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) to_power k is_integrable_on M
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) to_power k is_integrable_on M
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) to_power k is_integrable_on M
let f be PartFunc of X,REAL; :: thesis: for k being positive Real st f in Lp_Functions (M,k) holds
(abs f) to_power k is_integrable_on M
let k be positive Real; :: thesis: ( f in Lp_Functions (M,k) implies (abs f) to_power k is_integrable_on M )
assume f in Lp_Functions (M,k) ; :: thesis: (abs f) to_power k is_integrable_on M
then ex f2 being PartFunc of X,REAL st
( f = f2 & ex E being Element of S st
( M . (E `) = 0 & dom f2 = E & f2 is E -measurable & (abs f2) to_power k is_integrable_on M ) ) ;
hence (abs f) to_power k is_integrable_on M ; :: thesis: verum