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 st f a.e.= g,M holds
a.e-eq-class_Lp (f,M,k) = a.e-eq-class_Lp (g,M,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 st f a.e.= g,M holds
a.e-eq-class_Lp (f,M,k) = a.e-eq-class_Lp (g,M,k)
let M be sigma_Measure of S; :: thesis: for f, g being PartFunc of X,REAL
for k being positive Real st f a.e.= g,M holds
a.e-eq-class_Lp (f,M,k) = a.e-eq-class_Lp (g,M,k)
let f, g be PartFunc of X,REAL; :: thesis: for k being positive Real st f a.e.= g,M holds
a.e-eq-class_Lp (f,M,k) = a.e-eq-class_Lp (g,M,k)
let k be positive Real; :: thesis: ( f a.e.= g,M implies a.e-eq-class_Lp (f,M,k) = a.e-eq-class_Lp (g,M,k) )
assume A1: f a.e.= g,M ; :: thesis: a.e-eq-class_Lp (f,M,k) = a.e-eq-class_Lp (g,M,k)
now :: thesis: for x being object st x in a.e-eq-class_Lp (f,M,k) holds
x in a.e-eq-class_Lp (g,M,k)
let x be object ; :: thesis: ( x in a.e-eq-class_Lp (f,M,k) implies x in a.e-eq-class_Lp (g,M,k) )
assume x in a.e-eq-class_Lp (f,M,k) ; :: thesis: x in a.e-eq-class_Lp (g,M,k)
then consider r being PartFunc of X,REAL such that
A2: ( x = r & r in Lp_Functions (M,k) & f a.e.= r,M ) ;
r a.e.= f,M by A2;
then r a.e.= g,M by A1, LPSPACE1:30;
then g a.e.= r,M ;
hence x in a.e-eq-class_Lp (g,M,k) by A2; :: thesis: verum
end;
then A3: a.e-eq-class_Lp (f,M,k) c= a.e-eq-class_Lp (g,M,k) ;
now :: thesis: for x being object st x in a.e-eq-class_Lp (g,M,k) holds
x in a.e-eq-class_Lp (f,M,k)
let x be object ; :: thesis: ( x in a.e-eq-class_Lp (g,M,k) implies x in a.e-eq-class_Lp (f,M,k) )
assume x in a.e-eq-class_Lp (g,M,k) ; :: thesis: x in a.e-eq-class_Lp (f,M,k)
then consider r being PartFunc of X,REAL such that
A4: ( x = r & r in Lp_Functions (M,k) & g a.e.= r,M ) ;
( r a.e.= g,M & g a.e.= f,M ) by A1, A4;
then r a.e.= f,M by LPSPACE1:30;
then f a.e.= r,M ;
hence x in a.e-eq-class_Lp (f,M,k) by A4; :: thesis: verum
end;
then a.e-eq-class_Lp (g,M,k) c= a.e-eq-class_Lp (f,M,k) ;
hence a.e-eq-class_Lp (f,M,k) = a.e-eq-class_Lp (g,M,k) by A3; :: thesis: verum