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
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; :: thesis: 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; :: thesis: 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; :: thesis: 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; :: thesis: 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; :: thesis: ( f in Lp_Functions (M,k) implies a (#) f in Lp_Functions (M,k) )
assume f in Lp_Functions (M,k) ; :: thesis: a (#) f in Lp_Functions (M,k)
then consider f1 being PartFunc of X,REAL such that
A2: ( f1 = f & ex Ef1 being Element of S st
( M . (Ef1 `) = 0 & dom f1 = Ef1 & f1 is_measurable_on Ef1 & (abs f1) to_power k is_integrable_on M ) ) ;
consider Ef being Element of S such that
A3: ( M . (Ef `) = 0 & dom f1 = Ef & f1 is_measurable_on Ef & (abs f1) to_power k is_integrable_on M ) by A2;
A4: ( dom (a (#) f1) = Ef & a (#) f1 is_measurable_on Ef ) by A3, MESFUNC6:21, VALUED_1:def 5;
((abs a) to_power k) (#) ((abs f1) to_power k) is_integrable_on M by A2, MESFUNC6:102;
then (abs (a (#) f1)) to_power k is_integrable_on M by Lm001e;
hence a (#) f in Lp_Functions (M,k) by A2, A3, A4; :: thesis: verum