let X be non empty set ; :: thesis: for S being SigmaField of X

for M being sigma_Measure of S

for f being PartFunc of X,REAL st X = dom f & ( for x being Element of X st x in dom f holds

0 = f . x ) holds

( f is_integrable_on M & Integral (M,f) = 0 )

let S be SigmaField of X; :: thesis: for M being sigma_Measure of S

for f being PartFunc of X,REAL st X = dom f & ( for x being Element of X st x in dom f holds

0 = f . x ) holds

( f is_integrable_on M & Integral (M,f) = 0 )

let M be sigma_Measure of S; :: thesis: for f being PartFunc of X,REAL st X = dom f & ( for x being Element of X st x in dom f holds

0 = f . x ) holds

( f is_integrable_on M & Integral (M,f) = 0 )

let f be PartFunc of X,REAL; :: thesis: ( X = dom f & ( for x being Element of X st x in dom f holds

0 = f . x ) implies ( f is_integrable_on M & Integral (M,f) = 0 ) )

assume A1: ( X = dom f & ( for x being Element of X st x in dom f holds

0 = f . x ) ) ; :: thesis: ( f is_integrable_on M & Integral (M,f) = 0 )

X is Element of S by MEASURE1:7;

then ( R_EAL f is_integrable_on M & Integral (M,(R_EAL f)) = 0 ) by A1, LPSPACE1:22;

hence ( f is_integrable_on M & Integral (M,f) = 0 ) ; :: thesis: verum

for M being sigma_Measure of S

for f being PartFunc of X,REAL st X = dom f & ( for x being Element of X st x in dom f holds

0 = f . x ) holds

( f is_integrable_on M & Integral (M,f) = 0 )

let S be SigmaField of X; :: thesis: for M being sigma_Measure of S

for f being PartFunc of X,REAL st X = dom f & ( for x being Element of X st x in dom f holds

0 = f . x ) holds

( f is_integrable_on M & Integral (M,f) = 0 )

let M be sigma_Measure of S; :: thesis: for f being PartFunc of X,REAL st X = dom f & ( for x being Element of X st x in dom f holds

0 = f . x ) holds

( f is_integrable_on M & Integral (M,f) = 0 )

let f be PartFunc of X,REAL; :: thesis: ( X = dom f & ( for x being Element of X st x in dom f holds

0 = f . x ) implies ( f is_integrable_on M & Integral (M,f) = 0 ) )

assume A1: ( X = dom f & ( for x being Element of X st x in dom f holds

0 = f . x ) ) ; :: thesis: ( f is_integrable_on M & Integral (M,f) = 0 )

X is Element of S by MEASURE1:7;

then ( R_EAL f is_integrable_on M & Integral (M,(R_EAL f)) = 0 ) by A1, LPSPACE1:22;

hence ( f is_integrable_on M & Integral (M,f) = 0 ) ; :: thesis: verum