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

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