUNCT_2:sch 9( A44);

      (F1 . 0 ) in ( Lp_Functions (M,k)) by A7;

      then

      reconsider RF0 = ( Integral (M,(( abs (F1 . 0 )) to_power k))) as Element of REAL by Th49;

      deffunc LAbsFMF( Nat) = ((RF0 to_power (1 / k)) + (( Partial_Sums PAbsFMF) . $1));

      

       A46: for x be Element of NAT holds LAbsFMF(x) is Element of REAL by XREAL_0:def 1;

      consider QAbsFMF be sequence of REAL such that

       A47: for x be Element of NAT holds (QAbsFMF . x) = LAbsFMF(x) from FUNCT_2:sch 9( A46);

      

       A48: for n be Nat holds ((Ir . n) to_power (1 / k)) <= (QAbsFMF . n)

      proof

        defpred PN[ Nat] means ((Ir . $1) to_power (1 / k)) <= (QAbsFMF . $1);

        

         A49: ( abs (F1 . 0 )) in ( Lp_Functions (M,k)) & (AbsFMF . 0 ) in ( Lp_Functions (M,k)) by A13, A7;

        (G . 0 ) = (( abs (F1 . 0 )) + (( Partial_Sums AbsFMF) . 0 )) by A16;

        then

         A50: (G . 0 ) = (( abs (F1 . 0 )) + (AbsFMF . 0 )) by MESFUN9C:def 2;

        (Ir . 0 ) = ( Integral (M,(Gp . 0 ))) by A38;

        then (Ir . 0 ) = ( Integral (M,((G . 0 ) to_power k))) by A23;

        then

         A51: (Ir . 0 ) = ( Integral (M,(( abs (( abs (F1 . 0 )) + (AbsFMF . 0 ))) to_power k))) by A17, A50;

        (KPAbsFMF . 0 ) = ( Integral (M,((AbsFMF . 0 ) to_power k))) by A43;

        then

         A52: (KPAbsFMF . 0 ) = ( Integral (M,(( abs (AbsFMF . 0 )) to_power k))) by A13;

        

         A53: RF0 = ( Integral (M,(( abs ( abs (F1 . 0 ))) to_power k)));

        (QAbsFMF . 0 ) = ((RF0 to_power (1 / k)) + (( Partial_Sums PAbsFMF) . 0 )) by A47;

        then (QAbsFMF . 0 ) = ((RF0 to_power (1 / k)) + (PAbsFMF . 0 )) by SERIES_1:def 1;

        then (QAbsFMF . 0 ) = ((RF0 to_power (1 / k)) + ((KPAbsFMF . 0 ) to_power (1 / k))) by A45;

        then

         A54: PN[ 0 ] by A1, A49, A51, A52, A53, Th61;

         A55:

        now

          let n be Nat;

          

           A56: n in NAT by ORDINAL1:def 12;

          assume PN[n];

          then

           A57: (((Ir . n) to_power (1 / k)) + (PAbsFMF . (n + 1))) <= ((QAbsFMF . n) + (PAbsFMF . (n + 1))) by XREAL_1: 6;

          (G . (n + 1)) = (( abs (F1 . 0 )) + (( Partial_Sums AbsFMF) . (n + 1))) by A16

          .= (( abs (F1 . 0 )) + ((( Partial_Sums AbsFMF) . n) + (AbsFMF . (n + 1)))) by MESFUN9C:def 2

          .= ((( abs (F1 . 0 )) + (( Partial_Sums AbsFMF) . n)) + (AbsFMF . (n + 1))) by RFUNCT_1: 8;

          then

           A58: (G . (n + 1)) = ((G . n) + (AbsFMF . (n + 1))) by A16;

          

           A59: (AbsFMF . (n + 1)) in ( Lp_Functions (M,k)) & (G . n) in ( Lp_Functions (M,k)) by A13, A17;

          (KPAbsFMF . (n + 1)) = ( Integral (M,((AbsFMF . (n + 1)) to_power k))) by A43;

          then

           A60: (KPAbsFMF . (n + 1)) = ( Integral (M,(( abs (AbsFMF . (n + 1))) to_power k))) by A13;

          

           A61: (PAbsFMF . (n + 1)) = ((KPAbsFMF . (n + 1)) to_power (1 / k)) by A45;

          (Ir . n) = ( Integral (M,(Gp . n))) & (Ir . (n + 1)) = ( Integral (M,(Gp . (n + 1)))) by A38;

          then (Ir . n) = ( Integral (M,((G . n) to_power k))) & (Ir . (n + 1)) = ( Integral (M,((G . (n + 1)) to_power k))) by A23;

          then (Ir . n) = ( Integral (M,(( abs (G . n)) to_power k))) & (Ir . (n + 1)) = ( Integral (M,(( abs ((G . n) + (AbsFMF . (n + 1)))) to_power k))) by A58, A17;

          then ((Ir . (n + 1)) to_power (1 / k)) <= (((Ir . n) to_power (1 / k)) + (PAbsFMF . (n + 1))) by A1, A59, A60, A61, Th61;

          then

           A62: ((Ir . (n + 1)) to_power (1 / k)) <= ((QAbsFMF . n) + (PAbsFMF . (n + 1))) by A57, XXREAL_0: 2;

          ((QAbsFMF . n) + (PAbsFMF . (n + 1))) = (((RF0 to_power (1 / k)) + (( Partial_Sums PAbsFMF) . n)) + (PAbsFMF . (n + 1))) by A47, A56

          .= ((RF0 to_power (1 / k)) + ((( Partial_Sums PAbsFMF) . n) + (PAbsFMF . (n + 1))))

          .= ((RF0 to_power (1 / k)) + (( Partial_Sums PAbsFMF) . (n + 1))) by SERIES_1:def 1;

          hence PN[(n + 1)] by A62, A47;

        end;

        for n be Nat holds PN[n] from NAT_1:sch 2( A54, A55);

        hence thesis;

      end;

      

       A63: for n be Nat holds (PAbsFMF . n) = ||.((Sq . (N . (n + 1))) - (Sq . (N . n))).||

      proof

        let n be Nat;

        

         A64: n in NAT by ORDINAL1:def 12;

        set m = (N . n);

        set m1 = (N . (n + 1));

        

         A65: (F1 . (n + 1)) = (Fsq . (N . (n + 1))) & (F1 . n) = (Fsq . (N . n)) by A6;

        (AbsFMF . n) = ( abs (FMF . n)) by SEQFUNC:def 4;

        then

         A66: (AbsFMF . n) = ( abs ((Fsq . (N . (n + 1))) - (Fsq . (N . n)))) by A65, A10;

        

         A67: (Fsq . (N . (n + 1))) in ( Lp_Functions (M,k)) & (Fsq . (N . (n + 1))) in (Sq . (N . (n + 1))) & (Fsq . (N . n)) in ( Lp_Functions (M,k)) & (Fsq . (N . n)) in (Sq . m) by A3;

        then (( - 1) (#) (Fsq . m)) in (( - 1) * (Sq . m)) by Th54;

        then ((Fsq . m1) - (Fsq . m)) in ((Sq . m1) + (( - 1) * (Sq . m))) by Th54, A67;

        then ((Fsq . m1) - (Fsq . m)) in ((Sq . m1) - (Sq . m)) by RLVECT_1: 16;

        then

         A68: ex r be Real st 0 <= r & r = ( Integral (M,(( abs ((Fsq . m1) - (Fsq . m))) to_power k))) & ||.((Sq . m1) - (Sq . m)).|| = (r to_power (1 / k)) by Th53;

        (PAbsFMF . n) = ((KPAbsFMF . n) to_power (1 / k)) by A45, A64;

        hence thesis by A68, A66, A43, A64;

      end;

      (1 / 2) < 1;

      then |.(1 / 2).| < 1 by ABSVALUE:def 1;

      then

       A69: ((1 / 2) GeoSeq ) is summable & ( Sum ((1 / 2) GeoSeq )) = (1 / (1 - (1 / 2))) by SERIES_1: 24;

      for n be Nat holds 0 <= (PAbsFMF . n) & (PAbsFMF . n) <= (((1 / 2) GeoSeq ) . n)

      proof

        let n be Nat;

        

         A70: (PAbsFMF . n) = ||.((Sq . (N . (n + 1))) - (Sq . (N . n))).|| by A63;

        hence 0 <= (PAbsFMF . n);

        (((1 / 2) GeoSeq ) . n) = ((1 / 2) |^ n) by PREPOWER:def 1

        .= ((1 / 2) to_power n) by POWER: 41;

        then

         A71: (((1 / 2) GeoSeq ) . n) = (2 to_power ( - n)) by POWER: 32;

        N is Real_Sequence by FUNCT_2: 7, NUMBERS: 19;

        then (N . n) < (N . (n + 1)) by SEQM_3:def 6;

        hence (PAbsFMF . n) <= (((1 / 2) GeoSeq ) . n) by A5, A70, A71;

      end;

      then PAbsFMF is summable & ( Sum PAbsFMF) <= ( Sum ((1 / 2) GeoSeq )) by A69, SERIES_1: 20;

      then ( Partial_Sums PAbsFMF) is convergent by SERIES_1:def 2;

      then ( Partial_Sums PAbsFMF) is bounded;

      then

      consider Br be Real such that

       A72: for n be Nat holds (( Partial_Sums PAbsFMF) . n) < Br by SEQ_2:def 3;

      for n be Nat holds (Ir . n) < (((RF0 to_power (1 / k)) + Br) to_power k)

      proof

        let n be Nat;

        

         A73: n in NAT by ORDINAL1:def 12;

        ((Ir . n) to_power (1 / k)) <= (QAbsFMF . n) by A48;

        then

         A74: ((Ir . n) to_power (1 / k)) <= ((RF0 to_power (1 / k)) + (( Partial_Sums PAbsFMF) . n)) by A47, A73;

        ((RF0 to_power (1 / k)) + (( Partial_Sums PAbsFMF) . n)) < ((RF0 to_power (1 / k)) + Br) by A72, XREAL_1: 8;

        then

         A75: ((Ir . n) to_power (1 / k)) < ((RF0 to_power (1 / k)) + Br) by A74, XXREAL_0: 2;

        (Ir . n) = ( Integral (M,(Gp . n))) by A38;

        then (Ir . n) = ( Integral (M,((G . n) to_power k))) by A23;

        then

         A76: (Ir . n) = ( Integral (M,(( abs (G . n)) to_power k))) by A17;

        

         A77: (G . n) in ( Lp_Functions (M,k)) by A17;

        then 0 <= ((Ir . n) to_power (1 / k)) by Th49, A76, Th4;

        then (((Ir . n) to_power (1 / k)) to_power k) < (((RF0 to_power (1 / k)) + Br) to_power k) by A75, Th3;

        then ((Ir . n) to_power ((1 / k) * k)) < (((RF0 to_power (1 / k)) + Br) to_power k) by A77, Th49, A76, HOLDER_1: 2;

        then ((Ir . n) to_power 1) < (((RF0 to_power (1 / k)) + Br) to_power k) by XCMPLX_1: 106;

        hence thesis by POWER: 25;

      end;

      then

       A78: Ir is bounded_above by SEQ_2:def 3;

      for n,m be Nat st n <= m holds (Ir . n) <= (Ir . m)

      proof

        let n,m be Nat;

        assume n <= m;

        then

         A79: for x be Element of X st x in E holds ((ExtGp . n) . x) <= ((ExtGp . m) . x) by A28;

        

         A80: (ExtGp . n) is E -measurable & (ExtGp . m) is E -measurable & (ExtGp . n) is nonnegative & (ExtGp . m) is nonnegative by A26;

        

         A81: ( dom (ExtGp . n)) = E & ( dom (ExtGp . m)) = E by A26;

        then

         A82: ((ExtGp . n) | E) = (ExtGp . n) & ((ExtGp . m) | E) = (ExtGp . m) by RELAT_1: 68;

        (I . n) = ( Integral (M,(ExtGp . n))) & (I . m) = ( Integral (M,(ExtGp . m))) by A38;

        hence thesis by A79, A81, A80, A82, MESFUNC9: 15;

      end;

      then Ir is non-decreasing by SEQM_3: 6;

      then

       A83: I is convergent_to_finite_number & ( lim I) = ( lim Ir) by A78, RINFSUP2: 14;

      reconsider LExtGp = ( lim ExtGp) as PartFunc of X, ExtREAL ;

      

       A84: E = ( dom LExtGp) & LExtGp is E -measurable by A26, A27, A37, MESFUNC8: 25, MESFUNC8:def 9;

      

       A85: for x be object st x in ( dom LExtGp) holds 0 <= (LExtGp . x)

      proof

        let x be object;

        assume

         A86: x in ( dom LExtGp);

        then

        reconsider x1 = x as Element of X;

        

         A87: x1 in E by A27, A86, MESFUNC8:def 9;

        now

          let k1 be Nat;

          reconsider k = k1 as Nat;

          (ExtGp # x1) is non-decreasing by A35, A87;

          then

           A88: ((ExtGp # x1) . 0 ) <= ((ExtGp # x1) . k) by RINFSUP2: 7;

           0 <= ((ExtGp . 0 ) . x1) by A27, SUPINF_2: 39;

          hence 0 <= ((ExtGp # x1) . k1) by A88, MESFUNC5:def 13;

        end;

        then 0 <= ( lim (ExtGp # x1)) by A87, A37, MESFUNC9: 10;

        hence thesis by A86, MESFUNC8:def 9;

      end;

      

       A89: ( eq_dom (LExtGp, +infty )) = (E /\ ( eq_dom (LExtGp, +infty ))) by A84, RELAT_1: 132, XBOOLE_1: 28;

      then

      reconsider EE = ( eq_dom (LExtGp, +infty )) as Element of S by A84, MESFUNC1: 33;

      reconsider E0 = (E \ EE) as Element of S;

      (E0 ` ) = ((X \ E) \/ (X /\ EE)) by XBOOLE_1: 52;

      then

       A90: (E0 ` ) = ((E ` ) \/ EE) by XBOOLE_1: 28;

      (M . EE) = 0 by A38, A83, A84, A85, A89, MESFUNC9: 13, SUPINF_2: 52;

      then

       A91: EE is measure_zero of M by MEASURE1:def 7;

      (E ` ) is Element of S by MEASURE1: 34;

      then (E ` ) is measure_zero of M by A4, MEASURE1:def 7;

      then (E0 ` ) is measure_zero of M by A90, A91, MEASURE1: 37;

      then

       A92: (M . (E0 ` )) = 0 by MEASURE1:def 7;

      

       A93: for x be Element of X st x in E0 holds (LExtGp . x) in REAL

      proof

        let x be Element of X;

        assume x in E0;

        then x in E & not x in EE by XBOOLE_0:def 5;

        then (LExtGp . x) <> +infty & 0 <= (LExtGp . x) by A84, A85, MESFUNC1:def 15;

        hence (LExtGp . x) in REAL by XXREAL_0: 14;

      end;

      

       A94: for x be Element of X st x in E0 holds (Gp # x) is convergent & ( lim (Gp # x)) = ( lim (ExtGp # x))

      proof

        let x be Element of X;

        assume

         A95: x in E0;

        then

         A96: x in E by XBOOLE_0:def 5;

        then (LExtGp . x) = ( lim (ExtGp # x)) by A84, MESFUNC8:def 9;

        then

         A97: ( lim (ExtGp # x)) in REAL by A93, A95;

        (ExtGp # x) is convergent by A37, A96;

        then

         A98: ex g be Real st ( lim (ExtGp # x)) = g & (for p be Real st 0 < p holds ex n be Nat st for m be Nat st n <= m holds |.(((ExtGp # x) . m) - ( lim (ExtGp # x))).| < p) & (ExtGp # x) is convergent_to_finite_number by A97, MESFUNC5:def 12;

        (ExtGp # x) = (Gp # x) by MESFUN7C: 1;

        hence thesis by A98, RINFSUP2: 15;

      end;

      

       A99: for x be Element of X st x in E0 holds for n be Nat holds ((Gp # x) . n) = (((G # x) . n) to_power k)

      proof

        let x be Element of X;

        assume

         A100: x in E0;

        hereby

          let n be Nat;

          x in E by A100, XBOOLE_0:def 5;

          then x in ( dom (G . n)) by A17;

          then

           A101: x in ( dom ((G . n) to_power k)) by MESFUN6C:def 4;

          ((Gp # x) . n) = ((Gp . n) . x) by SEQFUNC:def 10

          .= (((G . n) to_power k) . x) by A23

          .= (((G . n) . x) to_power k) by A101, MESFUN6C:def 4;

          hence ((Gp # x) . n) = (((G # x) . n) to_power k) by SEQFUNC:def 10;

        end;

      end;

      

       A102: for x be Element of X st x in E0 holds (( Partial_Sums AbsFMF) # x) is convergent

      proof

        let x be Element of X;

        assume

         A103: x in E0;

        then

         A104: (Gp # x) is convergent by A94;

         A105:

        now

          let n be Nat;

          (G . n) is nonnegative by A17;

          then 0 <= ((G . n) . x) by MESFUNC6: 51;

          hence 0 <= ((G # x) . n) by SEQFUNC:def 10;

        end;

        for n be Nat holds ((Gp # x) . n) = (((G # x) . n) to_power k) by A103, A99;

        then

         A106: (G # x) is convergent by A104, A105, Th9;

        now

          let s be Real;

          assume 0 < s;

          then

          consider n be Nat such that

           A107: for m be Nat st n <= m holds |.(((G # x) . m) - ((G # x) . n)) qua Complex.| < s by A106, SEQ_4: 41;

          now

            let m be Nat;

            assume

             A108: n <= m;

            x in E by A103, XBOOLE_0:def 5;

            then

             A109: x in ( dom (G . n)) & x in ( dom (G . m)) by A17;

            (G . m) = (( abs (F1 . 0 )) + (( Partial_Sums AbsFMF) . m)) & (G . n) = (( abs (F1 . 0 )) + (( Partial_Sums AbsFMF) . n)) by A16;

            then ((G . m) . x) = ((( abs (F1 . 0 )) . x) + ((( Partial_Sums AbsFMF) . m) . x)) & ((G . n) . x) = ((( abs (F1 . 0 )) . x) + ((( Partial_Sums AbsFMF) . n) . x)) by A109, VALUED_1:def 1;

            then ((G # x) . m) = ((( abs (F1 . 0 )) . x) + ((( Partial_Sums AbsFMF) . m) . x)) & ((G # x) . n) = ((( abs (F1 . 0 )) . x) + ((( Partial_Sums AbsFMF) . n) . x)) by SEQFUNC:def 10;

            then

             A110: (((G # x) . m) - ((G # x) . n)) = (((( Partial_Sums AbsFMF) . m) . x) - ((( Partial_Sums AbsFMF) . n) . x));

            ((( Partial_Sums AbsFMF) # x) . m) = ((( Partial_Sums AbsFMF) . m) . x) & ((( Partial_Sums AbsFMF) # x) . n) = ((( Partial_Sums AbsFMF) . n) . x) by SEQFUNC:def 10;

            hence |.(((( Partial_Sums AbsFMF) # x) . m) - ((( Partial_Sums AbsFMF) # x) . n)) qua Complex.| < s by A107, A108, A110;

          end;

          hence ex n be Nat st for m be Nat st n <= m holds |.(((( Partial_Sums AbsFMF) # x) . m) - ((( Partial_Sums AbsFMF) # x) . n)) qua Complex.| < s;

        end;

        hence thesis by SEQ_4: 41;

      end;

      

       A111: for x be Element of X st x in E0 holds ( Partial_Sums ( abs (FMF # x))) = (( Partial_Sums AbsFMF) # x)

      proof

        let x be Element of X;

        assume x in E0;

        then

         A112: x in E by XBOOLE_0:def 5;

        defpred PQ[ Nat] means (( Partial_Sums ( abs (FMF # x))) . $1) = ((( Partial_Sums AbsFMF) # x) . $1);

        (( Partial_Sums ( abs (FMF # x))) . 0 ) = (( abs (FMF # x)) . 0 ) by SERIES_1:def 1

        .= |.((FMF # x) . 0 ).| by VALUED_1: 18

        .= |.((FMF . 0 ) . x).| by SEQFUNC:def 10

        .= (( abs (FMF . 0 )) . x) by VALUED_1: 18

        .= ((AbsFMF . 0 ) . x) by SEQFUNC:def 4

        .= ((( Partial_Sums AbsFMF) . 0 ) . x) by MESFUN9C:def 2

        .= ((( Partial_Sums AbsFMF) # x) . 0 ) by SEQFUNC:def 10;

        then

         A113: PQ[ 0 ];

         A114:

        now

          let n be Nat;

          assume

           A115: PQ[n];

          

           A116: (( Partial_Sums AbsFMF) . (n + 1)) = ((( Partial_Sums AbsFMF) . n) + (AbsFMF . (n + 1))) by MESFUN9C:def 2;

          ( dom (AbsFMF . 0 )) = E by A13;

          then

           A117: x in ( dom (( Partial_Sums AbsFMF) . (n + 1))) by A112, MESFUN9C: 11;

          

           A118: (( abs (FMF # x)) . (n + 1)) = |.((FMF # x) . (n + 1)).| by VALUED_1: 18

          .= |.((FMF . (n + 1)) . x).| by SEQFUNC:def 10

          .= (( abs (FMF . (n + 1))) . x) by VALUED_1: 18

          .= ((AbsFMF . (n + 1)) . x) by SEQFUNC:def 4;

          (( Partial_Sums ( abs (FMF # x))) . (n + 1)) = ((( Partial_Sums ( abs (FMF # x))) . n) + (( abs (FMF # x)) . (n + 1))) by SERIES_1:def 1

          .= (((( Partial_Sums AbsFMF) . n) . x) + ((AbsFMF . (n + 1)) . x)) by A115, A118, SEQFUNC:def 10

          .= ((( Partial_Sums AbsFMF) . (n + 1)) . x) by A116, A117, VALUED_1:def 1

          .= ((( Partial_Sums AbsFMF) # x) . (n + 1)) by SEQFUNC:def 10;

          hence PQ[(n + 1)];

        end;

        for n be Nat holds PQ[n] from NAT_1:sch 2( A113, A114);

        then for n be Element of NAT holds PQ[n];

        hence thesis by FUNCT_2: 63;

      end;

      

       A119: for x be Element of X st x in E0 holds for n be Nat holds ((F1 # x) . (n + 1)) = (((F1 # x) . 0 ) + (( Partial_Sums (FMF # x)) . n))

      proof

        let x be Element of X;

        assume x in E0;

        then

         A120: x in E by XBOOLE_0:def 5;

        defpred PQ[ Nat] means ((F1 # x) . ($1 + 1)) = (((F1 # x) . 0 ) + (( Partial_Sums (FMF # x)) . $1));

        ( dom (FMF . 0 )) = E by A11;

        then

         A121: x in ( dom ((F1 . ( 0 + 1)) - (F1 . 0 ))) by A10, A120;

        (( Partial_Sums (FMF # x)) . 0 ) = ((FMF # x) . 0 ) by SERIES_1:def 1

        .= ((FMF . 0 ) . x) by SEQFUNC:def 10

        .= (((F1 . ( 0 + 1)) - (F1 . 0 )) . x) by A10;

        then

         A122: (( Partial_Sums (FMF # x)) . 0 ) = (((F1 . ( 0 + 1)) . x) - ((F1 . 0 ) . x)) by A121, VALUED_1: 13;

        (((F1 # x) . 0 ) + (( Partial_Sums (FMF # x)) . 0 )) = (((F1 . 0 ) . x) + (( Partial_Sums (FMF # x)) . 0 )) by SEQFUNC:def 10;

        then

         A123: PQ[ 0 ] by A122, SEQFUNC:def 10;

         A124:

        now

          let n be Nat;

          assume

           A125: PQ[n];

          ( dom (FMF . (n + 1))) = E by A11;

          then

           A126: x in ( dom ((F1 . ((n + 1) + 1)) - (F1 . (n + 1)))) by A10, A120;

          ((FMF # x) . (n + 1)) = ((FMF . (n + 1)) . x) by SEQFUNC:def 10

          .= (((F1 . ((n + 1) + 1)) - (F1 . (n + 1))) . x) by A10;

          then

           A127: ((FMF # x) . (n + 1)) = (((F1 . ((n + 1) + 1)) . x) - ((F1 . (n + 1)) . x)) by A126, VALUED_1: 13;

          (((F1 # x) . 0 ) + (( Partial_Sums (FMF # x)) . (n + 1))) = (((F1 # x) . 0 ) + ((( Partial_Sums (FMF # x)) . n) + ((FMF # x) . (n + 1)))) by SERIES_1:def 1

          .= ((((F1 # x) . 0 ) + (( Partial_Sums (FMF # x)) . n)) + ((FMF # x) . (n + 1)))

          .= (((F1 . (n + 1)) . x) + ((FMF # x) . (n + 1))) by A125, SEQFUNC:def 10;

          hence PQ[(n + 1)] by A127, SEQFUNC:def 10;

        end;

        for n be Nat holds PQ[n] from NAT_1:sch 2( A123, A124);

        hence thesis;

      end;

      

       A128: for x be Element of X st x in E0 holds (F1 # x) is convergent

      proof

        let x be Element of X;

        assume

         A129: x in E0;

        then ( Partial_Sums ( abs (FMF # x))) = (( Partial_Sums AbsFMF) # x) by A111;

        then ( Partial_Sums ( abs (FMF # x))) is convergent by A129, A102;

        then ( abs (FMF # x)) is summable by SERIES_1:def 2;

        then (FMF # x) is absolutely_summable by SERIES_1:def 4;

        then (FMF # x) is summable;

        then

         A130: ( Partial_Sums (FMF # x)) is convergent by SERIES_1:def 2;

        now

          let s be Real;

          assume 0 < s;

          then

          consider n be Nat such that

           A131: for m be Nat st n <= m holds |.((( Partial_Sums (FMF # x)) . m) - (( Partial_Sums (FMF # x)) . n)) qua Complex.| < s by A130, SEQ_4: 41;

          set n1 = (n + 1);

          now

            let m1 be Nat;

            assume

             A132: n1 <= m1;

            1 <= n1 by NAT_1: 11;

            then

            reconsider m = (m1 - 1) as Nat by A132, NAT_1: 21, XXREAL_0: 2;

            (n1 - 1) <= (m1 - 1) by A132, XREAL_1: 9;

            then

             A133: |.((( Partial_Sums (FMF # x)) . m) - (( Partial_Sums (FMF # x)) . n)).| < s by A131;

            m1 = (m + 1);

            then ((F1 # x) . n1) = (((F1 # x) . 0 ) + (( Partial_Sums (FMF # x)) . n)) & ((F1 # x) . m1) = (((F1 # x) . 0 ) + (( Partial_Sums (FMF # x)) . m)) by A119, A129;

            hence |.(((F1 # x) . m1) - ((F1 # x) . n1)) qua Complex.| < s by A133;

          end;

          hence ex n be Nat st for m be Nat st n <= m holds |.(((F1 # x) . m) - ((F1 # x) . n)) qua Complex.| < s;

        end;

        hence thesis by SEQ_4: 41;

      end;

      set F2 = (F1 || E0);

      

       A134: for x be Element of X st x in E0 holds (F2 # x) is convergent

      proof

        let x be Element of X;

        assume

         A135: x in E0;

        then (F1 # x) is convergent by A128;

        hence thesis by A135, MESFUN9C: 1;

      end;

      

       A136: for x be Element of X st x in E0 holds (F2 # x) = (F1 # x)

      proof

        let x be Element of X;

        assume

         A137: x in E0;

        now

          let n be Element of NAT ;

          ((F2 # x) . n) = ((F2 . n) . x) by SEQFUNC:def 10

          .= (((F1 . n) | E0) . x) by MESFUN9C:def 1

          .= ((F1 . n) . x) by A137, FUNCT_1: 49;

          hence ((F2 # x) . n) = ((F1 # x) . n) by SEQFUNC:def 10;

        end;

        hence thesis by FUNCT_2: 63;

      end;

      

       A138: for n be Nat holds ( dom (F2 . n)) = E0 & (F2 . n) is E0 -measurable

      proof

        let n be Nat;

        

         A139: ( dom (F1 . 0 )) = E by A7;

        ( dom (F2 . n)) = ( dom ((F1 . n) | E0)) by MESFUN9C:def 1;

        then ( dom (F2 . n)) = (( dom (F1 . n)) /\ E0) by RELAT_1: 61;

        then ( dom (F2 . n)) = (E /\ E0) by A7;

        hence ( dom (F2 . n)) = E0 by XBOOLE_1: 28, XBOOLE_1: 36;

        for m be Nat holds (F1 . m) is E0 -measurable

        proof

          let m be Nat;

          (F1 . m) is E -measurable by A7;

          hence (F1 . m) is E0 -measurable by MESFUNC6: 16, XBOOLE_1: 36;

        end;

        hence (F2 . n) is E0 -measurable by A139, MESFUN9C: 4, XBOOLE_1: 36;

      end;

      reconsider F2 as with_the_same_dom Functional_Sequence of X, REAL by MESFUN9C: 2;

      

       A140: for n be Nat holds (F2 . n) in ( Lp_Functions (M,k)) & (F2 . n) in (Sq . (N . n))

      proof

        let n1 be Nat;

        (F2 . n1) = ((F1 . n1) | E0) by MESFUN9C:def 1;

        then ( abs (F2 . n1)) = (( abs (F1 . n1)) | E0) by Th13;

        then

         A141: ((( abs (F1 . n1)) to_power k) | E0) = (( abs (F2 . n1)) to_power k) by Th20;

        

         A142: (F2 . n1) is E0 -measurable & ( dom (F2 . n1)) = E0 by A138;

        (F1 . n1) in ( Lp_Functions (M,k)) by A7;

        then ex FMF be PartFunc of X, REAL st (F1 . n1) = FMF & ex ND be Element of S st (M . (ND ` )) = 0 & ( dom FMF) = ND & FMF is ND -measurable & (( abs FMF) to_power k) is_integrable_on M;

        then (( abs (F2 . n1)) to_power k) is_integrable_on M by A141, MESFUNC6: 91;

        hence

         A143: (F2 . n1) in ( Lp_Functions (M,k)) by A142, A92;

        reconsider n = n1 as Nat;

        set m = (N . n);

        (F1 . n) = (Fsq . m) by A6;

        then

         A144: (F1 . n) in (Sq . (N . n)) & (Sq . (N . n)) = ( a.e-eq-class_Lp ((F1 . n),M,k)) by A3;

        reconsider EB = (E0 ` ) as Element of S by MEASURE1: 34;

        ((F2 . n) | (EB ` )) = (F2 . n) by A142, RELAT_1: 68;

        then ((F2 . n) | (EB ` )) = ((F1 . n) | (EB ` )) by MESFUN9C:def 1;

        then (F2 . n) a.e.= ((F1 . n),M) by A92;

        hence thesis by A143, A144, Th36;

      end;

      

       A145: ( dom ( lim F2)) = ( dom (F2 . 0 )) by MESFUNC8:def 9;

      then

       A146: ( dom ( lim F2)) = E0 by A138;

      

       A147: for x be Element of X st x in E0 holds (( lim F2) . x) = ( lim (F2 # x))

      proof

        let x be Element of X;

        assume x in E0;

        then (( lim F2) . x) = ( lim ( R_EAL (F2 # x))) & (F2 # x) is convergent by A146, A134, MESFUN7C: 14;

        hence (( lim F2) . x) = ( lim (F2 # x)) by RINFSUP2: 14;

      end;

      now

        let y be object;

        assume y in ( rng ( lim F2));

        then

        consider x be Element of X such that

         A148: x in ( dom ( lim F2)) & y = (( lim F2) . x) by PARTFUN1: 3;

        y = ( lim (F2 # x)) by A148, A146, A147;

        hence y in REAL by XREAL_0:def 1;

      end;

      then ( rng ( lim F2)) c= REAL ;

      then

      reconsider F = ( lim F2) as PartFunc of X, REAL by A145, RELSET_1: 4;

      

       A149: ( dom (LExtGp | E0)) = (E /\ E0) by A84, RELAT_1: 61;

      then

       A150: ( dom (LExtGp | E0)) = E0 by XBOOLE_1: 28, XBOOLE_1: 36;

      now

        let y be object;

        assume y in ( rng (LExtGp | E0));

        then

        consider x be Element of X such that

         A151: x in ( dom (LExtGp | E0)) & y = ((LExtGp | E0) . x) by PARTFUN1: 3;

        y = (LExtGp . x) by A150, A151, FUNCT_1: 49;

        hence y in REAL by A150, A151, A93;

      end;

      then ( rng (LExtGp | E0)) c= REAL ;

      then

      reconsider gp = (LExtGp | E0) as PartFunc of X, REAL by A149, RELSET_1: 4;

      

       A152: for x be Element of X st x in E0 holds (gp . x) = ( lim (Gp # x))

      proof

        let x be Element of X;

        assume

         A153: x in E0;

        then x in ( dom LExtGp) by A84, XBOOLE_0:def 5;

        then (LExtGp . x) = ( lim (ExtGp # x)) by MESFUNC8:def 9;

        then (gp . x) = ( lim (ExtGp # x)) by A153, FUNCT_1: 49;

        hence (gp . x) = ( lim (Gp # x)) by A94, A153;

      end;

      

       A154: LExtGp is nonnegative by A85, SUPINF_2: 52;

      ( Integral (M,LExtGp)) in REAL by A83, A38, XREAL_0:def 1;

      then LExtGp is_integrable_on M by A154, A84, Th2;

      then ( R_EAL gp) is_integrable_on M by MESFUNC5: 97;

      then

       A155: gp is_integrable_on M;

      

       A156: ( dom (F2 . 0 )) = E0 by A138;

      then

       A157: ( dom F) = E0 by MESFUNC8:def 9;

      then

       A158: E0 = ( dom ( abs F)) by VALUED_1:def 11;

      then

       A159: E0 = ( dom (( abs F) to_power k)) by MESFUN6C:def 4;

      

       A160: for x be Element of X, n be Nat st x in E0 holds ( |.((F1 # x) . 0 ) qua Complex.| + |.(( Partial_Sums (FMF # x)) . n) qua Complex.|) <= ((G # x) . n)

      proof

        let x be Element of X, n be Nat;

        assume

         A161: x in E0;

        then x in E by XBOOLE_0:def 5;

        then

         A162: x in ( dom (G . n)) by A17;

        (G . n) = (( abs (F1 . 0 )) + (( Partial_Sums AbsFMF) . n)) by A16;

        then ((G . n) . x) = ((( abs (F1 . 0 )) . x) + ((( Partial_Sums AbsFMF) . n) . x)) by A162, VALUED_1:def 1;

        then

         A163: ((G . n) . x) = ( |.((F1 . 0 ) . x) qua Complex.| + ((( Partial_Sums AbsFMF) . n) . x)) by VALUED_1: 18;

        ((G # x) . n) = ((G . n) . x) by SEQFUNC:def 10

        .= ( |.((F1 . 0 ) . x) qua Complex.| + ((( Partial_Sums AbsFMF) # x) . n)) by A163, SEQFUNC:def 10

        .= ( |.((F1 # x) . 0 ) qua Complex.| + ((( Partial_Sums AbsFMF) # x) . n)) by SEQFUNC:def 10;

        then

         A164: ((G # x) . n) = ( |.((F1 # x) . 0 ) qua Complex.| + (( Partial_Sums ( abs (FMF # x))) . n)) by A111, A161;

         |.(( Partial_Sums (FMF # x)) . n) qua Complex.| <= (( Partial_Sums ( abs (FMF # x))) . n) by Lm1;

        hence thesis by A164, XREAL_1: 6;

      end;

      

       A165: for x be Element of X, n be Nat st x in E0 holds ( |.(((F1 # x) . 0 ) + (( Partial_Sums (FMF # x)) . n)) qua Complex.| to_power k) <= ((Gp # x) . n)

      proof

        let x be Element of X, n be Nat;

        assume

         A166: x in E0;

        then

         A167: ((Gp # x) . n) = (((G # x) . n) to_power k) by A99;

        

         A168: ( |.((F1 # x) . 0 ) qua Complex.| + |.(( Partial_Sums (FMF # x)) . n) qua Complex.|) <= ((G # x) . n) by A160, A166;

         |.(((F1 # x) . 0 ) + (( Partial_Sums (FMF # x)) . n)) qua Complex.| <= ( |.((F1 # x) . 0 ) qua Complex.| + |.(( Partial_Sums (FMF # x)) . n) qua Complex.|) by COMPLEX1: 56;

        then

         A169: |.(((F1 # x) . 0 ) + (( Partial_Sums (FMF # x)) . n)).| <= ((G # x) . n) by A168, XXREAL_0: 2;

         0 <= |.(((F1 # x) . 0 ) + (( Partial_Sums (FMF # x)) . n)).| by COMPLEX1: 46;

        hence thesis by A167, A169, HOLDER_1: 3;

      end;

      

       A170: for x be Element of X, n be Nat st x in E0 holds ( |.((F2 # x) . n) qua Complex.| to_power k) <= ((Gp # x) . n)

      proof

        let x be Element of X, n be Nat;

        assume

         A171: x in E0;

        then

         A172: (F1 # x) = (F2 # x) by A136;

        per cases ;

          suppose

           A173: n = 0 ;

          

           A174: ((Gp # x) . n) = (((G # x) . n) to_power k) by A171, A99;

          

           A175: ( |.((F1 # x) . 0 ) qua Complex.| + |.(( Partial_Sums (FMF # x)) . n) qua Complex.|) <= ((G # x) . n) by A160, A171;

           0 <= |.(( Partial_Sums (FMF # x)) . n).| by COMPLEX1: 46;

          then ( 0 + |.((F1 # x) . 0 ) qua Complex.|) <= ( |.((F1 # x) . 0 ) qua Complex.| + |.(( Partial_Sums (FMF # x)) . n) qua Complex.|) by XREAL_1: 6;

          then

           A176: |.((F1 # x) . 0 ) qua Complex.| <= ((G # x) . n) by A175, XXREAL_0: 2;

           0 <= |.((F1 # x) . 0 ).| by COMPLEX1: 46;

          hence ( |.((F2 # x) . n) qua Complex.| to_power k) <= ((Gp # x) . n) by A172, A173, A174, A176, HOLDER_1: 3;

        end;

          suppose n <> 0 ;

          then

          consider m be Nat such that

           A177: n = (m + 1) by NAT_1: 6;

          reconsider m as Nat;

          ((F1 # x) . (m + 1)) = (((F1 # x) . 0 ) + (( Partial_Sums (FMF # x)) . m)) by A119, A171;

          then

           A178: ( |.((F1 # x) . (m + 1)) qua Complex.| to_power k) <= ((Gp # x) . m) by A165, A171;

          x in E by A171, XBOOLE_0:def 5;

          then

           A179: (ExtGp # x) is non-decreasing by A35;

          

           A180: ((ExtGp # x) . m) <= ((ExtGp # x) . (m + 1)) by A179;

          (ExtGp # x) = (Gp # x) by MESFUN7C: 1;

          hence ( |.((F2 # x) . n) qua Complex.| to_power k) <= ((Gp # x) . n) by A172, A177, A178, A180, XXREAL_0: 2;

        end;

      end;

      

       A181: for x be Element of X st x in E0 holds |.((( abs F) to_power k) . x) qua Complex.| <= (gp . x)

      proof

        let x be Element of X;

        assume

         A182: x in E0;

        then

         A183: (Gp # x) is convergent by A94;

        deffunc ABSF2( set) = ((( abs (F2 # x)) . $1) to_power k);

        consider s be Real_Sequence such that

         A184: for n be Nat holds (s . n) = ABSF2(n) from SEQ_1:sch 1;

        

         A185: (gp . x) = ( lim (Gp # x)) by A152, A182;

        

         A186: ((( abs F) to_power k) . x) = ((( abs F) . x) to_power k) by A159, A182, MESFUN6C:def 4

        .= ( |.(F . x) qua Complex.| to_power k) by A158, A182, VALUED_1:def 11

        .= ( |.( lim (F2 # x)) qua Complex.| to_power k) by A182, A147

        .= (( lim ( abs (F2 # x))) to_power k) by A134, A182, SEQ_4: 14;

         A187:

        now

          let n be Nat;

           0 <= |.((F2 # x) . n).| by COMPLEX1: 46;

          hence 0 <= (( abs (F2 # x)) . n) by VALUED_1: 18;

        end;

        ( abs (F2 # x)) is convergent by A182, A134, SEQ_4: 13;

        then

         A188: s is convergent & (( lim ( abs (F2 # x))) to_power k) = ( lim s) by A187, A184, HOLDER_1: 10;

        now

          let n be Nat;

          ( |.((F2 # x) . n) qua Complex.| to_power k) <= ((Gp # x) . n) by A170, A182;

          then ((( abs (F2 # x)) . n) to_power k) <= ((Gp # x) . n) by VALUED_1: 18;

          hence (s . n) <= ((Gp # x) . n) by A184;

        end;

        then

         A189: ((( abs F) to_power k) . x) <= (gp . x) by A188, A185, A186, A183, SEQ_2: 18;

         0 <= ((( abs F) to_power k) . x) by MESFUNC6: 51;

        hence |.((( abs F) to_power k) . x) qua Complex.| <= (gp . x) by A189, ABSVALUE:def 1;

      end;

      ( R_EAL F) is E0 -measurable by A138, A156, A134, MESFUN7C: 21;

      then

       A190: F is E0 -measurable;

      then

       A191: ( abs F) is E0 -measurable by A157, MESFUNC6: 48;

      ( dom ( abs F)) = E0 by A157, VALUED_1:def 11;

      then (( abs F) to_power k) is E0 -measurable by A191, MESFUN6C: 29;

      then (( abs F) to_power k) is_integrable_on M by A150, A155, A159, A181, MESFUNC6: 96;

      then

       A192: F in ( Lp_Functions (M,k)) by A92, A157, A190;

      

       A193: for x be Element of X, n,m be Nat st x in E0 & m <= n holds ( |.(((F1 # x) . n) - ((F1 # x) . m)) qua Complex.| to_power k) <= ((Gp # x) . n)

      proof

        let x be Element of X, n1,m1 be Nat;

        assume

         A194: x in E0 & m1 <= n1;

        now

          per cases ;

            suppose

             A195: m1 = 0 ;

            now

              per cases ;

                suppose

                 A196: n1 = 0 ;

                ((G . n1) to_power k) is nonnegative by A24;

                then (Gp . n1) is nonnegative by A23;

                then 0 <= ((Gp . n1) . x) by MESFUNC6: 51;

                then 0 <= ((Gp # x) . n1) by SEQFUNC:def 10;

                hence ( |.(((F1 # x) . n1) - ((F1 # x) . m1)) qua Complex.| to_power k) <= ((Gp # x) . n1) by A195, A196, COMPLEX1: 44, POWER:def 2;

              end;

                suppose n1 <> 0 ;

                then

                consider n be Nat such that

                 A197: n1 = (n + 1) by NAT_1: 6;

                reconsider n as Nat;

                

                 A198: ((F1 # x) . (n + 1)) = (((F1 # x) . 0 ) + (( Partial_Sums (FMF # x)) . n)) by A194, A119;

                

                 A199: ( |.((F1 # x) . 0 ) qua Complex.| + |.(( Partial_Sums (FMF # x)) . n) qua Complex.|) <= ((G # x) . n) by A160, A194;

                 0 <= |.((F1 # x) . 0 ).| by COMPLEX1: 46;

                then ( |.(( Partial_Sums (FMF # x)) . n) qua Complex.| + 0 ) <= ( |.((F1 # x) . 0 ) qua Complex.| + |.(( Partial_Sums (FMF # x)) . n) qua Complex.|) by XREAL_1: 6;

                then

                 A200: |.(( Partial_Sums (FMF # x)) . n) qua Complex.| <= ((G # x) . n) by A199, XXREAL_0: 2;

                 0 <= |.(( Partial_Sums (FMF # x)) . n).| by COMPLEX1: 46;

                then

                 A201: ( |.(( Partial_Sums (FMF # x)) . n) qua Complex.| to_power k) <= (((G # x) . n) to_power k) by A200, HOLDER_1: 3;

                

                 A202: ((Gp # x) . n) = (((G # x) . n) to_power k) by A194, A99;

                x in E by A194, XBOOLE_0:def 5;

                then

                 A203: (ExtGp # x) is non-decreasing by A35;

                

                 A204: ((ExtGp # x) . n) <= ((ExtGp # x) . (n + 1)) by A203;

                (ExtGp # x) = (Gp # x) by MESFUN7C: 1;

                hence ( |.(((F1 # x) . n1) - ((F1 # x) . m1)) qua Complex.| to_power k) <= ((Gp # x) . n1) by A195, A197, A204, A201, A202, A198, XXREAL_0: 2;

              end;

            end;

            hence ( |.(((F1 # x) . n1) - ((F1 # x) . m1)) qua Complex.| to_power k) <= ((Gp # x) . n1);

          end;

            suppose

             A205: m1 <> 0 ;

            then

            consider m be Nat such that

             A206: m1 = (m + 1) by NAT_1: 6;

            reconsider m as Nat;

             0 < n1 by A194, A205;

            then

            consider n be Nat such that

             A207: n1 = (n + 1) by NAT_1: 6;

            reconsider n as Element of NAT by ORDINAL1:def 12;

            

             A208: (m1 - 1) <= (n1 - 1) by A194, XREAL_1: 9;

            x in E by A194, XBOOLE_0:def 5;

            then

             A209: x in ( dom (G . n)) by A17;

            then

             A210: x in ( dom ((G . n) to_power k)) by MESFUN6C:def 4;

            ((Gp # x) . n) = ((Gp . n) . x) by SEQFUNC:def 10;

            then ((Gp # x) . n) = (((G . n) to_power k) . x) by A23;

            then

             A211: ((Gp # x) . n) = (((G . n) . x) to_power k) by A210, MESFUN6C:def 4;

            (G . n) = (( abs (F1 . 0 )) + (( Partial_Sums AbsFMF) . n)) by A16;

            

            then ((G . n) . x) = ((( abs (F1 . 0 )) . x) + ((( Partial_Sums AbsFMF) . n) . x)) by A209, VALUED_1:def 1

            .= ( |.((F1 . 0 ) . x) qua Complex.| + ((( Partial_Sums AbsFMF) . n) . x)) by VALUED_1: 18;

            then

             A212: ((G . n) . x) = ( |.((F1 . 0 ) . x) qua Complex.| + ((( Partial_Sums AbsFMF) # x) . n)) by SEQFUNC:def 10;

            

             A213: ((F1 # x) . (n + 1)) = (((F1 # x) . 0 ) + (( Partial_Sums (FMF # x)) . n)) & ((F1 # x) . (m + 1)) = (((F1 # x) . 0 ) + (( Partial_Sums (FMF # x)) . m)) by A194, A119;

            

             A214: |.((( Partial_Sums (FMF # x)) . n) - (( Partial_Sums (FMF # x)) . m)).| <= (( Partial_Sums ( abs (FMF # x))) . n) by Th10, A206, A207, A208;

            

             A215: (( Partial_Sums ( abs (FMF # x))) . n) = ((( Partial_Sums AbsFMF) # x) . n) by A111, A194;

             0 <= |.((F1 . 0 ) . x).| by COMPLEX1: 46;

            then ( 0 + (( Partial_Sums ( abs (FMF # x))) . n)) <= ( |.((F1 . 0 ) . x) qua Complex.| + ((( Partial_Sums AbsFMF) # x) . n)) by A215, XREAL_1: 6;

            then

             A216: |.(((F1 # x) . (n + 1)) - ((F1 # x) . (m + 1))).| <= ((G . n) . x) by A212, A213, A214, XXREAL_0: 2;

             0 <= |.(((F1 # x) . (n + 1)) - ((F1 # x) . (m + 1))).| by COMPLEX1: 46;

            then

             A217: ( |.(((F1 # x) . (n + 1)) - ((F1 # x) . (m + 1))) qua Complex.| to_power k) <= ((Gp # x) . n) by A211, A216, HOLDER_1: 3;

            x in E by A194, XBOOLE_0:def 5;

            then

             A218: (ExtGp # x) is non-decreasing by A35;

            

             A219: ((ExtGp # x) . n) <= ((ExtGp # x) . (n + 1)) by A218;

            (ExtGp # x) = (Gp # x) by MESFUN7C: 1;

            hence ( |.(((F1 # x) . n1) - ((F1 # x) . m1)) qua Complex.| to_power k) <= ((Gp # x) . n1) by A206, A207, A219, A217, XXREAL_0: 2;

          end;

        end;

        hence thesis;

      end;

      

       A220: for x be Element of X, n be Nat st x in E0 holds ( |.((F . x) - ((F2 # x) . n)) qua Complex.| to_power k) <= (gp . x)

      proof

        let x be Element of X, n1 be Nat;

        assume

         A221: x in E0;

        then

         A222: (Gp # x) is convergent by A94;

        

         A223: (F1 # x) = (F2 # x) by A136, A221;

        

         A224: (F2 # x) is convergent by A221, A134;

        reconsider n = n1 as Nat;

        deffunc AF2F20( Nat) = (((F2 # x) . $1) - ((F2 # x) . n));

        consider s0 be Real_Sequence such that

         A225: for j be Nat holds (s0 . j) = AF2F20(j) from SEQ_1:sch 1;

         A226:

        now

          let p be Real;

          assume 0 < p;

          then

          consider n1 be Nat such that

           A227: for m be Nat st n1 <= m holds |.(((F2 # x) . m) - ( lim (F2 # x))) qua Complex.| < p by A224, SEQ_2:def 7;

          take n1;

          thus for m be Nat st n1 <= m holds |.((s0 . m) - (( lim (F2 # x)) - ((F2 # x) . n))) qua Complex.| < p

          proof

            let m be Nat;

            assume

             A228: n1 <= m;

            ((s0 . m) - (( lim (F2 # x)) - ((F2 # x) . n))) = ((((F2 # x) . m) - ((F2 # x) . n)) - (( lim (F2 # x)) - ((F2 # x) . n))) by A225;

            then ((s0 . m) - (( lim (F2 # x)) - ((F2 # x) . n))) = (((F2 # x) . m) - ( lim (F2 # x)));

            hence |.((s0 . m) - (( lim (F2 # x)) - ((F2 # x) . n))) qua Complex.| < p by A228, A227;

          end;

        end;

        then

         A229: s0 is convergent by SEQ_2:def 6;

        then ( lim s0) = (( lim (F2 # x)) - ((F2 # x) . n)) by A226, SEQ_2:def 7;

        then

         A230: ( lim ( abs s0)) = |.(( lim (F2 # x)) - ((F2 # x) . n)).| by A229, SEQ_4: 14;

        

         A231: ( abs s0) is convergent by A229;

        deffunc AF2F2( Nat) = ( |.(((F2 # x) . $1) - ((F2 # x) . n)) qua Complex.| to_power k);

        consider s be Real_Sequence such that

         A232: for j be Nat holds (s . j) = AF2F2(j) from SEQ_1:sch 1;

        

         A233: for j be Nat st n <= j holds (s . j) <= ((Gp # x) . j)

        proof

          let j be Nat;

          assume n <= j;

          then ( |.(((F2 # x) . j) - ((F2 # x) . n)) qua Complex.| to_power k) <= ((Gp # x) . j) by A223, A221, A193;

          hence thesis by A232;

        end;

         A234:

        now

          let n be Nat;

           0 <= |.(s0 . n).| by COMPLEX1: 46;

          hence 0 <= (( abs s0) . n) by VALUED_1: 18;

        end;

        now

          let j be Nat;

          

          thus (s . j) = ( |.(((F2 # x) . j) - ((F2 # x) . n)) qua Complex.| to_power k) by A232

          .= ( |.(s0 . j) qua Complex.| to_power k) by A225

          .= ((( abs s0) . j) to_power k) by VALUED_1: 18;

        end;

        then

         A235: s is convergent & ( lim s) = (( lim ( abs s0)) to_power k) by A234, A231, HOLDER_1: 10;

        then

         A236: (s ^\ n) is convergent & ( lim (s ^\ n)) = ( lim s) by SEQ_4: 20;

        (gp . x) = ( lim (Gp # x)) by A152, A221;

        then

         A237: ((Gp # x) ^\ n) is convergent & ( lim ((Gp # x) ^\ n)) = (gp . x) by A222, SEQ_4: 20;

        for j be Nat holds ((s ^\ n) . j) <= (((Gp # x) ^\ n) . j)

        proof

          let j be Nat;

          ((s ^\ n) . j) = (s . (n + j)) & (((Gp # x) ^\ n) . j) = ((Gp # x) . (n + j)) by NAT_1:def 3;

          hence thesis by A233, NAT_1: 11;

        end;

        then ( lim s) <= (gp . x) by A236, A237, SEQ_2: 18;

        hence thesis by A230, A235, A147, A221;

      end;

      deffunc FX3( Nat) = ( |.(F - (F2 . $1)).| to_power k);

      consider FP be Functional_Sequence of X, REAL such that

       A238: for n be Nat holds (FP . n) = FX3(n) from SEQFUNC:sch 1;

      

       A239: for n be Nat holds ( dom (FP . n)) = E0

      proof

        let n1 be Nat;

        reconsider n = n1 as Nat;

        

         A240: ( dom (F2 . n)) = E0 by A138;

        ( dom (FP . n1)) = ( dom (( abs (F - (F2 . n))) to_power k)) by A238;

        then ( dom (FP . n1)) = ( dom ( abs (F - (F2 . n)))) by MESFUN6C:def 4;

        then ( dom (FP . n1)) = ( dom (F - (F2 . n))) by VALUED_1:def 11;

        then ( dom (FP . n1)) = (E0 /\ E0) by A240, A146, VALUED_1: 12;

        hence ( dom (FP . n1)) = E0;

      end;

      then

       A241: E0 = ( dom (FP . 0 ));

      then

       A242: ( dom ( lim FP)) = E0 by MESFUNC8:def 9;

      for n,m be Nat holds ( dom (FP . n)) = ( dom (FP . m))

      proof

        let n,m be Nat;

        

        thus ( dom (FP . n)) = E0 by A239

        .= ( dom (FP . m)) by A239;

      end;

      then

      reconsider FP as with_the_same_dom Functional_Sequence of X, REAL by MESFUNC8:def 2;

      

       A243: for n be Nat holds (FP . n) is E0 -measurable

      proof

        let n1 be Nat;

        reconsider n = n1 as Nat;

        ( dom (F2 . n)) = E0 by A138;

        then

         A244: ( dom (F - (F2 . n))) = (E0 /\ E0) by A146, VALUED_1: 12;

        (F2 . n) is E0 -measurable & ( dom (F2 . n)) = E0 by A138;

        then (F - (F2 . n)) is E0 -measurable by A190, MESFUNC6: 29;

        then

         A245: ( abs (F - (F2 . n))) is E0 -measurable by A244, MESFUNC6: 48;

        ( dom ( abs (F - (F2 . n)))) = E0 by A244, VALUED_1:def 11;

        then (( abs (F - (F2 . n))) to_power k) is E0 -measurable by A245, MESFUN6C: 29;

        hence thesis by A238;

      end;

      for x be Element of X, n be Nat st x in E0 holds ( |.(FP . n).| . x) <= (gp . x)

      proof

        let x be Element of X, n1 be Nat;

        reconsider n = n1 as Element of NAT by ORDINAL1:def 12;

        assume

         A246: x in E0;

        then

         A247: x in ( dom (FP . n)) by A239;

        then x in ( dom ( |.(F - (F2 . n)).| to_power k)) by A238;

        then x in ( dom |.(F - (F2 . n)).|) by MESFUN6C:def 4;

        then

         A248: x in ( dom (F - (F2 . n))) by VALUED_1:def 11;

        

         A249: (FP . n1) = ( |.(F - (F2 . n1)).| to_power k) by A238;

        

         A250: 0 <= ( |.((F . x) - ((F2 . n1) . x)) qua Complex.| to_power k) by Th4, COMPLEX1: 46;

        ( |.(FP . n).| . x) = |.((FP . n) . x).| by VALUED_1: 18

        .= |.(( |.(F - (F2 . n1)).| . x) to_power k).| by A247, A249, MESFUN6C:def 4

        .= |.( |.((F - (F2 . n1)) . x) qua Complex.| to_power k).| by VALUED_1: 18

        .= |.( |.((F . x) - ((F2 . n1) . x)) qua Complex.| to_power k).| by A248, VALUED_1: 13

        .= ( |.((F . x) - ((F2 . n1) . x)) qua Complex.| to_power k) by A250, ABSVALUE:def 1

        .= ( |.((F . x) - ((F2 # x) . n)) qua Complex.| to_power k) by SEQFUNC:def 10;

        hence thesis by A220, A246;

      end;

      then

      consider Ip be Real_Sequence such that

       A251: (for n be Nat holds (Ip . n) = ( Integral (M,(FP . n)))) & ((for x be Element of X st x in E0 holds (FP # x) is convergent) implies Ip is convergent & ( lim Ip) = ( Integral (M,( lim FP)))) by A150, A155, A241, A243, MESFUN9C: 48;

      

       A252: for x be Element of X st x in E0 holds (FP # x) is convergent & ( lim (FP # x)) = 0

      proof

        let x be Element of X;

        assume

         A253: x in E0;

        

         A254: for n be Nat holds ((FP # x) . n) = ( |.(( lim (F2 # x)) - ((F2 # x) . n)) qua Complex.| to_power k)

        proof

          let n be Nat;

          x in ( dom (FP . n)) by A253, A239;

          then

           A255: x in ( dom ( |.(F - (F2 . n)).| to_power k)) by A238;

          then x in ( dom |.(F - (F2 . n)).|) by MESFUN6C:def 4;

          then

           A256: x in ( dom (F - (F2 . n))) by VALUED_1:def 11;

          

          thus ((FP # x) . n) = ((FP . n) . x) by SEQFUNC:def 10

          .= (( |.(F - (F2 . n)).| to_power k) . x) by A238

          .= (( |.(F - (F2 . n)).| . x) to_power k) by A255, MESFUN6C:def 4

          .= ( |.((F - (F2 . n)) . x) qua Complex.| to_power k) by VALUED_1: 18

          .= ( |.((F . x) - ((F2 . n) . x)) qua Complex.| to_power k) by A256, VALUED_1: 13

          .= ( |.(( lim (F2 # x)) - ((F2 . n) . x)) qua Complex.| to_power k) by A147, A253

          .= ( |.(( lim (F2 # x)) - ((F2 # x) . n)) qua Complex.| to_power k) by SEQFUNC:def 10;

        end;

        (F2 # x) is convergent by A253, A134;

        hence thesis by A254, Th11;

      end;

      

       A257: for x be Element of X st x in ( dom ( lim FP)) holds 0 = (( lim FP) . x)

      proof

        let x be Element of X;

        assume

         A258: x in ( dom ( lim FP));

        then

         A259: ( lim (FP # x)) = 0 & (FP # x) is convergent by A252, A242;

        (( lim FP) . x) = ( lim ( R_EAL (FP # x))) by A258, MESFUN7C: 14;

        hence thesis by A259, RINFSUP2: 14;

      end;

      ( a.e-eq-class_Lp (F,M,k)) in ( CosetSet (M,k)) by A192;

      then

      reconsider Sq0 = ( a.e-eq-class_Lp (F,M,k)) as Point of ( Lp-Space (M,k)) by Def11;

      

       A260: for n be Nat holds (Ip . n) = ( ||.(Sq0 - (Sq . (N . n))).|| to_power k)

      proof

        let n be Nat;

        set m = (N . n);

        reconsider n1 = n as Nat;

        

         A261: (FP . n) = (( abs (F - (F2 . n1))) to_power k) by A238;

        

         A262: F in ( Lp_Functions (M,k)) & F in Sq0 by A192, Th38;

        (F2 . n1) in ( Lp_Functions (M,k)) & (F2 . n1) in (Sq . m) by A140;

        then (( - 1) (#) (F2 . n1)) in (( - 1) * (Sq . m)) by Th54;

        then (F - (F2 . n1)) in (Sq0 + (( - 1) * (Sq . m))) by Th54, A262;

        then (F - (F2 . n1)) in (Sq0 - (Sq . m)) by RLVECT_1: 16;

        then

        consider r be Real such that

         A263: 0 <= r & r = ( Integral (M,(( abs (F - (F2 . n1))) to_power k))) & ||.(Sq0 - (Sq . m)).|| = (r to_power (1 / k)) by Th53;

        ( ||.(Sq0 - (Sq . m)).|| to_power k) = (r to_power ((1 / k) * k)) by A263, HOLDER_1: 2

        .= (r to_power 1) by XCMPLX_1: 106

        .= r by POWER: 25;

        hence thesis by A263, A261, A251;

      end;

      deffunc UZ( Nat) = ||.(Sq0 - (Sq . (N . $1))).||;

      consider Iq be Real_Sequence such that

       A264: for n be Nat holds (Iq . n) = UZ(n) from SEQ_1:sch 1;

      

       A265: for n be Nat holds (Iq . n) = ||.(Sq0 - (Sq . (N . n))).|| by A264;

      Iq is convergent & ( lim Iq) = 0

      proof

        

         A266: for n holds (Ip . n) >= 0

        proof

          let n;

          ( ||.(Sq0 - (Sq . (N . n))).|| to_power k) >= 0 by Th4;

          hence (Ip . n) >= 0 by A260;

        end;

        

         A267: for n be Nat holds (Iq . n) = ((Ip . n) to_power (1 / k))

        proof

          let n be Nat;

          

          thus ((Ip . n) to_power (1 / k)) = (( ||.(Sq0 - (Sq . (N . n))).|| to_power k) to_power (1 / k)) by A260

          .= ( ||.(Sq0 - (Sq . (N . n))).|| to_power (k * (1 / k))) by HOLDER_1: 2

          .= ( ||.(Sq0 - (Sq . (N . n))).|| to_power 1) by XCMPLX_1: 106

          .= ||.(Sq0 - (Sq . (N . n))).|| by POWER: 25

          .= (Iq . n) by A265;

        end;

        hence Iq is convergent by A266, A252, A251, HOLDER_1: 10;

        ( lim Iq) = (( lim Ip) to_power (1 / k)) by A252, A251, A266, A267, HOLDER_1: 10;

        then ( lim Iq) = ( 0 to_power (1 / k)) by A252, A251, A257, A242, LPSPACE1: 22;

        hence ( lim Iq) = 0 by POWER:def 2;

      end;

      hence thesis by A2, A265, Lm7;

    end;

    registration

      let X, S, M;

      let k be geq_than_1 Real;

      cluster ( Lp-Space (M,k)) -> complete;

      coherence

      proof

        for Sq be sequence of ( Lp-Space (M,k)) st Sq is Cauchy_sequence_by_Norm holds Sq is convergent by Th70;

        hence thesis by LOPBAN_1:def 15;

      end;

    end

    begin

    

     Lm8: f in ( L1_Functions M) implies f is_integrable_on M & (ex E be Element of S st (M . (E ` )) = 0 & E = ( dom f) & f is E -measurable)

    proof

      assume f in ( L1_Functions M);

      then ex f2 be PartFunc of X, REAL st f = f2 & ex E be Element of S st (M . E) = 0 & ( dom f2) = (E ` ) & f2 is_integrable_on M;

      then

      consider D be Element of S such that

       A1: (M . D) = 0 & ( dom f) = (D ` ) & f is_integrable_on M;

      thus f is_integrable_on M by A1;

      reconsider E = (D ` ) as Element of S by MEASURE1: 34;

      take E;

      thus (M . (E ` )) = 0 & ( dom f) = E by A1;

      ( R_EAL f) is_integrable_on M by A1;

      then ex B be Element of S st B = ( dom ( R_EAL f)) & ( R_EAL f) is B -measurable;

      hence f is E -measurable by A1;

    end;

    

     Lm9: f in ( Lp_Functions (M,k)) implies (( abs f) to_power k) is_integrable_on M

    proof

      assume f in ( Lp_Functions (M,k));

      then ex f2 be PartFunc of X, REAL st f = f2 & ex E be Element of S st (M . (E ` )) = 0 & ( dom f2) = E & f2 is E -measurable & (( abs f2) to_power k) is_integrable_on M;

      hence thesis;

    end;

    

     Lm10: (ex E be Element of S st (M . (E ` )) = 0 & E = ( dom f) & f is E -measurable) implies ( a.e-eq-class_Lp (f,M,1)) c= ( a.e-eq-class (f,M))

    proof

      assume

       A1: ex E be Element of S st (M . (E ` )) = 0 & E = ( dom f) & f is E -measurable;

      let x be object;

      assume x in ( a.e-eq-class_Lp (f,M,1));

      then

      consider h be PartFunc of X, REAL such that

       A2: x = h & h in ( Lp_Functions (M,1)) & f a.e.= (h,M);

      

       A3: ex g be PartFunc of X, REAL st h = g & (ex E be Element of S st (M . (E ` )) = 0 & ( dom g) = E & g is E -measurable & (( abs g) to_power 1) is_integrable_on M) by A2;

      then

      consider Eh be Element of S such that

       A4: (M . (Eh ` )) = 0 & ( dom h) = Eh & h is Eh -measurable & (( abs h) to_power 1) is_integrable_on M;

      

       A5: ( dom (( abs h) to_power 1)) = ( dom ( abs h)) by MESFUN6C:def 4;

      for x be Element of X st x in ( dom (( abs h) to_power 1)) holds ((( abs h) to_power 1) . x) = (( abs h) . x)

      proof

        let x be Element of X;

        assume x in ( dom (( abs h) to_power 1));

        then ((( abs h) to_power 1) . x) = ((( abs h) . x) to_power 1) by MESFUN6C:def 4;

        hence thesis by POWER: 25;

      end;

      then (( abs h) to_power 1) = ( abs h) by A5, PARTFUN1: 5;

      then

       A6: h is_integrable_on M by A3, MESFUNC6: 94;

      reconsider ND = (Eh ` ) as Element of S by MEASURE1: 34;

      (M . ND) = 0 & ( dom h) = (ND ` ) by A4;

      then

       A7: h in ( L1_Functions M) by A6;

      ex E be Element of S st (M . E) = 0 & ( dom f) = (E ` ) & f is_integrable_on M

      proof

        consider Ef be Element of S such that

         A8: (M . (Ef ` )) = 0 & Ef = ( dom f) & f is Ef -measurable by A1;

        reconsider E = (Ef ` ) as Element of S by MEASURE1: 34;

        take E;

        consider EE be Element of S such that

         A9: (M . EE) = 0 & (f | (EE ` )) = (h | (EE ` )) by A2;

        reconsider E1 = (ND \/ EE) as Element of S;

        ND is measure_zero of M & EE is measure_zero of M by A4, A9, MEASURE1:def 7;

        then E1 is measure_zero of M by MEASURE1: 37;

        then

         A10: (M . E1) = 0 by MEASURE1:def 7;

        EE c= E1 by XBOOLE_1: 7;

        then (E1 ` ) c= (EE ` ) by SUBSET_1: 12;

        then

         A11: (f | (E1 ` )) = ((f | (EE ` )) | (E1 ` )) & (h | (E1 ` )) = ((h | (EE ` )) | (E1 ` )) by FUNCT_1: 51;

        

         A12: ( dom ( max+ ( R_EAL f))) = Ef & ( dom ( max- ( R_EAL f))) = Ef by A8, MESFUNC2:def 2, MESFUNC2:def 3;

        

         A13: Ef = ( dom ( R_EAL f)) & ( R_EAL f) is Ef -measurable by A8;

        then

         A14: ( max+ ( R_EAL f)) is Ef -measurable & ( max- ( R_EAL f)) is Ef -measurable by MESFUNC2: 25, MESFUNC2: 26;

        (for x be Element of X holds 0. <= (( max+ ( R_EAL f)) . x)) & (for x be Element of X holds 0. <= (( max- ( R_EAL f)) . x)) by MESFUNC2: 12, MESFUNC2: 13;

        then

         A15: ( max+ ( R_EAL f)) is nonnegative & ( max- ( R_EAL f)) is nonnegative by SUPINF_2: 39;

        

         A16: Ef = ((Ef /\ E1) \/ (Ef \ E1)) by XBOOLE_1: 51;

        reconsider E0 = (Ef /\ E1) as Element of S;

        

         A17: (Ef \ E1) = (Ef /\ (E1 ` )) by SUBSET_1: 13;

        reconsider E2 = (Ef \ E1) as Element of S;

        ( max+ ( R_EAL f)) = (( max+ ( R_EAL f)) | ( dom ( max+ ( R_EAL f)))) & ( max- ( R_EAL f)) = (( max- ( R_EAL f)) | ( dom ( max- ( R_EAL f)))) by RELAT_1: 69;

        then

         A18: ( integral+ (M,( max+ ( R_EAL f)))) = (( integral+ (M,(( max+ ( R_EAL f)) | E0))) + ( integral+ (M,(( max+ ( R_EAL f)) | E2)))) & ( integral+ (M,( max- ( R_EAL f)))) = (( integral+ (M,(( max- ( R_EAL f)) | E0))) + ( integral+ (M,(( max- ( R_EAL f)) | E2)))) by A12, A15, A16, A14, MESFUNC5: 81, XBOOLE_1: 89;

        

         A19: ( integral+ (M,(( max+ ( R_EAL f)) | E0))) >= 0 & ( integral+ (M,(( max- ( R_EAL f)) | E0))) >= 0 by A15, A14, A12, MESFUNC5: 80;

        ( integral+ (M,(( max+ ( R_EAL f)) | E1))) = 0 & ( integral+ (M,(( max- ( R_EAL f)) | E1))) = 0 by A10, A12, A15, A14, MESFUNC5: 82;

        then ( integral+ (M,(( max+ ( R_EAL f)) | E0))) = 0 & ( integral+ (M,(( max- ( R_EAL f)) | E0))) = 0 by A19, A12, A15, A14, MESFUNC5: 83, XBOOLE_1: 17;

        then

         A20: ( integral+ (M,( max+ ( R_EAL f)))) = ( integral+ (M,(( max+ ( R_EAL f)) | E2))) & ( integral+ (M,( max- ( R_EAL f)))) = ( integral+ (M,(( max- ( R_EAL f)) | E2))) by A18, XXREAL_3: 4;

        

         A21: E2 c= (E1 ` ) by A17, XBOOLE_1: 17;

        then (f | E2) = ((h | (E1 ` )) | E2) by A9, A11, FUNCT_1: 51;

        then

         A22: (( R_EAL f) | E2) = (( R_EAL h) | E2) by A21, FUNCT_1: 51;

        

         A23: (( max+ ( R_EAL f)) | E2) = ( max+ (( R_EAL f) | E2)) & (( max+ ( R_EAL h)) | E2) = ( max+ (( R_EAL h) | E2)) & (( max- ( R_EAL f)) | E2) = ( max- (( R_EAL f) | E2)) & (( max- ( R_EAL h)) | E2) = ( max- (( R_EAL h) | E2)) by MESFUNC5: 28;

        

         A24: ( R_EAL h) is_integrable_on M by A6;

        then

         A25: ( integral+ (M,( max+ ( R_EAL h)))) < +infty & ( integral+ (M,( max- ( R_EAL h)))) < +infty ;

        ( integral+ (M,( max+ (( R_EAL h) | E2)))) <= ( integral+ (M,( max+ ( R_EAL h)))) & ( integral+ (M,( max- (( R_EAL h) | E2)))) <= ( integral+ (M,( max- ( R_EAL h)))) by A24, MESFUNC5: 97;

        then ( integral+ (M,( max+ ( R_EAL f)))) < +infty & ( integral+ (M,( max- ( R_EAL f)))) < +infty by A20, A25, A23, A22, XXREAL_0: 2;

        then ( R_EAL f) is_integrable_on M by A13;

        hence thesis by A8;

      end;

      then f in ( L1_Functions M);

      hence x in ( a.e-eq-class (f,M)) by A2, A7;

    end;

    

     Lm11: ( a.e-eq-class (f,M)) c= ( a.e-eq-class_Lp (f,M,1))

    proof

      let x be object;

      assume x in ( a.e-eq-class (f,M));

      then

      consider g be PartFunc of X, REAL such that

       A1: x = g & g in ( L1_Functions M) & f in ( L1_Functions M) & f a.e.= (g,M);

      

       A2: ex h be PartFunc of X, REAL st g = h & ex D be Element of S st (M . D) = 0 & ( dom h) = (D ` ) & h is_integrable_on M by A1;

      then ( R_EAL g) is_integrable_on M;

      then

      consider A be Element of S such that

       A3: A = ( dom ( R_EAL g)) & ( R_EAL g) is A -measurable;

      

       A4: A = ( dom g) & g is A -measurable by A3;

      

       A5: (M . (A ` )) = 0 by A2, A3;

      (( abs g) to_power 1) = ( abs g) by Th8;

      then (( abs g) to_power 1) is_integrable_on M by A2, A4, MESFUNC6: 94;

      then g in { p where p be PartFunc of X, REAL : ex Ep be Element of S st (M . (Ep ` )) = 0 & ( dom p) = Ep & p is Ep -measurable & (( abs p) to_power 1) is_integrable_on M } by A4, A5;

      hence x in ( a.e-eq-class_Lp (f,M,1)) by A1;

    end;

    

     Lm12: (ex E be Element of S st (M . (E ` )) = 0 & E = ( dom f) & f is E -measurable) implies ( a.e-eq-class_Lp (f,M,1)) = ( a.e-eq-class (f,M)) by Lm10, Lm11;

    theorem :: LPSPACE2:71

    

     Th71: for X be non empty set, S be SigmaField of X, M be sigma_Measure of S holds ( CosetSet M) = ( CosetSet (M,1))

    proof

      let X be non empty set;

      let S be SigmaField of X;

      let M be sigma_Measure of S;

      now

        let x be object;

        assume x in ( CosetSet M);

        then

        consider g be PartFunc of X, REAL such that

         A1: x = ( a.e-eq-class (g,M)) & g in ( L1_Functions M);

        

         A2: g is_integrable_on M & ex E be Element of S st (M . (E ` )) = 0 & E = ( dom g) & g is E -measurable by A1, Lm8;

        then

         A3: x = ( a.e-eq-class_Lp (g,M,1)) by A1, Lm12;

        (( abs g) to_power 1) = ( abs g) by Th8;

        then (( abs g) to_power 1) is_integrable_on M by A2, MESFUNC6: 94;

        then g in ( Lp_Functions (M,1)) by A2;

        hence x in ( CosetSet (M,1)) by A3;

      end;

      then

       A4: ( CosetSet M) c= ( CosetSet (M,1));

      now

        let x be object;

        assume x in ( CosetSet (M,1));

        then

        consider g be PartFunc of X, REAL such that

         A5: x = ( a.e-eq-class_Lp (g,M,1)) & g in ( Lp_Functions (M,1));

        consider E be Element of S such that

         A6: (M . (E ` )) = 0 & ( dom g) = E & g is E -measurable by A5, Th35;

        

         A7: x = ( a.e-eq-class (g,M)) by A5, A6, Lm12;

        reconsider D = (E ` ) as Element of S by MEASURE1: 34;

        

         A8: (M . D) = 0 & ( dom g) = (D ` ) by A6;

        (( abs g) to_power 1) is_integrable_on M by A5, Lm9;

        then ( abs g) is_integrable_on M by Th8;

        then g is_integrable_on M by A6, MESFUNC6: 94;

        then g in ( L1_Functions M) by A8;

        hence x in ( CosetSet M) by A7;

      end;

      then ( CosetSet (M,1)) c= ( CosetSet M);

      hence thesis by A4;

    end;

    theorem :: LPSPACE2:72

    

     Th72: for X be non empty set, S be SigmaField of X, M be sigma_Measure of S holds ( addCoset M) = ( addCoset (M,1))

    proof

      let X be non empty set;

      let S be SigmaField of X;

      let M be sigma_Measure of S;

      

       A1: ( CosetSet M) = ( CosetSet (M,1)) by Th71;

      now

        let A,B be Element of ( CosetSet M);

        A in { ( a.e-eq-class (f,M)) where f be PartFunc of X, REAL : f in ( L1_Functions M) };

        then

        consider a be PartFunc of X, REAL such that

         A2: A = ( a.e-eq-class (a,M)) & a in ( L1_Functions M);

        B in { ( a.e-eq-class (f,M)) where f be PartFunc of X, REAL : f in ( L1_Functions M) };

        then

        consider b be PartFunc of X, REAL such that

         A3: B = ( a.e-eq-class (b,M)) & b in ( L1_Functions M);

        

         A4: A is Element of ( CosetSet (M,1)) & B is Element of ( CosetSet (M,1)) by Th71;

        

         A5: a in ( a.e-eq-class (a,M)) & b in ( a.e-eq-class (b,M)) by A2, A3, LPSPACE1: 38;

        then

         A6: (( addCoset M) . (A,B)) = ( a.e-eq-class ((a + b),M)) by A2, A3, LPSPACE1:def 15;

        (a + b) in ( L1_Functions M) by A2, A3, LPSPACE1: 23;

        then ex E be Element of S st (M . (E ` )) = 0 & E = ( dom (a + b)) & (a + b) is E -measurable by Lm8;

        then (( addCoset M) . (A,B)) = ( a.e-eq-class_Lp ((a + b),M,1)) by A6, Lm12;

        hence (( addCoset M) . (A,B)) = (( addCoset (M,1)) . (A,B)) by A4, A5, A2, A3, Def8;

      end;

      hence ( addCoset M) = ( addCoset (M,1)) by A1, BINOP_1: 2;

    end;

    theorem :: LPSPACE2:73

    

     Th73: for X be non empty set, S be SigmaField of X, M be sigma_Measure of S holds ( zeroCoset M) = ( zeroCoset (M,1))

    proof

      let X be non empty set;

      let S be SigmaField of X;

      let M be sigma_Measure of S;

      reconsider z = ( zeroCoset (M,1)) as Element of ( CosetSet M) by Th71;

      (X --> 0 ) in ( Lp_Functions (M,1)) by Th23;

      then ex E be Element of S st (M . (E ` )) = 0 & ( dom (X --> 0 )) = E & (X --> 0 ) is E -measurable by Th35;

      then

       A1: z = ( a.e-eq-class ((X --> 0 ),M)) by Lm12;

      (X --> 0 ) in ( L1_Functions M) by Th56;

      hence thesis by A1, LPSPACE1:def 16;

    end;

    theorem :: LPSPACE2:74

    

     Th74: for X be non empty set, S be SigmaField of X, M be sigma_Measure of S holds ( lmultCoset M) = ( lmultCoset (M,1))

    proof

      let X be non empty set;

      let S be SigmaField of X;

      let M be sigma_Measure of S;

      

       A1: ( CosetSet M) = ( CosetSet (M,1)) by Th71;

      now

        let z be Element of REAL , A be Element of ( CosetSet M);

        A in { ( a.e-eq-class (f,M)) where f be PartFunc of X, REAL : f in ( L1_Functions M) };

        then

        consider a be PartFunc of X, REAL such that

         A2: A = ( a.e-eq-class (a,M)) & a in ( L1_Functions M);

        

         A3: A is Element of ( CosetSet (M,1)) by Th71;

        

         A4: a in A by A2, LPSPACE1: 38;

        then

         A5: (( lmultCoset M) . (z,A)) = ( a.e-eq-class ((z (#) a),M)) by LPSPACE1:def 17;

        (z (#) a) in ( L1_Functions M) by A2, LPSPACE1: 24;

        then ex E be Element of S st (M . (E ` )) = 0 & E = ( dom (z (#) a)) & (z (#) a) is E -measurable by Lm8;

        then (( lmultCoset M) . (z,A)) = ( a.e-eq-class_Lp ((z (#) a),M,1)) by A5, Lm12;

        hence (( lmultCoset M) . (z,A)) = (( lmultCoset (M,1)) . (z,A)) by A3, A4, Def10;

      end;

      hence ( lmultCoset M) = ( lmultCoset (M,1)) by A1, BINOP_1: 2;

    end;

    theorem :: LPSPACE2:75

    

     Th75: for X be non empty set, S be SigmaField of X, M be sigma_Measure of S holds ( Pre-L-Space M) = ( Pre-Lp-Space (M,1))

    proof

      let X be non empty set, S be SigmaField of X, M be sigma_Measure of S;

      

       A1: the carrier of ( Pre-L-Space M) = ( CosetSet M) & the addF of ( Pre-L-Space M) = ( addCoset M) & ( 0. ( Pre-L-Space M)) = ( zeroCoset M) & the Mult of ( Pre-L-Space M) = ( lmultCoset M) by LPSPACE1:def 18;

      ( CosetSet M) = ( CosetSet (M,1)) & ( addCoset M) = ( addCoset (M,1)) & ( zeroCoset M) = ( zeroCoset (M,1)) & ( lmultCoset M) = ( lmultCoset (M,1)) by Th71, Th72, Th73, Th74;

      hence thesis by A1, Def11;

    end;

    theorem :: LPSPACE2:76

    

     Th76: for X be non empty set, S be SigmaField of X, M be sigma_Measure of S holds ( L-1-Norm M) = ( Lp-Norm (M,1))

    proof

      let X be non empty set, S be SigmaField of X, M be sigma_Measure of S;

      

       A1: the carrier of ( Pre-L-Space M) = the carrier of ( Pre-Lp-Space (M,1)) by Th75;

      now

        let x be Element of the carrier of ( Pre-L-Space M);

        x in the carrier of ( Pre-L-Space M);

        then x in ( CosetSet M) by LPSPACE1:def 18;

        then

        consider g be PartFunc of X, REAL such that

         A2: x = ( a.e-eq-class (g,M)) & g in ( L1_Functions M);

        consider a be PartFunc of X, REAL such that

         A3: a in x & (( L-1-Norm M) . x) = ( Integral (M, |.a.|)) by LPSPACE1:def 19;

        

         A4: ex p be PartFunc of X, REAL st a = p & p in ( L1_Functions M) & g in ( L1_Functions M) & g a.e.= (p,M) by A2, A3;

        consider b be PartFunc of X, REAL such that

         A5: b in x & ex r be Real st r = ( Integral (M,( |.b.| to_power 1))) & (( Lp-Norm (M,1)) . x) = (r to_power (1 / 1)) by A1, Def12;

        

         A6: ex q be PartFunc of X, REAL st b = q & q in ( L1_Functions M) & g in ( L1_Functions M) & g a.e.= (q,M) by A2, A5;

        a a.e.= (g,M) by A4;

        then a a.e.= (b,M) by A6, LPSPACE1: 30;

        then

         A7: ( Integral (M, |.a.|)) = ( Integral (M, |.b.|)) by A2, A3, A5, LPSPACE1: 45;

        ( |.b.| to_power 1) = |.b.| by Th8;

        hence (( L-1-Norm M) . x) = (( Lp-Norm (M,1)) . x) by A3, A5, A7, POWER: 25;

      end;

      hence thesis by A1, FUNCT_2: 63;

    end;

    theorem :: LPSPACE2:77

    for X be non empty set, S be SigmaField of X, M be sigma_Measure of S holds ( L-1-Space M) = ( Lp-Space (M,1))

    proof

      let X be non empty set, S be SigmaField of X, M be sigma_Measure of S;

      ( Pre-L-Space M) = ( Pre-Lp-Space (M,1)) by Th75;

      hence thesis by Th76;

    end;