let X1, X2 be non empty set ; for S1 being SigmaField of X1
for S2 being SigmaField of X2
for M1 being sigma_Measure of S1
for M2 being sigma_Measure of S2
for A being Element of sigma (measurable_rectangles (S1,S2))
for f being PartFunc of [:X1,X2:],ExtREAL st M1 is sigma_finite & M2 is sigma_finite & f is nonpositive & A = dom f & f is A -measurable holds
Integral ((Prod_Measure (M1,M2)),f) = Integral (M1,(Integral2 (M2,f)))
let S1 be SigmaField of X1; for S2 being SigmaField of X2
for M1 being sigma_Measure of S1
for M2 being sigma_Measure of S2
for A being Element of sigma (measurable_rectangles (S1,S2))
for f being PartFunc of [:X1,X2:],ExtREAL st M1 is sigma_finite & M2 is sigma_finite & f is nonpositive & A = dom f & f is A -measurable holds
Integral ((Prod_Measure (M1,M2)),f) = Integral (M1,(Integral2 (M2,f)))
let S2 be SigmaField of X2; for M1 being sigma_Measure of S1
for M2 being sigma_Measure of S2
for A being Element of sigma (measurable_rectangles (S1,S2))
for f being PartFunc of [:X1,X2:],ExtREAL st M1 is sigma_finite & M2 is sigma_finite & f is nonpositive & A = dom f & f is A -measurable holds
Integral ((Prod_Measure (M1,M2)),f) = Integral (M1,(Integral2 (M2,f)))
let M1 be sigma_Measure of S1; for M2 being sigma_Measure of S2
for A being Element of sigma (measurable_rectangles (S1,S2))
for f being PartFunc of [:X1,X2:],ExtREAL st M1 is sigma_finite & M2 is sigma_finite & f is nonpositive & A = dom f & f is A -measurable holds
Integral ((Prod_Measure (M1,M2)),f) = Integral (M1,(Integral2 (M2,f)))
let M2 be sigma_Measure of S2; for A being Element of sigma (measurable_rectangles (S1,S2))
for f being PartFunc of [:X1,X2:],ExtREAL st M1 is sigma_finite & M2 is sigma_finite & f is nonpositive & A = dom f & f is A -measurable holds
Integral ((Prod_Measure (M1,M2)),f) = Integral (M1,(Integral2 (M2,f)))
let A be Element of sigma (measurable_rectangles (S1,S2)); for f being PartFunc of [:X1,X2:],ExtREAL st M1 is sigma_finite & M2 is sigma_finite & f is nonpositive & A = dom f & f is A -measurable holds
Integral ((Prod_Measure (M1,M2)),f) = Integral (M1,(Integral2 (M2,f)))
let f be PartFunc of [:X1,X2:],ExtREAL; ( M1 is sigma_finite & M2 is sigma_finite & f is nonpositive & A = dom f & f is A -measurable implies Integral ((Prod_Measure (M1,M2)),f) = Integral (M1,(Integral2 (M2,f))) )
assume that
A1:
M1 is sigma_finite
and
A2:
M2 is sigma_finite
and
A3:
f is nonpositive
and
A4:
A = dom f
and
A5:
f is A -measurable
; Integral ((Prod_Measure (M1,M2)),f) = Integral (M1,(Integral2 (M2,f)))
reconsider XX1 = X1 as Element of S1 by MEASURE1:7;
reconsider g = - f as nonnegative PartFunc of [:X1,X2:],ExtREAL by A3;
A6:
g = (- 1) (#) f
by MESFUNC2:9;
( - (Integral2 (M2,f)) = (- 1) (#) (Integral2 (M2,f)) & dom (Integral2 (M2,f)) = XX1 & Integral2 (M2,f) is nonpositive & Integral2 (M2,f) is XX1 -measurable )
by A2, A3, A4, A5, Th67, Th60, MESFUNC2:9, FUNCT_2:def 1;
then A7:
Integral (M1,(- (Integral2 (M2,f)))) = (- 1) * (Integral (M1,(Integral2 (M2,f))))
by Lm2;
( A = dom g & g is A -measurable )
by A4, A5, MESFUNC1:def 7, MEASUR11:63;
then
Integral ((Prod_Measure (M1,M2)),g) = Integral (M1,(Integral2 (M2,g)))
by A1, A2, Lm17;
then
(- 1) * (Integral ((Prod_Measure (M1,M2)),f)) = Integral (M1,(Integral2 (M2,g)))
by A3, A4, A5, A6, Lm2;
then
(- 1) * (Integral ((Prod_Measure (M1,M2)),f)) = Integral (M1,(- (Integral2 (M2,f))))
by A4, A5, Th73;
hence
Integral ((Prod_Measure (M1,M2)),f) = Integral (M1,(Integral2 (M2,f)))
by A7, XXREAL_3:68; verum