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 ?



Borel Sets on $\mathbb{R}^n$
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).


