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 E -measurable ) & f is nonnegative & A misses B holds
Integral (M,(f | (A \/ B))) = (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 E -measurable ) & f is nonnegative & A misses B holds
Integral (M,(f | (A \/ B))) = (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 E -measurable ) & f is nonnegative & A misses B holds
Integral (M,(f | (A \/ B))) = (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 E -measurable ) & f is nonnegative & A misses B holds
Integral (M,(f | (A \/ B))) = (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 E -measurable ) & f is nonnegative & A misses B implies Integral (M,(f | (A \/ B))) = (Integral (M,(f | A))) + (Integral (M,(f | B))) )
assume that
A1: ex E being Element of S st
( E = dom f & f is E -measurable ) and
A2: f is nonnegative and
A3: A misses B ; :: thesis: Integral (M,(f | (A \/ B))) = (Integral (M,(f | A))) + (Integral (M,(f | B)))
consider E being Element of S such that
A4: E = dom f and
A5: f is E -measurable by A1;
ex C being Element of S st
( C = dom (f | A) & f | A is C -measurable )
proof
take C = E /\ A; :: thesis: ( C = dom (f | A) & f | A is C -measurable )
thus dom (f | A) = C by A4, RELAT_1:61; :: thesis: f | A is C -measurable
A6: C = (dom f) /\ C by A4, XBOOLE_1:17, XBOOLE_1:28;
A7: dom (f | A) = C by A4, RELAT_1:61
.= dom (f | C) by A6, RELAT_1:61 ;
for x being object st x in dom (f | A) holds
(f | A) . x = (f | C) . x
proof
let x be object ; :: thesis: ( x in dom (f | A) implies (f | A) . x = (f | C) . x )
assume A8: x in dom (f | A) ; :: thesis: (f | A) . x = (f | C) . x
then (f | A) . x = f . x by FUNCT_1:47;
hence (f | A) . x = (f | C) . x by A7, A8, FUNCT_1:47; :: thesis: verum
end;
then A9: f | C = f | A by A7, FUNCT_1:2;
f is C -measurable by A5, MESFUNC1:30, XBOOLE_1:17;
hence f | A is C -measurable by A6, A9, Th42; :: thesis: verum
end;
then A10: Integral (M,(f | A)) = integral+ (M,(f | A)) by A2, Th15, Th88;
ex C being Element of S st
( C = dom (f | (A \/ B)) & f | (A \/ B) is C -measurable )
proof
reconsider C = E /\ (A \/ B) as Element of S ;
take C ; :: thesis: ( C = dom (f | (A \/ B)) & f | (A \/ B) is C -measurable )
thus dom (f | (A \/ B)) = C by A4, RELAT_1:61; :: thesis: f | (A \/ B) is C -measurable
A11: C = (dom f) /\ C by A4, XBOOLE_1:17, XBOOLE_1:28;
A12: dom (f | (A \/ B)) = C by A4, RELAT_1:61
.= dom (f | C) by A11, RELAT_1:61 ;
A13: for x being object st x in dom (f | (A \/ B)) holds
(f | (A \/ B)) . x = (f | C) . x
proof
let x be object ; :: thesis: ( x in dom (f | (A \/ B)) implies (f | (A \/ B)) . x = (f | C) . x )
assume A14: x in dom (f | (A \/ B)) ; :: thesis: (f | (A \/ B)) . x = (f | C) . x
then (f | (A \/ B)) . x = f . x by FUNCT_1:47;
hence (f | (A \/ B)) . x = (f | C) . x by A12, A14, FUNCT_1:47; :: thesis: verum
end;
f is C -measurable by A5, MESFUNC1:30, XBOOLE_1:17;
then f | C is C -measurable by A11, Th42;
hence f | (A \/ B) is C -measurable by A12, A13, FUNCT_1:2; :: thesis: verum
end;
then A15: Integral (M,(f | (A \/ B))) = integral+ (M,(f | (A \/ B))) by A2, Th15, Th88;
A16: ex C being Element of S st
( C = dom (f | B) & f | B is C -measurable )
proof
take C = E /\ B; :: thesis: ( C = dom (f | B) & f | B is C -measurable )
thus dom (f | B) = C by A4, RELAT_1:61; :: thesis: f | B is C -measurable
A17: C = (dom f) /\ C by A4, XBOOLE_1:17, XBOOLE_1:28;
A18: dom (f | B) = C by A4, RELAT_1:61
.= dom (f | C) by A17, RELAT_1:61 ;
for x being object st x in dom (f | B) holds
(f | B) . x = (f | C) . x
proof
let x be object ; :: thesis: ( x in dom (f | B) implies (f | B) . x = (f | C) . x )
assume A19: x in dom (f | B) ; :: thesis: (f | B) . x = (f | C) . x
then (f | B) . x = f . x by FUNCT_1:47;
hence (f | B) . x = (f | C) . x by A18, A19, FUNCT_1:47; :: thesis: verum
end;
then A20: f | C = f | B by A18, FUNCT_1:2;
f is C -measurable by A5, MESFUNC1:30, XBOOLE_1:17;
hence f | B is C -measurable by A17, A20, Th42; :: thesis: verum
end;
integral+ (M,(f | (A \/ B))) = (integral+ (M,(f | A))) + (integral+ (M,(f | B))) by A1, A2, A3, Th81;
hence Integral (M,(f | (A \/ B))) = (Integral (M,(f | A))) + (Integral (M,(f | B))) by A2, A15, A10, A16, Th15, Th88; :: thesis: verum