let X be non empty set ; for S being SigmaField of X
for M being sigma_Measure of S
for f being PartFunc of X,REAL
for a being Real
for k being positive Real st f in Lp_Functions (M,k) holds
a (#) f in Lp_Functions (M,k)
let S be SigmaField of X; for M being sigma_Measure of S
for f being PartFunc of X,REAL
for a being Real
for k being positive Real st f in Lp_Functions (M,k) holds
a (#) f in Lp_Functions (M,k)
let M be sigma_Measure of S; for f being PartFunc of X,REAL
for a being Real
for k being positive Real st f in Lp_Functions (M,k) holds
a (#) f in Lp_Functions (M,k)
let f be PartFunc of X,REAL; for a being Real
for k being positive Real st f in Lp_Functions (M,k) holds
a (#) f in Lp_Functions (M,k)
let a be Real; for k being positive Real st f in Lp_Functions (M,k) holds
a (#) f in Lp_Functions (M,k)
let k be positive Real; ( f in Lp_Functions (M,k) implies a (#) f in Lp_Functions (M,k) )
assume
f in Lp_Functions (M,k)
; a (#) f in Lp_Functions (M,k)
then consider f1 being PartFunc of X,REAL such that
A1:
( f1 = f & ex Ef1 being Element of S st
( M . (Ef1 `) = 0 & dom f1 = Ef1 & f1 is Ef1 -measurable & (abs f1) to_power k is_integrable_on M ) )
;
consider Ef being Element of S such that
A2:
( M . (Ef `) = 0 & dom f1 = Ef & f1 is Ef -measurable & (abs f1) to_power k is_integrable_on M )
by A1;
A3:
( dom (a (#) f1) = Ef & a (#) f1 is Ef -measurable )
by A2, MESFUNC6:21, VALUED_1:def 5;
(|.a.| to_power k) (#) ((abs f1) to_power k) is_integrable_on M
by A1, MESFUNC6:102;
then
(abs (a (#) f1)) to_power k is_integrable_on M
by Th18;
hence
a (#) f in Lp_Functions (M,k)
by A1, A2, A3; verum