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
for x being Point of (Pre-Lp-Space (M,k)) st f in x holds
( (abs f) to_power k is_integrable_on M & 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
for x being Point of (Pre-Lp-Space (M,k)) st f in x holds
( (abs f) to_power k is_integrable_on M & 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
for x being Point of (Pre-Lp-Space (M,k)) st f in x holds
( (abs f) to_power k is_integrable_on M & f in Lp_Functions (M,k) )
let f be PartFunc of X,REAL; :: thesis: for k being positive Real
for x being Point of (Pre-Lp-Space (M,k)) st f in x holds
( (abs f) to_power k is_integrable_on M & f in Lp_Functions (M,k) )
let k be positive Real; :: thesis: for x being Point of (Pre-Lp-Space (M,k)) st f in x holds
( (abs f) to_power k is_integrable_on M & f in Lp_Functions (M,k) )
let x be Point of (Pre-Lp-Space (M,k)); :: thesis: ( f in x implies ( (abs f) to_power k is_integrable_on M & f in Lp_Functions (M,k) ) )
assume A1: f in x ; :: thesis: ( (abs f) to_power k is_integrable_on M & f in Lp_Functions (M,k) )
x in the carrier of (Pre-Lp-Space (M,k)) ;
then x in CosetSet (M,k) by Def11;
then consider h being PartFunc of X,REAL such that
A2: ( x = a.e-eq-class_Lp (h,M,k) & h in Lp_Functions (M,k) ) ;
ex g being PartFunc of X,REAL st
( f = g & g in Lp_Functions (M,k) & h a.e.= g,M ) by A1, A2;
then ex f0 being PartFunc of X,REAL st
( f = f0 & ex ND being Element of S st
( M . (ND `) = 0 & dom f0 = ND & f0 is ND -measurable & (abs f0) to_power k is_integrable_on M ) ) ;
hence ( (abs f) to_power k is_integrable_on M & f in Lp_Functions (M,k) ) ; :: thesis: verum