Question: Show that the collection of measurable rectangles form an algebra.
Denote $\Sigma$ by $\sigma$-algebra.
I found the following set operations. Let $A_1, A_2 \in \Sigma_A$, and $B_1, B_2 \in \Sigma_B$. Then, $(A_1 \times B_1) \setminus(A_2 \times B_2) = [(A_1 \cap A_2) \times (B_1 \setminus B_2)] \cup [(A_1 \setminus A_2) \times B_2)]$.
In addition, $(A_1 \times B_1) \cap (A_2 \times B_2) = (A_1 \cap A_2) \times (B_1 \cap B_2)$.
Then, by De Morgan, I can show the finite union as well.
But, two set operations is intuitively understandable, but is there a simple way to prove these operations formally?
Also, how can we show that the product of empty sets is in the collection? Can we still use the caratheodory criterion in this case?