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