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 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 f, g 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 f, g 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 f, g 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

A1: g in Lp_Functions (M,k) and

A2: g a.e.= f,M ; :: thesis: g in a.e-eq-class_Lp (f,M,k)

f a.e.= g,M by A2;

hence g in a.e-eq-class_Lp (f,M,k) by A1; :: thesis: verum

for M being sigma_Measure of S

for f, g 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 f, g 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 f, g 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 f, g 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

A1: g in Lp_Functions (M,k) and

A2: g a.e.= f,M ; :: thesis: g in a.e-eq-class_Lp (f,M,k)

f a.e.= g,M by A2;

hence g in a.e-eq-class_Lp (f,M,k) by A1; :: thesis: verum