Answer to Question #63677 in Real Analysis for Bright

Question

4.4. Let (X1, E1), (X2, E2), (Y1, K1) and (Y2, K2) be measurable spaces.
Let
f1 : (X1,E1) → (Y1,K1), f2 : (X2,E2) → (Y2,K2), be measurable maps.

Construct the map f1 × f2 : X1 × X2 → Y1 × Y2 given by
f1 × f2 (x1, x2) = f1(x1), f2(x2) for all (x1, x2) ∈ X1 × X2
.
Show that f1 × f2 is E1 ⊗E2 −K1 ⊗K2 measurable.