theorem Th82: :: MESFUN12:82

for X1, X2 being 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 E being Element of sigma (measurable_rectangles (S1,S2)) st M1 is sigma_finite & M2 is sigma_finite holds
( Integral ((Prod_Measure (M1,M2)),(chi (E,[:X1,X2:]))) = Integral (M2,(Integral1 (M1,(chi (E,[:X1,X2:]))))) & Integral ((Prod_Measure (M1,M2)),((chi (E,[:X1,X2:])) | E)) = Integral (M2,(Integral1 (M1,((chi (E,[:X1,X2:])) | E)))) & Integral ((Prod_Measure (M1,M2)),(chi (E,[:X1,X2:]))) = Integral (M1,(Integral2 (M2,(chi (E,[:X1,X2:]))))) & Integral ((Prod_Measure (M1,M2)),((chi (E,[:X1,X2:])) | E)) = Integral (M1,(Integral2 (M2,((chi (E,[:X1,X2:])) | E)))) )