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 f being b₇ -measurable PartFunc of [:X1,X2:],ExtREAL st M1 is sigma_finite & M2 is sigma_finite & E = dom f holds
( f is_integrable_on Prod_Measure (M1,M2) iff Integral (M2,(Integral1 (M1,|.f.|))) < +infty ) by Lm1, Lm2;