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,ExtREAL
for A, B being Element of S st ex E being Element of S st
( E = dom f & f is_measurable_on E ) & f is nonnegative & A c= B holds
Integral M,(f | A) <= Integral M,(f | B)
let S be SigmaField of X; :: thesis: for M being sigma_Measure of S
for f being PartFunc of X,ExtREAL
for A, B being Element of S st ex E being Element of S st
( E = dom f & f is_measurable_on E ) & f is nonnegative & A c= B holds
Integral M,(f | A) <= Integral M,(f | B)
let M be sigma_Measure of S; :: thesis: for f being PartFunc of X,ExtREAL
for A, B being Element of S st ex E being Element of S st
( E = dom f & f is_measurable_on E ) & f is nonnegative & A c= B holds
Integral M,(f | A) <= Integral M,(f | B)
let f be PartFunc of X,ExtREAL ; :: thesis: for A, B being Element of S st ex E being Element of S st
( E = dom f & f is_measurable_on E ) & f is nonnegative & A c= B holds
Integral M,(f | A) <= Integral M,(f | B)
let A, B be Element of S; :: thesis: ( ex E being Element of S st
( E = dom f & f is_measurable_on E ) & f is nonnegative & A c= B implies Integral M,(f | A) <= Integral M,(f | B) )
assume that
A1: ex E being Element of S st
( E = dom f & f is_measurable_on E ) and
A2: f is nonnegative and
A3: A c= B ; :: thesis: Integral M,(f | A) <= Integral M,(f | B)
consider E being Element of S such that
A4: ( E = dom f & f is_measurable_on E ) by A1;
A5: ex C being Element of S st
( C = dom (f | A) & f | A is_measurable_on C )

proof

take C = E /\ A; :: thesis: ( C = dom (f | A) & f | A is_measurable_on C )
thus dom (f | A) = C by A4, RELAT_1:90; :: thesis: f | A is_measurable_on C
C c= E by XBOOLE_1:17;
then A6: ( f is_measurable_on C & C = (dom f) /\ C ) by A4, MESFUNC1:34, XBOOLE_1:28;
then A7: f | C is_measurable_on C by Th48;
A8: dom (f | A) = C by A4, RELAT_1:90
.= dom (f | C) by A6, RELAT_1:90 ;
for x being set st x in dom (f | A) holds
(f | A) . x = (f | C) . x

proof

let x be set ; :: thesis: ( x in dom (f | A) implies (f | A) . x = (f | C) . x )
assume A9: x in dom (f | A) ; :: thesis: (f | A) . x = (f | C) . x
then (f | A) . x = f . x by FUNCT_1:70;
hence (f | A) . x = (f | C) . x by A8, A9, FUNCT_1:70; :: thesis: verum

end;

hence f | A is_measurable_on C by A7, A8, FUNCT_1:9; :: thesis: verum

end;

A10: ex C being Element of S st
( C = dom (f | B) & f | B is_measurable_on C )

proof

take C = E /\ B; :: thesis: ( C = dom (f | B) & f | B is_measurable_on C )
thus dom (f | B) = C by A4, RELAT_1:90; :: thesis: f | B is_measurable_on C
C c= E by XBOOLE_1:17;
then A11: ( f is_measurable_on C & C = (dom f) /\ C ) by A4, MESFUNC1:34, XBOOLE_1:28;
then A12: f | C is_measurable_on C by Th48;
A13: dom (f | B) = C by A4, RELAT_1:90
.= dom (f | C) by A11, RELAT_1:90 ;
for x being set st x in dom (f | B) holds
(f | B) . x = (f | C) . x

proof

let x be set ; :: thesis: ( x in dom (f | B) implies (f | B) . x = (f | C) . x )
assume A14: x in dom (f | B) ; :: thesis: (f | B) . x = (f | C) . x
then (f | B) . x = f . x by FUNCT_1:70;
hence (f | B) . x = (f | C) . x by A13, A14, FUNCT_1:70; :: thesis: verum

end;

hence f | B is_measurable_on C by A12, A13, FUNCT_1:9; :: thesis: verum

end;

integral+ M,(f | A) <= integral+ M,(f | B) by A1, A2, A3, Th89;
then Integral M,(f | A) <= integral+ M,(f | B) by A2, A5, Th21, Th94;
hence Integral M,(f | A) <= Integral M,(f | B) by A2, A10, Th21, Th94; :: thesis: verum