... thatΩ 1 ×Ω2f(ω 1 , ω2)d( 1 ×2)=Ω 1 1IM 1 (ω 1 )Ω2f(ω 1 , ω2)d2(ω2)d 1 (ω 1 )=Ω21IM2(ω2)Ω 1 f(ω 1 , ω2)d 1 (ω 1 )d2(ω2).Remark 3.5.6 (1) Our ... follows fromΩ 1 ×Ω2f(ω 1 , ω2)d( 1 ×2) = ( 1 ×2)(A × B) = 1 (A)2(B)and, for example,Ω 1 Ω2f(ω 1 , ω2)d2(ω2)d 1 (ω 1 ) =Ω 1 1IA(ω 1 )2(B)d 1 (ω 1 )= 1 (A)2(B).Applying ... ’S THE OREM 55(3) The mapsω 1 → 1IM 1 (ω 1 )Ω2f(ω 1 , ω2)d2(ω2)andω2→ 1IM2(ω2)Ω 1 f(ω 1 , ω2)d 1 (ω 1 )are F 1 -measurable and F2-measurable, respectively, random...