theorem :: MEASUR11:118

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 (M2,(X-vol (E,M1))) = (product_sigma_Measure (M1,M2)) . E