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 E -measurable )

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 E -measurable )

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 E -measurable )

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 E -measurable )

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 E -measurable ) )

assume f in Lp_Functions (M,k) ; :: thesis: ex E being Element of S st
( M . (E `) = 0 & dom f = E & f is E -measurable )

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 E -measurable & (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 E -measurable ) ; :: thesis: verum