let X be non empty set ; :: thesis: for S being SigmaField of X
for M being sigma_Measure of S
for g, f 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_measurable_on E ) & 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 g, f 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_measurable_on E ) & 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 g, f 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_measurable_on E ) & 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 g, f 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_measurable_on E ) & 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_measurable_on E ) & 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
A02: ex E being Element of S st
( M . (E ` ) = 0 & E = dom g & g is_measurable_on E ) and
A1: a.e-eq-class_Lp f,M,k <> {} and
A2: 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 set such that
A3: x in a.e-eq-class_Lp f,M,k by A1, XBOOLE_0:def 1;
consider r being PartFunc of X,REAL such that
A4: ( x = r & r in Lp_Functions M,k & f a.e.= r,M ) by A3;
r a.e.= g,M by A02, A2, A3, A4, EQC00c;
hence f a.e.= g,M by A4, LPSPACE1:30; :: thesis: verum