:: deftheorem defines product_Measure MEASUR11:def 2 :
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 b₇ being induced_Measure of measurable_rectangles (S1,S2), product-pre-Measure (M1,M2) holds
( b₇ = product_Measure (M1,M2) iff for E being set st E in Field_generated_by (measurable_rectangles (S1,S2)) holds
for F being disjoint_valued FinSequence of measurable_rectangles (S1,S2) st E = Union F holds
b₇ . E = Sum ((product-pre-Measure (M1,M2)) * F) );