theorem Th81: :: MESFUN12:81

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))
for er being ExtReal holds
( Integral1 (M1,((chi (er,E,[:X1,X2:])) | E)) = Integral1 (M1,(chi (er,E,[:X1,X2:]))) & Integral2 (M2,((chi (er,E,[:X1,X2:])) | E)) = Integral2 (M2,(chi (er,E,[:X1,X2:]))) )