How to show that the set of all cross products of a first element from one sigma algebra and a second from a second sigma algebra is a semi-algebra

Question

Let $F_1$ and $F_2$ be sigma-algebras and let

$$J = \{ A \times B; A \in F_1, B \in F_2 \}.$$

I have managed to show that such a set would be closed under finite intersections, however, I am finding it difficult to see why complements of the set would be finite disjoint unions of elements of the set.

Answer 1

Note that

$$(A \times B)^c = (A^c \times B) \cup (A \times B^c) \cup (A^c \times B^c).$$

The union is disjoint and each set is a member of $J$.