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 E -measurable ) & 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 E -measurable ) & 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 E -measurable ) & 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 E -measurable ) & 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 E -measurable ) & 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 E -measurable ) 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, Th38;
hence f a.e.= g,M by A2, A3, Th39; :: thesis: verum