theorem :: MESFUN12:70

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 f being PartFunc of [:X1,X2:],ExtREAL
for E being Element of sigma (measurable_rectangles (S1,S2))
for y being Element of X2 st E = dom f & ( f is nonnegative or f is nonpositive ) & f is E -measurable & ( for x being Element of X1 st x in dom (ProjPMap2 (f,y)) holds
(ProjPMap2 (f,y)) . x = 0 ) holds
(Integral1 (M1,f)) . y = 0