:: On $L^1$ Space Formed by Real-valued Partial Functions
:: by Yasushige Watase , Noboru Endou and Yasunari Shidama
::
:: Copyright (c) 2008-2021 Association of Mizar Users

definition
let V be non empty RLSStruct ;
let V1 be Subset of V;
attr V1 is multi-closed means :Def1: :: LPSPACE1:def 1
for a being Real
for v being VECTOR of V st v in V1 holds
a * v in V1;
end;

:: deftheorem Def1 defines multi-closed LPSPACE1:def 1 :
for V being non empty RLSStruct
for V1 being Subset of V holds
( V1 is multi-closed iff for a being Real
for v being VECTOR of V st v in V1 holds
a * v in V1 );

theorem :: LPSPACE1:1
for V being RealLinearSpace
for V1 being Subset of V holds
( V1 is linearly-closed iff ( V1 is add-closed & V1 is multi-closed ) ) by RLSUB_1:def 1;

registration
let V be non empty RLSStruct ;
cluster non empty add-closed multi-closed for Element of bool the carrier of V;
existence
ex b1 being Subset of V st
( b1 is add-closed & b1 is multi-closed & not b1 is empty )
proof end;
end;

definition
let X be non empty RLSStruct ;
let X1 be non empty multi-closed Subset of X;
func Mult_ X1 -> Function of [:REAL,X1:],X1 equals :: LPSPACE1:def 2
the Mult of X | [:REAL,X1:];
correctness
coherence
the Mult of X | [:REAL,X1:] is Function of [:REAL,X1:],X1
;
proof end;
end;

:: deftheorem defines Mult_ LPSPACE1:def 2 :
for X being non empty RLSStruct
for X1 being non empty multi-closed Subset of X holds Mult_ X1 = the Mult of X | [:REAL,X1:];

theorem Th2: :: LPSPACE1:2
for V being non empty Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
for V1 being non empty Subset of V
for d1 being Element of V1
for A being BinOp of V1
for M being Function of [:REAL,V1:],V1 st d1 = 0. V & A = the addF of V || V1 & M = the Mult of V | [:REAL,V1:] holds
( RLSStruct(# V1,d1,A,M #) is Abelian & RLSStruct(# V1,d1,A,M #) is add-associative & RLSStruct(# V1,d1,A,M #) is right_zeroed & RLSStruct(# V1,d1,A,M #) is vector-distributive & RLSStruct(# V1,d1,A,M #) is scalar-distributive & RLSStruct(# V1,d1,A,M #) is scalar-associative & RLSStruct(# V1,d1,A,M #) is scalar-unital )
proof end;

theorem Th3: :: LPSPACE1:3
for V being non empty Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
for V1 being non empty add-closed multi-closed Subset of V st 0. V in V1 holds
( RLSStruct(# V1,(In ((0. V),V1)),(add| (V1,V)),(Mult_ V1) #) is Abelian & RLSStruct(# V1,(In ((0. V),V1)),(add| (V1,V)),(Mult_ V1) #) is add-associative & RLSStruct(# V1,(In ((0. V),V1)),(add| (V1,V)),(Mult_ V1) #) is right_zeroed & RLSStruct(# V1,(In ((0. V),V1)),(add| (V1,V)),(Mult_ V1) #) is vector-distributive & RLSStruct(# V1,(In ((0. V),V1)),(add| (V1,V)),(Mult_ V1) #) is scalar-distributive & RLSStruct(# V1,(In ((0. V),V1)),(add| (V1,V)),(Mult_ V1) #) is scalar-associative & RLSStruct(# V1,(In ((0. V),V1)),(add| (V1,V)),(Mult_ V1) #) is scalar-unital )
proof end;

theorem Th4: :: LPSPACE1:4
for V being non empty RLSStruct
for V1 being non empty add-closed multi-closed Subset of V
for v, u being VECTOR of V
for w1, w2 being VECTOR of RLSStruct(# V1,(In ((0. V),V1)),(add| (V1,V)),(Mult_ V1) #) st w1 = v & w2 = u holds
w1 + w2 = v + u by ;

theorem Th5: :: LPSPACE1:5
for V being non empty RLSStruct
for V1 being non empty add-closed multi-closed Subset of V
for a being Real
for v being VECTOR of V
for w being VECTOR of RLSStruct(# V1,(In ((0. V),V1)),(add| (V1,V)),(Mult_ V1) #) st w = v holds
a * w = a * v
proof end;

definition
let A, B be non empty set ;
let F be BinOp of (PFuncs (A,B));
let f, g be Element of PFuncs (A,B);
:: original: .
redefine func F . (f,g) -> Element of PFuncs (A,B);
coherence
F . (f,g) is Element of PFuncs (A,B)
proof end;
end;

definition
let A be non empty set ;
func multpfunc A -> BinOp of (PFuncs (A,REAL)) means :Def3: :: LPSPACE1:def 3
for f, g being Element of PFuncs (A,REAL) holds it . (f,g) = f (#) g;
existence
ex b1 being BinOp of (PFuncs (A,REAL)) st
for f, g being Element of PFuncs (A,REAL) holds b1 . (f,g) = f (#) g
proof end;
uniqueness
for b1, b2 being BinOp of (PFuncs (A,REAL)) st ( for f, g being Element of PFuncs (A,REAL) holds b1 . (f,g) = f (#) g ) & ( for f, g being Element of PFuncs (A,REAL) holds b2 . (f,g) = f (#) g ) holds
b1 = b2
proof end;
end;

:: deftheorem Def3 defines multpfunc LPSPACE1:def 3 :
for A being non empty set
for b2 being BinOp of (PFuncs (A,REAL)) holds
( b2 = multpfunc A iff for f, g being Element of PFuncs (A,REAL) holds b2 . (f,g) = f (#) g );

definition
let A be non empty set ;
func multrealpfunc A -> Function of [:REAL,(PFuncs (A,REAL)):],(PFuncs (A,REAL)) means :Def4: :: LPSPACE1:def 4
for a being Real
for f being Element of PFuncs (A,REAL) holds it . (a,f) = a (#) f;
existence
ex b1 being Function of [:REAL,(PFuncs (A,REAL)):],(PFuncs (A,REAL)) st
for a being Real
for f being Element of PFuncs (A,REAL) holds b1 . (a,f) = a (#) f
proof end;
uniqueness
for b1, b2 being Function of [:REAL,(PFuncs (A,REAL)):],(PFuncs (A,REAL)) st ( for a being Real
for f being Element of PFuncs (A,REAL) holds b1 . (a,f) = a (#) f ) & ( for a being Real
for f being Element of PFuncs (A,REAL) holds b2 . (a,f) = a (#) f ) holds
b1 = b2
proof end;
end;

:: deftheorem Def4 defines multrealpfunc LPSPACE1:def 4 :
for A being non empty set
for b2 being Function of [:REAL,(PFuncs (A,REAL)):],(PFuncs (A,REAL)) holds
( b2 = multrealpfunc A iff for a being Real
for f being Element of PFuncs (A,REAL) holds b2 . (a,f) = a (#) f );

definition
let A be non empty set ;
func RealPFuncZero A -> Element of PFuncs (A,REAL) equals :: LPSPACE1:def 5
A --> 0;
coherence
A --> 0 is Element of PFuncs (A,REAL)
proof end;
end;

:: deftheorem defines RealPFuncZero LPSPACE1:def 5 :
for A being non empty set holds RealPFuncZero A = A --> 0;

reconsider jj = 1 as Element of REAL by XREAL_0:def 1;

definition
let A be non empty set ;
func RealPFuncUnit A -> Element of PFuncs (A,REAL) equals :: LPSPACE1:def 6
A --> 1;
coherence
A --> 1 is Element of PFuncs (A,REAL)
proof end;
end;

:: deftheorem defines RealPFuncUnit LPSPACE1:def 6 :
for A being non empty set holds RealPFuncUnit A = A --> 1;

theorem Th6: :: LPSPACE1:6
for A being non empty set
for f, g, h being Element of PFuncs (A,REAL) holds
( h = () . (f,g) iff ( dom h = (dom f) /\ (dom g) & ( for x being Element of A st x in dom h holds
h . x = (f . x) + (g . x) ) ) )
proof end;

theorem Th7: :: LPSPACE1:7
for A being non empty set
for f, g, h being Element of PFuncs (A,REAL) holds
( h = () . (f,g) iff ( dom h = (dom f) /\ (dom g) & ( for x being Element of A st x in dom h holds
h . x = (f . x) * (g . x) ) ) )
proof end;

theorem :: LPSPACE1:8
for A being non empty set holds RealPFuncZero A <> RealPFuncUnit A
proof end;

theorem Th9: :: LPSPACE1:9
for a being Real
for A being non empty set
for f, h being Element of PFuncs (A,REAL) holds
( h = () . (a,f) iff ( dom h = dom f & ( for x being Element of A st x in dom f holds
h . x = a * (f . x) ) ) )
proof end;

theorem Th10: :: LPSPACE1:10
for A being non empty set
for f, g being Element of PFuncs (A,REAL) holds () . (f,g) = () . (g,f) by ;

theorem Th11: :: LPSPACE1:11
for A being non empty set
for f, g, h being Element of PFuncs (A,REAL) holds () . (f,(() . (g,h))) = () . ((() . (f,g)),h) by ;

theorem :: LPSPACE1:12
for A being non empty set
for f, g being Element of PFuncs (A,REAL) holds () . (f,g) = () . (g,f)
proof end;

theorem :: LPSPACE1:13
for A being non empty set
for f, g, h being Element of PFuncs (A,REAL) holds () . (f,(() . (g,h))) = () . ((() . (f,g)),h)
proof end;

theorem :: LPSPACE1:14
for A being non empty set
for f being Element of PFuncs (A,REAL) holds () . ((),f) = f
proof end;

theorem Th15: :: LPSPACE1:15
for A being non empty set
for f being Element of PFuncs (A,REAL) holds () . ((),f) = f
proof end;

theorem Th16: :: LPSPACE1:16
for A being non empty set
for f being Element of PFuncs (A,REAL) holds () . (f,(() . ((- 1),f))) = () | (dom f)
proof end;

theorem Th17: :: LPSPACE1:17
for A being non empty set
for f being Element of PFuncs (A,REAL) holds () . (1,f) = f
proof end;

theorem Th18: :: LPSPACE1:18
for a, b being Real
for A being non empty set
for f being Element of PFuncs (A,REAL) holds () . (a,(() . (b,f))) = () . ((a * b),f)
proof end;

theorem Th19: :: LPSPACE1:19
for a, b being Real
for A being non empty set
for f being Element of PFuncs (A,REAL) holds () . ((() . (a,f)),(() . (b,f))) = () . ((a + b),f)
proof end;

Lm1: for a being Real
for A being non empty set
for f, g being Element of PFuncs (A,REAL) holds () . ((() . (a,f)),(() . (a,g))) = () . (a,(() . (f,g)))

proof end;

theorem :: LPSPACE1:20
for A being non empty set
for f, g, h being Element of PFuncs (A,REAL) holds () . (f,(() . (g,h))) = () . ((() . (f,g)),(() . (f,h)))
proof end;

theorem :: LPSPACE1:21
for a being Real
for A being non empty set
for f, g being Element of PFuncs (A,REAL) holds () . ((() . (a,f)),g) = () . (a,(() . (f,g)))
proof end;

definition
let A be non empty set ;
func RLSp_PFunct A -> non empty RLSStruct equals :: LPSPACE1:def 7
RLSStruct(# (PFuncs (A,REAL)),(),(),() #);
coherence
RLSStruct(# (PFuncs (A,REAL)),(),(),() #) is non empty RLSStruct
;
end;

:: deftheorem defines RLSp_PFunct LPSPACE1:def 7 :
for A being non empty set holds RLSp_PFunct A = RLSStruct(# (PFuncs (A,REAL)),(),(),() #);

registration
let A be non empty set ;
coherence
proof end;
end;

theorem Th22: :: LPSPACE1:22
for X being non empty set
for S being SigmaField of X
for M being sigma_Measure of S
for f being PartFunc of X,ExtREAL st ex E being Element of S st E = dom f & ( for x being Element of X st x in dom f holds
0 = f . x ) holds
( f is_integrable_on M & Integral (M,f) = 0 )
proof end;

definition
let X be non empty set ;
let r be Real;
:: original: -->
redefine func X --> r -> PartFunc of X,REAL;
coherence
X --> r is PartFunc of X,REAL
proof end;
end;

Lm2: for X being non empty set
for S being SigmaField of X
for M being sigma_Measure of S
for f being PartFunc of X,REAL st X = dom f & ( for x being Element of X st x in dom f holds
0 = f . x ) holds
( f is_integrable_on M & Integral (M,f) = 0 )

proof end;

definition
let X be non empty set ;
let S be SigmaField of X;
let M be sigma_Measure of S;
func L1_Functions M -> non empty Subset of () equals :: LPSPACE1:def 8
{ f where f is PartFunc of X,REAL : ex ND being Element of S st
( M . ND = 0 & dom f = ND  & f is_integrable_on M )
}
;
correctness
coherence
{ f where f is PartFunc of X,REAL : ex ND being Element of S st
( M . ND = 0 & dom f = ND  & f is_integrable_on M )
}
is non empty Subset of ()
;
proof end;
end;

:: deftheorem defines L1_Functions LPSPACE1:def 8 :
for X being non empty set
for S being SigmaField of X
for M being sigma_Measure of S holds L1_Functions M = { f where f is PartFunc of X,REAL : ex ND being Element of S st
( M . ND = 0 & dom f = ND  & f is_integrable_on M )
}
;

Lm3: for X being non empty set
for S being SigmaField of X
for M being sigma_Measure of S holds
( X --> 0 is PartFunc of X,REAL & X --> 0 in L1_Functions M )

proof end;

Lm4: for X being non empty set
for S being SigmaField of X
for M being sigma_Measure of S
for E1, E2 being Element of S st M . E1 = 0 & M . E2 = 0 holds
M . (E1 \/ E2) = 0

proof end;

theorem Th23: :: LPSPACE1:23
for X being non empty set
for S being SigmaField of X
for M being sigma_Measure of S
for f, g being PartFunc of X,REAL st f in L1_Functions M & g in L1_Functions M holds
f + g in L1_Functions M
proof end;

theorem Th24: :: LPSPACE1:24
for a being Real
for X being non empty set
for S being SigmaField of X
for M being sigma_Measure of S
for f being PartFunc of X,REAL st f in L1_Functions M holds
a (#) f in L1_Functions M
proof end;

Lm5: for X being non empty set
for S being SigmaField of X
for M being sigma_Measure of S holds
( L1_Functions M is add-closed & L1_Functions M is multi-closed )

proof end;

registration
let X be non empty set ;
let S be SigmaField of X;
let M be sigma_Measure of S;
coherence by Lm5;
end;

definition
let X be non empty set ;
let S be SigmaField of X;
let M be sigma_Measure of S;
func RLSp_L1Funct M -> non empty RLSStruct equals :: LPSPACE1:def 9
RLSStruct(# (),(In ((0. ()),())),(add| ((),())),() #);
coherence
RLSStruct(# (),(In ((0. ()),())),(add| ((),())),() #) is non empty RLSStruct
;
end;

:: deftheorem defines RLSp_L1Funct LPSPACE1:def 9 :
for X being non empty set
for S being SigmaField of X
for M being sigma_Measure of S holds RLSp_L1Funct M = RLSStruct(# (),(In ((0. ()),())),(add| ((),())),() #);

registration
let X be non empty set ;
let S be SigmaField of X;
let M be sigma_Measure of S;
coherence
proof end;
end;

theorem Th25: :: LPSPACE1:25
for X being non empty set
for S being SigmaField of X
for M being sigma_Measure of S
for f, g being PartFunc of X,REAL
for v, u being VECTOR of () st f = v & g = u holds
f + g = v + u
proof end;

theorem Th26: :: LPSPACE1:26
for a being Real
for X being non empty set
for S being SigmaField of X
for M being sigma_Measure of S
for f being PartFunc of X,REAL
for u being VECTOR of () st f = u holds
a (#) f = a * u
proof end;

definition
let X be non empty set ;
let S be SigmaField of X;
let M be sigma_Measure of S;
let f, g be PartFunc of X,REAL;
pred f a.e.= g,M means :: LPSPACE1:def 10
ex E being Element of S st
( M . E = 0 & f | (E ) = g | (E ) );
end;

:: deftheorem defines a.e.= LPSPACE1:def 10 :
for X being non empty set
for S being SigmaField of X
for M being sigma_Measure of S
for f, g being PartFunc of X,REAL holds
( f a.e.= g,M iff ex E being Element of S st
( M . E = 0 & f | (E ) = g | (E `) ) );

theorem Th27: :: LPSPACE1:27
for X being non empty set
for S being SigmaField of X
for M being sigma_Measure of S
for f being PartFunc of X,REAL
for u being VECTOR of () st f = u holds
( u + ((- 1) * u) = (X --> 0) | (dom f) & ex v, g being PartFunc of X,REAL st
( v in L1_Functions M & g in L1_Functions M & v = u + ((- 1) * u) & g = X --> 0 & v a.e.= g,M ) )
proof end;

theorem Th28: :: LPSPACE1:28
for X being non empty set
for S being SigmaField of X
for M being sigma_Measure of S
for f being PartFunc of X,REAL holds f a.e.= f,M
proof end;

theorem :: LPSPACE1:29
for X being non empty set
for S being SigmaField of X
for M being sigma_Measure of S
for f, g being PartFunc of X,REAL st f a.e.= g,M holds
g a.e.= f,M ;

theorem Th30: :: LPSPACE1:30
for X being non empty set
for S being SigmaField of X
for M being sigma_Measure of S
for f, g, h being PartFunc of X,REAL st f a.e.= g,M & g a.e.= h,M holds
f a.e.= h,M
proof end;

theorem Th31: :: LPSPACE1:31
for X being non empty set
for S being SigmaField of X
for M being sigma_Measure of S
for f, g, f1, g1 being PartFunc of X,REAL st f a.e.= f1,M & g a.e.= g1,M holds
f + g a.e.= f1 + g1,M
proof end;

theorem Th32: :: LPSPACE1:32
for a being Real
for X being non empty set
for S being SigmaField of X
for M being sigma_Measure of S
for f, g being PartFunc of X,REAL st f a.e.= g,M holds
a (#) f a.e.= a (#) g,M
proof end;

definition
let X be non empty set ;
let S be SigmaField of X;
let M be sigma_Measure of S;
func AlmostZeroFunctions M -> non empty Subset of () equals :: LPSPACE1:def 11
{ f where f is PartFunc of X,REAL : ( f in L1_Functions M & f a.e.= X --> 0,M ) } ;
coherence
{ f where f is PartFunc of X,REAL : ( f in L1_Functions M & f a.e.= X --> 0,M ) } is non empty Subset of ()
proof end;
end;

:: deftheorem defines AlmostZeroFunctions LPSPACE1:def 11 :
for X being non empty set
for S being SigmaField of X
for M being sigma_Measure of S holds AlmostZeroFunctions M = { f where f is PartFunc of X,REAL : ( f in L1_Functions M & f a.e.= X --> 0,M ) } ;

theorem Th33: :: LPSPACE1:33
for a being Real
for X being non empty set holds
( (X --> 0) + (X --> 0) = X --> 0 & a (#) (X --> 0) = X --> 0 )
proof end;

Lm6: for X being non empty set
for S being SigmaField of X
for M being sigma_Measure of S holds
( AlmostZeroFunctions M is add-closed & AlmostZeroFunctions M is multi-closed )

proof end;

registration
let X be non empty set ;
let S be SigmaField of X;
let M be sigma_Measure of S;
coherence by Lm6;
end;

theorem :: LPSPACE1:34
for X being non empty set
for S being SigmaField of X
for M being sigma_Measure of S holds
( 0. () = X --> 0 & 0. () in AlmostZeroFunctions M )
proof end;

definition
let X be non empty set ;
let S be SigmaField of X;
let M be sigma_Measure of S;
func RLSp_AlmostZeroFunct M -> non empty RLSStruct equals :: LPSPACE1:def 12
RLSStruct(# ,(In ((0. ()),)),(add| (,())),() #);
coherence
RLSStruct(# ,(In ((0. ()),)),(add| (,())),() #) is non empty RLSStruct
;
end;

:: deftheorem defines RLSp_AlmostZeroFunct LPSPACE1:def 12 :
for X being non empty set
for S being SigmaField of X
for M being sigma_Measure of S holds RLSp_AlmostZeroFunct M = RLSStruct(# ,(In ((0. ()),)),(add| (,())),() #);

registration
let X be non empty set ;
let S be SigmaField of X;
let M be sigma_Measure of S;
coherence ;
end;

theorem :: LPSPACE1:35
for X being non empty set
for S being SigmaField of X
for M being sigma_Measure of S
for f, g being PartFunc of X,REAL
for v, u being VECTOR of st f = v & g = u holds
f + g = v + u
proof end;

theorem :: LPSPACE1:36
for a being Real
for X being non empty set
for S being SigmaField of X
for M being sigma_Measure of S
for f being PartFunc of X,REAL
for u being VECTOR of st f = u holds
a (#) f = a * u
proof end;

definition
let X be non empty set ;
let S be SigmaField of X;
let M be sigma_Measure of S;
let f be PartFunc of X,REAL;
func a.e-eq-class (f,M) -> Subset of () equals :: LPSPACE1:def 13
{ g where g is PartFunc of X,REAL : ( g in L1_Functions M & f in L1_Functions M & f a.e.= g,M ) } ;
correctness
coherence
{ g where g is PartFunc of X,REAL : ( g in L1_Functions M & f in L1_Functions M & f a.e.= g,M ) } is Subset of ()
;
proof end;
end;

:: deftheorem defines a.e-eq-class LPSPACE1:def 13 :
for X being non empty set
for S being SigmaField of X
for M being sigma_Measure of S
for f being PartFunc of X,REAL holds a.e-eq-class (f,M) = { g where g is PartFunc of X,REAL : ( g in L1_Functions M & f in L1_Functions M & f a.e.= g,M ) } ;

theorem Th37: :: LPSPACE1:37
for X being non empty set
for S being SigmaField of X
for M being sigma_Measure of S
for f, g being PartFunc of X,REAL st f in L1_Functions M & g in L1_Functions M holds
( g a.e.= f,M iff g in a.e-eq-class (f,M) )
proof end;

theorem Th38: :: LPSPACE1:38
for X being non empty set
for S being SigmaField of X
for M being sigma_Measure of S
for f being PartFunc of X,REAL st f in L1_Functions M holds
f in a.e-eq-class (f,M)
proof end;

theorem Th39: :: LPSPACE1:39
for X being non empty set
for S being SigmaField of X
for M being sigma_Measure of S
for f, g being PartFunc of X,REAL st f in L1_Functions M & g in L1_Functions M holds
( a.e-eq-class (f,M) = a.e-eq-class (g,M) iff f a.e.= g,M )
proof end;

theorem :: LPSPACE1:40
for X being non empty set
for S being SigmaField of X
for M being sigma_Measure of S
for f, g being PartFunc of X,REAL st f in L1_Functions M & g in L1_Functions M holds
( a.e-eq-class (f,M) = a.e-eq-class (g,M) iff g in a.e-eq-class (f,M) )
proof end;

theorem Th41: :: LPSPACE1:41
for X being non empty set
for S being SigmaField of X
for M being sigma_Measure of S
for f, g, f1, g1 being PartFunc of X,REAL st f in L1_Functions M & f1 in L1_Functions M & g in L1_Functions M & g1 in L1_Functions M & a.e-eq-class (f,M) = a.e-eq-class (f1,M) & a.e-eq-class (g,M) = a.e-eq-class (g1,M) holds
a.e-eq-class ((f + g),M) = a.e-eq-class ((f1 + g1),M)
proof end;

theorem Th42: :: LPSPACE1:42
for a being Real
for X being non empty set
for S being SigmaField of X
for M being sigma_Measure of S
for f, g being PartFunc of X,REAL st f in L1_Functions M & g in L1_Functions M & a.e-eq-class (f,M) = a.e-eq-class (g,M) holds
a.e-eq-class ((a (#) f),M) = a.e-eq-class ((a (#) g),M)
proof end;

definition
let X be non empty set ;
let S be SigmaField of X;
let M be sigma_Measure of S;
func CosetSet M -> non empty Subset-Family of () equals :: LPSPACE1:def 14
{ (a.e-eq-class (f,M)) where f is PartFunc of X,REAL : f in L1_Functions M } ;
correctness
coherence
{ (a.e-eq-class (f,M)) where f is PartFunc of X,REAL : f in L1_Functions M } is non empty Subset-Family of ()
;
proof end;
end;

:: deftheorem defines CosetSet LPSPACE1:def 14 :
for X being non empty set
for S being SigmaField of X
for M being sigma_Measure of S holds CosetSet M = { (a.e-eq-class (f,M)) where f is PartFunc of X,REAL : f in L1_Functions M } ;

definition
let X be non empty set ;
let S be SigmaField of X;
let M be sigma_Measure of S;
func addCoset M -> BinOp of () means :Def15: :: LPSPACE1:def 15
for A, B being Element of CosetSet M
for a, b being PartFunc of X,REAL st a in A & b in B holds
it . (A,B) = a.e-eq-class ((a + b),M);
existence
ex b1 being BinOp of () st
for A, B being Element of CosetSet M
for a, b being PartFunc of X,REAL st a in A & b in B holds
b1 . (A,B) = a.e-eq-class ((a + b),M)
proof end;
uniqueness
for b1, b2 being BinOp of () st ( for A, B being Element of CosetSet M
for a, b being PartFunc of X,REAL st a in A & b in B holds
b1 . (A,B) = a.e-eq-class ((a + b),M) ) & ( for A, B being Element of CosetSet M
for a, b being PartFunc of X,REAL st a in A & b in B holds
b2 . (A,B) = a.e-eq-class ((a + b),M) ) holds
b1 = b2
proof end;
end;

:: deftheorem Def15 defines addCoset LPSPACE1:def 15 :
for X being non empty set
for S being SigmaField of X
for M being sigma_Measure of S
for b4 being BinOp of () holds
( b4 = addCoset M iff for A, B being Element of CosetSet M
for a, b being PartFunc of X,REAL st a in A & b in B holds
b4 . (A,B) = a.e-eq-class ((a + b),M) );

definition
let X be non empty set ;
let S be SigmaField of X;
let M be sigma_Measure of S;
func zeroCoset M -> Element of CosetSet M means :Def16: :: LPSPACE1:def 16
ex f being PartFunc of X,REAL st
( f = X --> 0 & f in L1_Functions M & it = a.e-eq-class (f,M) );
correctness
existence
ex b1 being Element of CosetSet M ex f being PartFunc of X,REAL st
( f = X --> 0 & f in L1_Functions M & b1 = a.e-eq-class (f,M) )
;
uniqueness
for b1, b2 being Element of CosetSet M st ex f being PartFunc of X,REAL st
( f = X --> 0 & f in L1_Functions M & b1 = a.e-eq-class (f,M) ) & ex f being PartFunc of X,REAL st
( f = X --> 0 & f in L1_Functions M & b2 = a.e-eq-class (f,M) ) holds
b1 = b2
;
proof end;
end;

:: deftheorem Def16 defines zeroCoset LPSPACE1:def 16 :
for X being non empty set
for S being SigmaField of X
for M being sigma_Measure of S
for b4 being Element of CosetSet M holds
( b4 = zeroCoset M iff ex f being PartFunc of X,REAL st
( f = X --> 0 & f in L1_Functions M & b4 = a.e-eq-class (f,M) ) );

definition
let X be non empty set ;
let S be SigmaField of X;
let M be sigma_Measure of S;
func lmultCoset M -> Function of [:REAL,():],() means :Def17: :: LPSPACE1:def 17
for z being Real
for A being Element of CosetSet M
for f being PartFunc of X,REAL st f in A holds
it . (z,A) = a.e-eq-class ((z (#) f),M);
existence
ex b1 being Function of [:REAL,():],() st
for z being Real
for A being Element of CosetSet M
for f being PartFunc of X,REAL st f in A holds
b1 . (z,A) = a.e-eq-class ((z (#) f),M)
proof end;
uniqueness
for b1, b2 being Function of [:REAL,():],() st ( for z being Real
for A being Element of CosetSet M
for f being PartFunc of X,REAL st f in A holds
b1 . (z,A) = a.e-eq-class ((z (#) f),M) ) & ( for z being Real
for A being Element of CosetSet M
for f being PartFunc of X,REAL st f in A holds
b2 . (z,A) = a.e-eq-class ((z (#) f),M) ) holds
b1 = b2
proof end;
end;

:: deftheorem Def17 defines lmultCoset LPSPACE1:def 17 :
for X being non empty set
for S being SigmaField of X
for M being sigma_Measure of S
for b4 being Function of [:REAL,():],() holds
( b4 = lmultCoset M iff for z being Real
for A being Element of CosetSet M
for f being PartFunc of X,REAL st f in A holds
b4 . (z,A) = a.e-eq-class ((z (#) f),M) );

definition
let X be non empty set ;
let S be SigmaField of X;
let M be sigma_Measure of S;
existence
ex b1 being non empty right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct st
( the carrier of b1 = CosetSet M & the addF of b1 = addCoset M & 0. b1 = zeroCoset M & the Mult of b1 = lmultCoset M )
proof end;
uniqueness
for b1, b2 being non empty right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct st the carrier of b1 = CosetSet M & the addF of b1 = addCoset M & 0. b1 = zeroCoset M & the Mult of b1 = lmultCoset M & the carrier of b2 = CosetSet M & the addF of b2 = addCoset M & 0. b2 = zeroCoset M & the Mult of b2 = lmultCoset M holds
b1 = b2
;
end;

:: deftheorem Def18 defines Pre-L-Space LPSPACE1:def 18 :
for X being non empty set
for S being SigmaField of X
for M being sigma_Measure of S
for b4 being non empty right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct holds
( b4 = Pre-L-Space M iff ( the carrier of b4 = CosetSet M & the addF of b4 = addCoset M & 0. b4 = zeroCoset M & the Mult of b4 = lmultCoset M ) );

theorem Th43: :: LPSPACE1:43
for X being non empty set
for S being SigmaField of X
for M being sigma_Measure of S
for f, g being PartFunc of X,REAL st f in L1_Functions M & g in L1_Functions M & f a.e.= g,M holds
Integral (M,f) = Integral (M,g)
proof end;

theorem Th44: :: LPSPACE1:44
for X being non empty set
for S being SigmaField of X
for M being sigma_Measure of S
for f being PartFunc of X,REAL st f is_integrable_on M holds
( Integral (M,f) in REAL & Integral (M,(abs f)) in REAL & abs f is_integrable_on M )
proof end;

theorem Th45: :: LPSPACE1:45
for X being non empty set
for S being SigmaField of X
for M being sigma_Measure of S
for f, g being PartFunc of X,REAL st f in L1_Functions M & g in L1_Functions M & f a.e.= g,M holds
( abs f a.e.= abs g,M & Integral (M,(abs f)) = Integral (M,(abs g)) )
proof end;

theorem Th46: :: LPSPACE1:46
for X being non empty set
for S being SigmaField of X
for M being sigma_Measure of S
for f, g being PartFunc of X,REAL st ex x being VECTOR of () st
( f in x & g in x ) holds
( f a.e.= g,M & f in L1_Functions M & g in L1_Functions M )
proof end;

theorem Th47: :: LPSPACE1:47
for X being non empty set
for S being SigmaField of X
for M being sigma_Measure of S
for f being PartFunc of X,REAL
for x being Point of () st f in x holds
( f is_integrable_on M & f in L1_Functions M & abs f is_integrable_on M )
proof end;

theorem Th48: :: LPSPACE1:48
for X being non empty set
for S being SigmaField of X
for M being sigma_Measure of S
for f, g being PartFunc of X,REAL
for x being Point of () st f in x & g in x holds
( f a.e.= g,M & Integral (M,f) = Integral (M,g) & Integral (M,(abs f)) = Integral (M,(abs g)) )
proof end;

definition
let X be non empty set ;
let S be SigmaField of X;
let M be sigma_Measure of S;
func L-1-Norm M -> Function of the carrier of (),REAL means :Def19: :: LPSPACE1:def 19
for x being Point of () ex f being PartFunc of X,REAL st
( f in x & it . x = Integral (M,(abs f)) );
existence
ex b1 being Function of the carrier of (),REAL st
for x being Point of () ex f being PartFunc of X,REAL st
( f in x & b1 . x = Integral (M,(abs f)) )
proof end;
uniqueness
for b1, b2 being Function of the carrier of (),REAL st ( for x being Point of () ex f being PartFunc of X,REAL st
( f in x & b1 . x = Integral (M,(abs f)) ) ) & ( for x being Point of () ex f being PartFunc of X,REAL st
( f in x & b2 . x = Integral (M,(abs f)) ) ) holds
b1 = b2
proof end;
end;

:: deftheorem Def19 defines L-1-Norm LPSPACE1:def 19 :
for X being non empty set
for S being SigmaField of X
for M being sigma_Measure of S
for b4 being Function of the carrier of (),REAL holds
( b4 = L-1-Norm M iff for x being Point of () ex f being PartFunc of X,REAL st
( f in x & b4 . x = Integral (M,(abs f)) ) );

definition
let X be non empty set ;
let S be SigmaField of X;
let M be sigma_Measure of S;
func L-1-Space M -> non empty NORMSTR equals :: LPSPACE1:def 20
NORMSTR(# the carrier of (), the ZeroF of (), the addF of (), the Mult of (),() #);
coherence
NORMSTR(# the carrier of (), the ZeroF of (), the addF of (), the Mult of (),() #) is non empty NORMSTR
;
end;

:: deftheorem defines L-1-Space LPSPACE1:def 20 :
for X being non empty set
for S being SigmaField of X
for M being sigma_Measure of S holds L-1-Space M = NORMSTR(# the carrier of (), the ZeroF of (), the addF of (), the Mult of (),() #);

theorem Th49: :: LPSPACE1:49
for X being non empty set
for S being SigmaField of X
for M being sigma_Measure of S
for x being Point of () holds
( ex f being PartFunc of X,REAL st
( f in L1_Functions M & x = a.e-eq-class (f,M) & = Integral (M,(abs f)) ) & ( for f being PartFunc of X,REAL st f in x holds
Integral (M,(abs f)) = ) )
proof end;

theorem Th50: :: LPSPACE1:50
for X being non empty set
for S being SigmaField of X
for M being sigma_Measure of S
for f being PartFunc of X,REAL
for x being Point of () st f in x holds
( x = a.e-eq-class (f,M) & = Integral (M,(abs f)) )
proof end;

theorem Th51: :: LPSPACE1:51
for a being Real
for X being non empty set
for S being SigmaField of X
for M being sigma_Measure of S
for f, g being PartFunc of X,REAL
for x, y being Point of () holds
( ( f in x & g in y implies f + g in x + y ) & ( f in x implies a (#) f in a * x ) )
proof end;

theorem Th52: :: LPSPACE1:52
for r being Real
for X being non empty set
for S being SigmaField of X
for E being Element of S
for f being PartFunc of X,REAL st E = dom f & ( for x being set st x in dom f holds
f . x = r ) holds
f is E -measurable
proof end;

theorem Th53: :: LPSPACE1:53
for X being non empty set
for S being SigmaField of X
for M being sigma_Measure of S
for f being PartFunc of X,REAL st f in L1_Functions M & Integral (M,(abs f)) = 0 holds
f a.e.= X --> 0,M
proof end;

theorem Th54: :: LPSPACE1:54
for X being non empty set
for S being SigmaField of X
for M being sigma_Measure of S holds Integral (M,(abs (X --> 0))) = 0
proof end;

theorem Th55: :: LPSPACE1:55
for X being non empty set
for S being SigmaField of X
for M being sigma_Measure of S
for f, g being PartFunc of X,REAL st f is_integrable_on M & g is_integrable_on M holds
Integral (M,(abs (f + g))) <= (Integral (M,(abs f))) + (Integral (M,(abs g)))
proof end;

Lm7: for X being non empty set
for S being SigmaField of X
for M being sigma_Measure of S holds
( L-1-Space M is RealNormSpace-like & L-1-Space M is vector-distributive & L-1-Space M is scalar-distributive & L-1-Space M is scalar-associative & L-1-Space M is scalar-unital & L-1-Space M is Abelian & L-1-Space M is add-associative & L-1-Space M is right_zeroed & L-1-Space M is right_complementable )

proof end;

registration
let X be non empty set ;
let S be SigmaField of X;
let M be sigma_Measure of S;
coherence by Lm7;
end;