let X be non empty set ; :: thesis: for S being SigmaField of X
for M being sigma_Measure of S
for g, f being PartFunc of X,REAL
for k being positive Real st g in Lp_Functions M,k & g a.e.= f,M holds
g in a.e-eq-class_Lp f,M,k
let S be SigmaField of X; :: thesis: for M being sigma_Measure of S
for g, f being PartFunc of X,REAL
for k being positive Real st g in Lp_Functions M,k & g a.e.= f,M holds
g in a.e-eq-class_Lp f,M,k
let M be sigma_Measure of S; :: thesis: for g, f being PartFunc of X,REAL
for k being positive Real st g in Lp_Functions M,k & g a.e.= f,M holds
g in a.e-eq-class_Lp f,M,k
let g, f be PartFunc of X,REAL ; :: thesis: for k being positive Real st g in Lp_Functions M,k & g a.e.= f,M holds
g in a.e-eq-class_Lp f,M,k
let k be positive Real; :: thesis: ( g in Lp_Functions M,k & g a.e.= f,M implies g in a.e-eq-class_Lp f,M,k )
assume that
A2: g in Lp_Functions M,k and
A3: g a.e.= f,M ; :: thesis: g in a.e-eq-class_Lp f,M,k
f a.e.= g,M by A3, LPSPACE1:29;
hence g in a.e-eq-class_Lp f,M,k by A2; :: thesis: verum