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 st ex E being Element of S st

( E = dom f & E = dom g & f is E -measurable & g is E -measurable ) & f is nonnegative & g is nonnegative & ( for x being Element of X st x in dom g holds

g . x <= f . x ) holds

Integral (M,g) <= Integral (M,f)

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

for f, g being PartFunc of X,REAL st ex E being Element of S st

( E = dom f & E = dom g & f is E -measurable & g is E -measurable ) & f is nonnegative & g is nonnegative & ( for x being Element of X st x in dom g holds

g . x <= f . x ) holds

Integral (M,g) <= Integral (M,f)

let M be sigma_Measure of S; :: thesis: for f, g being PartFunc of X,REAL st ex E being Element of S st

( E = dom f & E = dom g & f is E -measurable & g is E -measurable ) & f is nonnegative & g is nonnegative & ( for x being Element of X st x in dom g holds

g . x <= f . x ) holds

Integral (M,g) <= Integral (M,f)

let f, g be PartFunc of X,REAL; :: thesis: ( ex E being Element of S st

( E = dom f & E = dom g & f is E -measurable & g is E -measurable ) & f is nonnegative & g is nonnegative & ( for x being Element of X st x in dom g holds

g . x <= f . x ) implies Integral (M,g) <= Integral (M,f) )

assume that

A1: ex A being Element of S st

( A = dom f & A = dom g & f is A -measurable & g is A -measurable ) and

A2: ( f is nonnegative & g is nonnegative ) and

A3: for x being Element of X st x in dom g holds

g . x <= f . x ; :: thesis: Integral (M,g) <= Integral (M,f)

A4: ( Integral (M,g) = integral+ (M,(R_EAL g)) & Integral (M,f) = integral+ (M,(R_EAL f)) ) by A1, A2, MESFUNC6:82;

consider A being Element of S such that

A5: ( A = dom f & A = dom g ) and

A6: ( f is A -measurable & g is A -measurable ) by A1;

( R_EAL f is A -measurable & R_EAL g is A -measurable ) by A6, MESFUNC6:def 1;

hence Integral (M,g) <= Integral (M,f) by A2, A3, A5, A4, MESFUNC5:85; :: thesis: verum

for M being sigma_Measure of S

for f, g being PartFunc of X,REAL st ex E being Element of S st

( E = dom f & E = dom g & f is E -measurable & g is E -measurable ) & f is nonnegative & g is nonnegative & ( for x being Element of X st x in dom g holds

g . x <= f . x ) holds

Integral (M,g) <= Integral (M,f)

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

for f, g being PartFunc of X,REAL st ex E being Element of S st

( E = dom f & E = dom g & f is E -measurable & g is E -measurable ) & f is nonnegative & g is nonnegative & ( for x being Element of X st x in dom g holds

g . x <= f . x ) holds

Integral (M,g) <= Integral (M,f)

let M be sigma_Measure of S; :: thesis: for f, g being PartFunc of X,REAL st ex E being Element of S st

( E = dom f & E = dom g & f is E -measurable & g is E -measurable ) & f is nonnegative & g is nonnegative & ( for x being Element of X st x in dom g holds

g . x <= f . x ) holds

Integral (M,g) <= Integral (M,f)

let f, g be PartFunc of X,REAL; :: thesis: ( ex E being Element of S st

( E = dom f & E = dom g & f is E -measurable & g is E -measurable ) & f is nonnegative & g is nonnegative & ( for x being Element of X st x in dom g holds

g . x <= f . x ) implies Integral (M,g) <= Integral (M,f) )

assume that

A1: ex A being Element of S st

( A = dom f & A = dom g & f is A -measurable & g is A -measurable ) and

A2: ( f is nonnegative & g is nonnegative ) and

A3: for x being Element of X st x in dom g holds

g . x <= f . x ; :: thesis: Integral (M,g) <= Integral (M,f)

A4: ( Integral (M,g) = integral+ (M,(R_EAL g)) & Integral (M,f) = integral+ (M,(R_EAL f)) ) by A1, A2, MESFUNC6:82;

consider A being Element of S such that

A5: ( A = dom f & A = dom g ) and

A6: ( f is A -measurable & g is A -measurable ) by A1;

( R_EAL f is A -measurable & R_EAL g is A -measurable ) by A6, MESFUNC6:def 1;

hence Integral (M,g) <= Integral (M,f) by A2, A3, A5, A4, MESFUNC5:85; :: thesis: verum