let X1, X2 be non empty set ; :: thesis: 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 (M2,(Integral1 (M1,f)))
let S1 be SigmaField of X1; :: thesis: 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 (M2,(Integral1 (M1,f)))
let S2 be SigmaField of X2; :: thesis: 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 (M2,(Integral1 (M1,f)))
let M1 be sigma_Measure of S1; :: thesis: 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 (M2,(Integral1 (M1,f)))
let M2 be sigma_Measure of S2; :: thesis: 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 (M2,(Integral1 (M1,f)))
let A be Element of sigma (measurable_rectangles (S1,S2)); :: thesis: 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 (M2,(Integral1 (M1,f)))
let f be PartFunc of [:X1,X2:],ExtREAL; :: thesis: ( 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 (M2,(Integral1 (M1,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 ; :: thesis: Integral ((Prod_Measure (M1,M2)),f) = Integral (M2,(Integral1 (M1,f)))
reconsider XX2 = X2 as Element of S2 by MEASURE1:7;
reconsider g = - f as nonnegative PartFunc of [:X1,X2:],ExtREAL by A3;
A6: g = (- 1) (#) f by MESFUNC2:9;
( - (Integral1 (M1,f)) = (- 1) (#) (Integral1 (M1,f)) & dom (Integral1 (M1,f)) = XX2 & Integral1 (M1,f) is nonpositive & Integral1 (M1,f) is XX2 -measurable ) by A1, A3, A4, A5, Th67, Th59, MESFUNC2:9, FUNCT_2:def 1;
then A7: Integral (M2,(- (Integral1 (M1,f)))) = (- 1) * (Integral (M2,(Integral1 (M1,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 (M2,(Integral1 (M1,g))) by A1, A2, Lm16;
then (- 1) * (Integral ((Prod_Measure (M1,M2)),f)) = Integral (M2,(Integral1 (M1,g))) by A3, A4, A5, A6, Lm2;
then (- 1) * (Integral ((Prod_Measure (M1,M2)),f)) = Integral (M2,(- (Integral1 (M1,f)))) by A4, A5, Th73;
hence Integral ((Prod_Measure (M1,M2)),f) = Integral (M2,(Integral1 (M1,f))) by A7, XXREAL_3:68; :: thesis: verum