I am learning about product measures and I was stuck on a detail of the proof. I would appreciate any assistance!
Suppose we have a measure spaces $(X_i, M_i, \mu_i), i=1, ..., N$, that are complete and $\sigma$-finite. We call sets measurable rectangles, if it has the following form $$ A_1 \times ... \times A_N, $$ where each $A_i \in M_i$.
Let $M_0$ denote the collection of sets that are finite disjoint union of measurable rectangles. Then $M_0$ turns out to be an algebra of subsets of $X_1 \times ... \times X_N$. I am having hard time seeing how $M_0$ is an algebra. Could someone please explain this to me? Thank you!