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

for M being sigma_Measure of S

for f, g being PartFunc of X,REAL

for k being positive Real st ex E being 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) holds

f a.e.= g,M

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

for f, g being PartFunc of X,REAL

for k being positive Real st ex E being 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) holds

f a.e.= g,M

let M be sigma_Measure of S; :: thesis: for f, g being PartFunc of X,REAL

for k being positive Real st ex E being 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) holds

f a.e.= g,M

let f, g be PartFunc of X,REAL; :: thesis: for k being positive Real st ex E being 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) holds

f a.e.= g,M

let k be positive Real; :: thesis: ( ex E being 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 )

assume that

A1: ex E being 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) ; :: thesis: f a.e.= g,M

consider x being object such that

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

consider r being 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 f a.e.= g,M by A5, LPSPACE1:30; :: thesis: verum

for M being sigma_Measure of S

for f, g being PartFunc of X,REAL

for k being positive Real st ex E being 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) holds

f a.e.= g,M

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

for f, g being PartFunc of X,REAL

for k being positive Real st ex E being 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) holds

f a.e.= g,M

let M be sigma_Measure of S; :: thesis: for f, g being PartFunc of X,REAL

for k being positive Real st ex E being 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) holds

f a.e.= g,M

let f, g be PartFunc of X,REAL; :: thesis: for k being positive Real st ex E being 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) holds

f a.e.= g,M

let k be positive Real; :: thesis: ( ex E being 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 )

assume that

A1: ex E being 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) ; :: thesis: f a.e.= g,M

consider x being object such that

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

consider r being 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 f a.e.= g,M by A5, LPSPACE1:30; :: thesis: verum