theorem Th107: :: MEASUR11:107

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 V being Element of sigma (measurable_rectangles (S1,S2))
for A being Element of S1
for B being Element of S2 st M1 is sigma_finite & V = [:A,B:] holds
Field_generated_by (measurable_rectangles (S1,S2)) c= { E where E is Element of sigma (measurable_rectangles (S1,S2)) : Integral (M2,(X-vol ((E /\ V),M1))) = (product_sigma_Measure (M1,M2)) . (E /\ V) }