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 f in Lp_Functions M,k & 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 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 f in Lp_Functions M,k & 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 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 f in Lp_Functions M,k & 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 g,M,k holds
f a.e.= g,M
let f, g be PartFunc of X,REAL ; :: thesis: for k being positive Real st f in Lp_Functions M,k & 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 g,M,k holds
f a.e.= g,M
let k be positive Real; :: thesis: ( f in Lp_Functions M,k & 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 g,M,k implies f a.e.= g,M )
assume that
A1: f in Lp_Functions M,k and
A2: ex E being Element of S st
( M . (E ` ) = 0 & E = dom g & g is_measurable_on E ) and
A3: a.e-eq-class_Lp f,M,k = a.e-eq-class_Lp g,M,k ; :: thesis: f a.e.= g,M
not a.e-eq-class_Lp f,M,k is empty by A1, EQC01;
hence f a.e.= g,M by A2, A3, EQC02a; :: thesis: verum