let X be non empty set ; 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 ex E being Element of S st
( M . (E `) = 0 & E = dom g & g is E -measurable ) & a.e-eq-class_Lp (f,M,k) <> {} & a.e-eq-class_Lp (f,M,k) = a.e-eq-class_Lp (g,M,k) holds
f a.e.= g,M
let S be SigmaField of X; for M being sigma_Measure of S
for f, g being PartFunc of X,REAL
for k being positive Real st ex E being Element of S st
( M . (E `) = 0 & E = dom g & g is E -measurable ) & a.e-eq-class_Lp (f,M,k) <> {} & a.e-eq-class_Lp (f,M,k) = a.e-eq-class_Lp (g,M,k) holds
f a.e.= g,M
let M be sigma_Measure of S; for f, g being PartFunc of X,REAL
for k being positive Real st ex E being Element of S st
( M . (E `) = 0 & E = dom g & g is E -measurable ) & a.e-eq-class_Lp (f,M,k) <> {} & a.e-eq-class_Lp (f,M,k) = a.e-eq-class_Lp (g,M,k) holds
f a.e.= g,M
let f, g be PartFunc of X,REAL; for k being positive Real st ex E being Element of S st
( M . (E `) = 0 & E = dom g & g is E -measurable ) & a.e-eq-class_Lp (f,M,k) <> {} & a.e-eq-class_Lp (f,M,k) = a.e-eq-class_Lp (g,M,k) holds
f a.e.= g,M
let k be positive Real; ( ex E being Element of S st
( M . (E `) = 0 & E = dom g & g is E -measurable ) & a.e-eq-class_Lp (f,M,k) <> {} & a.e-eq-class_Lp (f,M,k) = a.e-eq-class_Lp (g,M,k) implies f a.e.= g,M )
assume that
A1:
ex E being Element of S st
( M . (E `) = 0 & E = dom g & g is E -measurable )
and
A2:
a.e-eq-class_Lp (f,M,k) <> {}
and
A3:
a.e-eq-class_Lp (f,M,k) = a.e-eq-class_Lp (g,M,k)
; f a.e.= g,M
consider x being object such that
A4:
x in a.e-eq-class_Lp (f,M,k)
by A2, XBOOLE_0:def 1;
consider r being PartFunc of X,REAL such that
A5:
( x = r & r in Lp_Functions (M,k) & f a.e.= r,M )
by A4;
r a.e.= g,M
by A1, A3, A4, A5, Th37;
hence
f a.e.= g,M
by A5, LPSPACE1:30; verum