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 being Element of S st ex E being Element of S st
( E = dom f & f is_measurable_on E ) & f is nonnegative holds
0 <= Integral M,(f | A)
let S be SigmaField of X; :: thesis: for M being sigma_Measure of S
for f being PartFunc of X,ExtREAL
for A being Element of S st ex E being Element of S st
( E = dom f & f is_measurable_on E ) & f is nonnegative holds
0 <= Integral M,(f | A)
let M be sigma_Measure of S; :: thesis: for f being PartFunc of X,ExtREAL
for A being Element of S st ex E being Element of S st
( E = dom f & f is_measurable_on E ) & f is nonnegative holds
0 <= Integral M,(f | A)
let f be PartFunc of X,ExtREAL ; :: thesis: for A being Element of S st ex E being Element of S st
( E = dom f & f is_measurable_on E ) & f is nonnegative holds
0 <= Integral M,(f | A)
let A be Element of S; :: thesis: ( ex E being Element of S st
( E = dom f & f is_measurable_on E ) & f is nonnegative implies 0 <= Integral M,(f | A) )
assume that
A1: ex E being Element of S st
( E = dom f & f is_measurable_on E ) and
A2: f is nonnegative ; :: thesis: 0 <= Integral M,(f | A)
consider E being Element of S such that
A3: E = dom f and
A4: 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 A3, RELAT_1:90; :: thesis: f | A is_measurable_on C
A6: C = (dom f) /\ C by A3, XBOOLE_1:17, XBOOLE_1:28;
A7: dom (f | A) = C by A3, RELAT_1:90
.= dom (f | C) by A6, RELAT_1:90 ;
A8: 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 A7, A9, FUNCT_1:70; :: thesis: verum

end;

f is_measurable_on C by A4, MESFUNC1:34, XBOOLE_1:17;
then f | C is_measurable_on C by A6, Th48;
hence f | A is_measurable_on C by A7, A8, FUNCT_1:9; :: thesis: verum

end;

then 0 <= integral+ M,(f | A) by A2, Th21, Th85;
hence 0 <= Integral M,(f | A) by A2, A5, Th21, Th94; :: thesis: verum