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 & g in a.e-eq-class_Lp f,M,k holds
a.e-eq-class_Lp f,M,k = a.e-eq-class_Lp g,M,k
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 & g in a.e-eq-class_Lp f,M,k holds
a.e-eq-class_Lp f,M,k = a.e-eq-class_Lp g,M,k
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 & g in a.e-eq-class_Lp f,M,k holds
a.e-eq-class_Lp f,M,k = a.e-eq-class_Lp g,M,k
let f, g be PartFunc of X,REAL ; :: thesis: for k being positive Real st f in Lp_Functions M,k & g in a.e-eq-class_Lp f,M,k holds
a.e-eq-class_Lp f,M,k = a.e-eq-class_Lp g,M,k
let k be positive Real; :: thesis: ( f in Lp_Functions M,k & g in a.e-eq-class_Lp f,M,k implies a.e-eq-class_Lp f,M,k = a.e-eq-class_Lp g,M,k )
assume that
A1: f in Lp_Functions M,k and
A2: g in a.e-eq-class_Lp f,M,k ; :: thesis: a.e-eq-class_Lp f,M,k = a.e-eq-class_Lp g,M,k
ex E being Element of S st
( M . (E ` ) = 0 & dom f = E & f is_measurable_on E ) by A1, EQC00a;
hence a.e-eq-class_Lp f,M,k = a.e-eq-class_Lp g,M,k by EQC02b, A2, EQC00c; :: thesis: verum