Prove that $\mathcal B(\mathbb R)\times \mathcal B(\mathbb R)\subseteq \mathcal B (\mathbb R^2)$ I need to prove that
$$\mathcal A(\mathcal B(\mathbb R)\times \mathcal B(\mathbb R))= \mathcal B (\mathbb R^2)$$
Where $\mathcal B$ is the generated Borel algebra and $\mathcal A$ is the generated $\sigma$-algebra. I've reduced this to the problem of showing that $\mathcal B(\mathbb R)\times \mathcal B(\mathbb R)\subseteq \mathcal B (\mathbb R^2)$. However, I really don't know where to start on this.
There must be a solution not involving things like the Borel rank, by the way, since we didn't cover that in class.

Hints:


*

*Show that $$\mathcal{D} := \{A \in \mathcal{B}(\mathbb{R}); \forall O \subseteq \mathbb{R} \, \text{open}: A \times O \in \mathcal{B}(\mathbb{R}^2)\}$$ is a $\sigma$-algebra and conclude that $\mathcal{B}(\mathbb{R}) \times \mathcal{O} \subseteq \mathcal{B}(\mathbb{R}^2)$. ($\mathcal{O}$ denote the open sets in $\mathbb{R}$.)

*Prove in a similar way that $\mathcal{O} \times \mathcal{B}(\mathbb{R}) \subseteq \mathcal{B}(\mathbb{R}^2)$.

*Conclude from $$A \times B = \bigcup_{n \geq 1} (A \times B(0,n)) \cap (B(0,n) \times B)$$ that $\mathcal{B}(\mathbb{R}) \times \mathcal{B}(\mathbb{R}) \subseteq \mathcal{B}(\mathbb{R}^2)$.

