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 in Lp_Functions (M,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

abs f in Lp_Functions (M,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

abs f in Lp_Functions (M,k)

let f be PartFunc of X,REAL; :: thesis: for k being positive Real st f in Lp_Functions (M,k) holds

abs f in Lp_Functions (M,k)

let k be positive Real; :: thesis: ( f in Lp_Functions (M,k) implies abs f in Lp_Functions (M,k) )

set W = Lp_Functions (M,k);

assume f in Lp_Functions (M,k) ; :: thesis: abs f in Lp_Functions (M,k)

then consider f1 being PartFunc of X,REAL such that

A1: ( f1 = f & ex Ef1 being Element of S st

( M . (Ef1 `) = 0 & dom f1 = Ef1 & f1 is Ef1 -measurable & (abs f1) to_power k is_integrable_on M ) ) ;

consider Ef being Element of S such that

A2: ( M . (Ef `) = 0 & dom f1 = Ef & f1 is Ef -measurable & (abs f1) to_power k is_integrable_on M ) by A1;

dom (abs f1) = Ef by A2, VALUED_1:def 11;

then Z1: ( M . (Ef `) = 0 & dom (abs f1) = Ef & abs f1 is Ef -measurable & (abs (abs f1)) to_power k is_integrable_on M ) by A2, MESFUNC6:48;

thus abs f in Lp_Functions (M,k) by A1, Z1; :: 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 in Lp_Functions (M,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

abs f in Lp_Functions (M,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

abs f in Lp_Functions (M,k)

let f be PartFunc of X,REAL; :: thesis: for k being positive Real st f in Lp_Functions (M,k) holds

abs f in Lp_Functions (M,k)

let k be positive Real; :: thesis: ( f in Lp_Functions (M,k) implies abs f in Lp_Functions (M,k) )

set W = Lp_Functions (M,k);

assume f in Lp_Functions (M,k) ; :: thesis: abs f in Lp_Functions (M,k)

then consider f1 being PartFunc of X,REAL such that

A1: ( f1 = f & ex Ef1 being Element of S st

( M . (Ef1 `) = 0 & dom f1 = Ef1 & f1 is Ef1 -measurable & (abs f1) to_power k is_integrable_on M ) ) ;

consider Ef being Element of S such that

A2: ( M . (Ef `) = 0 & dom f1 = Ef & f1 is Ef -measurable & (abs f1) to_power k is_integrable_on M ) by A1;

dom (abs f1) = Ef by A2, VALUED_1:def 11;

then Z1: ( M . (Ef `) = 0 & dom (abs f1) = Ef & abs f1 is Ef -measurable & (abs (abs f1)) to_power k is_integrable_on M ) by A2, MESFUNC6:48;

thus abs f in Lp_Functions (M,k) by A1, Z1; :: thesis: verum