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 Lm10;
a.e-eq-class (f,M) c= a.e-eq-class_Lp (f,M,1) by Lm11;
hence a.e-eq-class_Lp (f,M,1) = a.e-eq-class (f,M) by A1, XBOOLE_0:def 10; :: thesis: verum