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
( Integral (M,((abs f) to_power k)) in REAL & 0 <= Integral (M,((abs f) to_power 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
( Integral (M,((abs f) to_power k)) in REAL & 0 <= Integral (M,((abs f) to_power 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
( Integral (M,((abs f) to_power k)) in REAL & 0 <= Integral (M,((abs f) to_power k)) )
let f be PartFunc of X,REAL; :: thesis: for k being positive Real st f in Lp_Functions (M,k) holds
( Integral (M,((abs f) to_power k)) in REAL & 0 <= Integral (M,((abs f) to_power k)) )
let k be positive Real; :: thesis: ( f in Lp_Functions (M,k) implies ( Integral (M,((abs f) to_power k)) in REAL & 0 <= Integral (M,((abs f) to_power k)) ) )
assume f in Lp_Functions (M,k) ; :: thesis: ( Integral (M,((abs f) to_power k)) in REAL & 0 <= Integral (M,((abs f) to_power k)) )
then A1: ex f1 being PartFunc of X,REAL st
( f = f1 & ex ND being Element of S st
( M . (ND `) = 0 & dom f1 = ND & f1 is ND -measurable & (abs f1) to_power k is_integrable_on M ) ) ;
then ( -infty < Integral (M,((abs f) to_power k)) & Integral (M,((abs f) to_power k)) < +infty ) by MESFUNC6:90;
hence Integral (M,((abs f) to_power k)) in REAL by XXREAL_0:14; :: thesis: 0 <= Integral (M,((abs f) to_power k))
R_EAL ((abs f) to_power k) is_integrable_on M by A1;
then consider A being Element of S such that
A2: ( A = dom (R_EAL ((abs f) to_power k)) & R_EAL ((abs f) to_power k) is A -measurable ) ;
( A = dom ((abs f) to_power k) & (abs f) to_power k is A -measurable ) by A2;
hence 0 <= Integral (M,((abs f) to_power k)) by MESFUNC6:84; :: thesis: verum