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
ex E being Element of S st
( M . (E ` ) = 0 & dom f = E & f is_measurable_on E )
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
ex E being Element of S st
( M . (E ` ) = 0 & dom f = E & f is_measurable_on E )
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
ex E being Element of S st
( M . (E ` ) = 0 & dom f = E & f is_measurable_on E )
let f be PartFunc of X,REAL ; :: thesis: for k being positive Real st f in Lp_Functions M,k holds
ex E being Element of S st
( M . (E ` ) = 0 & dom f = E & f is_measurable_on E )
let k be positive Real; :: thesis: ( f in Lp_Functions M,k implies ex E being Element of S st
( M . (E ` ) = 0 & dom f = E & f is_measurable_on E ) )
assume f in Lp_Functions M,k ; :: thesis: ex E being Element of S st
( M . (E ` ) = 0 & dom f = E & f is_measurable_on E )
then ex f1 being PartFunc of X,REAL st
( f = f1 & ex E being Element of S st
( M . (E ` ) = 0 & dom f1 = E & f1 is_measurable_on E & (abs f1) to_power k is_integrable_on M ) ) ;
hence ex E being Element of S st
( M . (E ` ) = 0 & dom f = E & f is_measurable_on E ) ; :: thesis: verum