theorem Th77: :: MESFUN12:77

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 (M1,(Y-vol (E,M2))) = Integral ((Prod_Measure (M1,M2)),(chi (E,[:X1,X2:]))) & Integral (M2,(X-vol (E,M1))) = Integral ((Prod_Measure (M1,M2)),(chi (E,[:X1,X2:]))) )