theorem Th55: :: MESFUN12:55

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 y being Element of X2
for E being Element of sigma (measurable_rectangles (S1,S2))
for r being Real st M1 is sigma_finite holds
( (r (#) (X-vol (E,M1))) . y = Integral (M1,(ProjPMap2 ((chi (r,E,[:X1,X2:])),y))) & ( r >= 0 implies (r (#) (X-vol (E,M1))) . y = integral+ (M1,(ProjPMap2 ((chi (r,E,[:X1,X2:])),y))) ) )