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 a being Real
for k being positive Real st f in Lp_Functions M,k & g in Lp_Functions M,k & a.e-eq-class_Lp f,M,k = a.e-eq-class_Lp g,M,k holds
a.e-eq-class_Lp (a (#) f),M,k = a.e-eq-class_Lp (a (#) 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 a being Real
for k being positive Real st f in Lp_Functions M,k & g in Lp_Functions M,k & a.e-eq-class_Lp f,M,k = a.e-eq-class_Lp g,M,k holds
a.e-eq-class_Lp (a (#) f),M,k = a.e-eq-class_Lp (a (#) g),M,k
let M be sigma_Measure of S; :: thesis: for f, g being PartFunc of X,REAL
for a being Real
for k being positive Real st f in Lp_Functions M,k & g in Lp_Functions M,k & a.e-eq-class_Lp f,M,k = a.e-eq-class_Lp g,M,k holds
a.e-eq-class_Lp (a (#) f),M,k = a.e-eq-class_Lp (a (#) g),M,k
let f, g be PartFunc of X,REAL ; :: thesis: for a being Real
for k being positive Real st f in Lp_Functions M,k & g in Lp_Functions M,k & a.e-eq-class_Lp f,M,k = a.e-eq-class_Lp g,M,k holds
a.e-eq-class_Lp (a (#) f),M,k = a.e-eq-class_Lp (a (#) g),M,k
let a be Real; :: thesis: for k being positive Real st f in Lp_Functions M,k & g in Lp_Functions M,k & a.e-eq-class_Lp f,M,k = a.e-eq-class_Lp g,M,k holds
a.e-eq-class_Lp (a (#) f),M,k = a.e-eq-class_Lp (a (#) g),M,k
let k be positive Real; :: thesis: ( f in Lp_Functions M,k & g in Lp_Functions M,k & a.e-eq-class_Lp f,M,k = a.e-eq-class_Lp g,M,k implies a.e-eq-class_Lp (a (#) f),M,k = a.e-eq-class_Lp (a (#) g),M,k )
assume A1: ( f in Lp_Functions M,k & g in Lp_Functions M,k & a.e-eq-class_Lp f,M,k = a.e-eq-class_Lp g,M,k ) ; :: thesis: a.e-eq-class_Lp (a (#) f),M,k = a.e-eq-class_Lp (a (#) g),M,k
then A2: ( ex E being Element of S st
( M . (E ` ) = 0 & dom f = E & f is_measurable_on E ) & ex E being Element of S st
( M . (E ` ) = 0 & dom g = E & g is_measurable_on E ) ) by EQC00a;
f in a.e-eq-class_Lp g,M,k by A1, EQC01;
then ( f a.e.= g,M & a (#) f in Lp_Functions M,k & a (#) g in Lp_Functions M,k ) by A2, EQC00c, Th01bLp;
hence a.e-eq-class_Lp (a (#) f),M,k = a.e-eq-class_Lp (a (#) g),M,k by EQC02bx, LPSPACE1:32; :: thesis: verum