let X be non empty set ; :: thesis: for S being SigmaField of X
for M being sigma_Measure of S
for f being PartFunc of X,REAL st ex E being Element of S st
( M . (E ` ) = 0 & E = dom f & f is_measurable_on E ) holds
a.e-eq-class_Lp f,M,1 = a.e-eq-class f,M
let S be SigmaField of X; :: thesis: for M being sigma_Measure of S
for f being PartFunc of X,REAL st ex E being Element of S st
( M . (E ` ) = 0 & E = dom f & f is_measurable_on E ) holds
a.e-eq-class_Lp f,M,1 = a.e-eq-class f,M
let M be sigma_Measure of S; :: thesis: for f being PartFunc of X,REAL st ex E being Element of S st
( M . (E ` ) = 0 & E = dom f & f is_measurable_on E ) holds
a.e-eq-class_Lp f,M,1 = a.e-eq-class f,M
let f be PartFunc of X,REAL ; :: thesis: ( ex E being Element of S st
( M . (E ` ) = 0 & E = dom f & f is_measurable_on E ) implies a.e-eq-class_Lp f,M,1 = a.e-eq-class f,M )
assume ex E being Element of S st
( M . (E ` ) = 0 & E = dom f & f is_measurable_on E ) ; :: thesis: a.e-eq-class_Lp f,M,1 = a.e-eq-class f,M
then A1: a.e-eq-class_Lp f,M,1 c= a.e-eq-class f,M by Lem01;
a.e-eq-class f,M c= a.e-eq-class_Lp f,M,1 by Lem02;
hence a.e-eq-class_Lp f,M,1 = a.e-eq-class f,M by A1, XBOOLE_0:def 10; :: thesis: verum