Let $(X,\mathcal{A})$ y $(Y,\mathcal{B})$ measure spaces. Let $R:=R(\mathcal{A},\mathcal{B})$ the collection of measurable rectangles $R:=\{A\times B:A\in \mathcal{A},B\in \mathcal{B}\}.$. Prove that the algebra of subsets of $X\times Y$ generated by $R$ is the collection of finite unions of elements of $R$. I have a hint and it is in which I call $C$ to be the collection of finite unions of elements of $R$, prove that $C$ is in the algebra generated by $R$, then prove that $C$ is an algebra that contains $R$.