Borel Sets on $\mathbb{R}^n$ Define the Borel sigma-algebra on $\mathbb{R}^n$ as the smallest sigma-algebra containing all $n$-rectangles
$(a_1, b_1) \times \cdots \times (a_n, b_n)$.
Is it true that the Borel sigma algebra contains all sets of the form $A_1 \times \cdots \times A_n$, where each $A_i$ is some Borel set in $\mathbb{R}$?
I thought this would be trivially true, but I had a lot of trouble trying to prove it, and I'm not even sure its true anymore.
If this is a well-known result, could you please refer me to a text where it has been (dis)proved ?
 A: A way to prove it:
1/ any set of the form $A_1 \times \mathbb R \ldots \times \mathbb R$, where $A_1$ is Borel, or more generally a "Borel rectangle" with only one slice not equal to the whole space, is in the Borel sigma-algebra (this is essentially a 1-dimensional Borel set, and those are generated by open intervals).
2/ any product $A_1 \times \ldots \times A_n$ (with each $A_i$ Borel) is a finite intersection of sets of the above form.
Not sure I should have answered this, it may be a homework problem... I'd have just written a comment but I'm not reputable enough to do so :)
Any standard reference on measure theory will provide a proof of the result you're asking about (say, Dudley's book).
A: I would like to generalize this, since I was stumped for a while by the original question too. Let $\mathcal A_i$ be any collection of sets in $X_i$ such that $X_i \in \mathcal A_i$. For any collection of sets $\mathcal A$, let $S(\mathcal A)$ be the smallest $\sigma$-algebra containing $\mathcal A$. Let $I$ be a finite index set. Then  $\Pi_{i \in I} \space S(\mathcal A_i) \subset S(\Pi_{i \in I} \space \mathcal A_i)$. The proof should be clear from the answer by Julien Melleray and the 'Ok, done' comment by Cosmonut above. It follows that $S(\Pi_{i \in I} \space S(\mathcal A_i)) = S(\Pi_{i \in I} \space \mathcal A_i)$.
