let X be non empty set ; :: thesis: for S being SigmaField of X

for M being sigma_Measure of S

for k being positive Real holds

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

let S be SigmaField of X; :: thesis: for M being sigma_Measure of S

for k being positive Real holds

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

let M be sigma_Measure of S; :: thesis: for k being positive Real holds

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

let k be positive Real; :: thesis: ( AlmostZeroLpFunctions (M,k) is add-closed & AlmostZeroLpFunctions (M,k) is multi-closed )

set Z = AlmostZeroLpFunctions (M,k);

set V = RLSp_LpFunct (M,k);

for M being sigma_Measure of S

for k being positive Real holds

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

let S be SigmaField of X; :: thesis: for M being sigma_Measure of S

for k being positive Real holds

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

let M be sigma_Measure of S; :: thesis: for k being positive Real holds

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

let k be positive Real; :: thesis: ( AlmostZeroLpFunctions (M,k) is add-closed & AlmostZeroLpFunctions (M,k) is multi-closed )

set Z = AlmostZeroLpFunctions (M,k);

set V = RLSp_LpFunct (M,k);

now :: thesis: for v, u being VECTOR of (RLSp_LpFunct (M,k)) st v in AlmostZeroLpFunctions (M,k) & u in AlmostZeroLpFunctions (M,k) holds

v + u in AlmostZeroLpFunctions (M,k)

hence
AlmostZeroLpFunctions (M,k) is add-closed
by IDEAL_1:def 1; :: thesis: AlmostZeroLpFunctions (M,k) is multi-closed v + u in AlmostZeroLpFunctions (M,k)

let v, u be VECTOR of (RLSp_LpFunct (M,k)); :: thesis: ( v in AlmostZeroLpFunctions (M,k) & u in AlmostZeroLpFunctions (M,k) implies v + u in AlmostZeroLpFunctions (M,k) )

assume A1: ( v in AlmostZeroLpFunctions (M,k) & u in AlmostZeroLpFunctions (M,k) ) ; :: thesis: v + u in AlmostZeroLpFunctions (M,k)

then consider v1 being PartFunc of X,REAL such that

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

consider u1 being 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 AlmostZeroLpFunctions (M,k) by A4; :: thesis: verum

end;assume A1: ( v in AlmostZeroLpFunctions (M,k) & u in AlmostZeroLpFunctions (M,k) ) ; :: thesis: v + u in AlmostZeroLpFunctions (M,k)

then consider v1 being PartFunc of X,REAL such that

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

consider u1 being 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 AlmostZeroLpFunctions (M,k) by A4; :: thesis: verum

now :: thesis: for a being Real

for u being VECTOR of (RLSp_LpFunct (M,k)) st u in AlmostZeroLpFunctions (M,k) holds

a * u in AlmostZeroLpFunctions (M,k)

hence
AlmostZeroLpFunctions (M,k) is multi-closed
; :: thesis: verumfor u being VECTOR of (RLSp_LpFunct (M,k)) st u in AlmostZeroLpFunctions (M,k) holds

a * u in AlmostZeroLpFunctions (M,k)

let a be Real; :: thesis: for u being VECTOR of (RLSp_LpFunct (M,k)) st u in AlmostZeroLpFunctions (M,k) holds

a * u in AlmostZeroLpFunctions (M,k)

let u be VECTOR of (RLSp_LpFunct (M,k)); :: thesis: ( u in AlmostZeroLpFunctions (M,k) implies a * u in AlmostZeroLpFunctions (M,k) )

assume u in AlmostZeroLpFunctions (M,k) ; :: thesis: a * u in AlmostZeroLpFunctions (M,k)

then consider u1 being 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 AlmostZeroLpFunctions (M,k) by A6; :: thesis: verum

end;a * u in AlmostZeroLpFunctions (M,k)

let u be VECTOR of (RLSp_LpFunct (M,k)); :: thesis: ( u in AlmostZeroLpFunctions (M,k) implies a * u in AlmostZeroLpFunctions (M,k) )

assume u in AlmostZeroLpFunctions (M,k) ; :: thesis: a * u in AlmostZeroLpFunctions (M,k)

then consider u1 being 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 AlmostZeroLpFunctions (M,k) by A6; :: thesis: verum