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 E -measurable ) by A1, Th35;
hence a.e-eq-class_Lp (f,M,k) = a.e-eq-class_Lp (g,M,k) by Th41, A2, Th37; :: thesis: verum