let X be non empty set ; :: thesis: for S being SigmaField of X
for M being sigma_Measure of S
for f, g being PartFunc of X,REAL
for k being positive Real
for x being Point of (Pre-Lp-Space (M,k)) st f in x & g in x holds
( f a.e.= g,M & Integral (M,((abs f) to_power k)) = Integral (M,((abs g) to_power k)) )
let S be SigmaField of X; :: thesis: for M being sigma_Measure of S
for f, g being PartFunc of X,REAL
for k being positive Real
for x being Point of (Pre-Lp-Space (M,k)) st f in x & g in x holds
( f a.e.= g,M & Integral (M,((abs f) to_power k)) = Integral (M,((abs g) to_power k)) )
let M be sigma_Measure of S; :: thesis: for f, g being PartFunc of X,REAL
for k being positive Real
for x being Point of (Pre-Lp-Space (M,k)) st f in x & g in x holds
( f a.e.= g,M & Integral (M,((abs f) to_power k)) = Integral (M,((abs g) to_power k)) )
let f, g 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 & g in x holds
( f a.e.= g,M & Integral (M,((abs f) to_power k)) = Integral (M,((abs g) to_power k)) )
let k be positive Real; :: thesis: for x being Point of (Pre-Lp-Space (M,k)) st f in x & g in x holds
( f a.e.= g,M & Integral (M,((abs f) to_power k)) = Integral (M,((abs g) to_power k)) )
let x be Point of (Pre-Lp-Space (M,k)); :: thesis: ( f in x & g in x implies ( f a.e.= g,M & Integral (M,((abs f) to_power k)) = Integral (M,((abs g) to_power k)) ) )
assume ( f in x & g in x ) ; :: thesis: ( f a.e.= g,M & Integral (M,((abs f) to_power k)) = Integral (M,((abs g) to_power k)) )
then ( f a.e.= g,M & f in Lp_Functions (M,k) & g in Lp_Functions (M,k) ) by Th50;
hence ( f a.e.= g,M & Integral (M,((abs f) to_power k)) = Integral (M,((abs g) to_power k)) ) by Th48; :: thesis: verum