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 A2: 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_measurable_on ND & (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 A2, MESFUNC6:def 9;
then consider A being Element of S such that
A3: ( A = dom (R_EAL ((abs f) to_power k)) & R_EAL ((abs f) to_power k) is_measurable_on A ) by MESFUNC5:def 17;
( A = dom ((abs f) to_power k) & (abs f) to_power k is_measurable_on A ) by A3, MESFUNC6:def 6;
hence 0 <= Integral M,((abs f) to_power k) by MESFUNC6:84; :: thesis: verum