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 E -measurable ) & ex E being Element of S st
( M . (E `) = 0 & dom g = E & g is E -measurable ) ) by Th35;
f in a.e-eq-class_Lp (g,M,k) by A1, Th38;
then ( f a.e.= g,M & a (#) f in Lp_Functions (M,k) & a (#) g in Lp_Functions (M,k) ) by A2, Th37, Th26;
hence a.e-eq-class_Lp ((a (#) f),M,k) = a.e-eq-class_Lp ((a (#) g),M,k) by Th42, LPSPACE1:32; :: thesis: verum