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 k being positive Real st f in Lp_Functions M,k holds
abs 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 k being positive Real st f in Lp_Functions M,k holds
abs f in Lp_Functions M,k
let M be sigma_Measure of S; :: thesis: for f being PartFunc of X,REAL
for k being positive Real st f in Lp_Functions M,k holds
abs f in Lp_Functions M,k
let f be PartFunc of X,REAL ; :: thesis: for k being positive Real st f in Lp_Functions M,k holds
abs f in Lp_Functions M,k
let k be positive Real; :: thesis: ( f in Lp_Functions M,k implies abs f in Lp_Functions M,k )
set W = Lp_Functions M,k;
assume f in Lp_Functions M,k ; :: thesis: abs 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;
dom (abs f1) = Ef by A3, VALUED_1:def 11;
then ex Ef being Element of S st
( M . (Ef ` ) = 0 & dom (abs f1) = Ef & abs f1 is_measurable_on Ef & (abs (abs f1)) to_power k is_integrable_on M ) by A3, MESFUNC6:48;
hence abs f in Lp_Functions M,k by A2; :: thesis: verum