# Thread: need help on Borel algebra

1. ## need help on Borel algebra

Hi,
I've a quick question about Borel algebra. The Borel algebra on set X is defined as the smallest σ-algebra containing all open sets (or, equivalently, all closed sets). Let R be the set of reals and B(R) denote the Borel algebra on R. Then my question is: whether B(R)*B(R) equals B(R2)? How do you prove or disprove this?

Thanks

2. ## Re: need help on Borel algebra

Could you remind what B(R)*B(R) is?

3. ## Re: need help on Borel algebra

By B(R)×B(R), I mean the product of two Borel algebras.

4. ## Re: need help on Borel algebra

And could you remind what a product of two sigma-algebras is?

5. ## Re: need help on Borel algebra

Never mind. I found the answer here:
Borel Sets on R^n - MathOverflow

6. ## Re: need help on Borel algebra

As the answer in link says, the set $\displaystyle A=\{X_1\times X_2\mid X_1,X_2\in B(\mathbb{R})\}$ is a subset of $\displaystyle B(\mathbb{R}\times\mathbb{R})$. If $\displaystyle B(\mathbb{R})\times B(\mathbb{R})$ is defines as A, then $\displaystyle B(\mathbb{R})\times B(\mathbb{R})\subsetneq B(\mathbb{R}\times\mathbb{R})$, but if $\displaystyle B(\mathbb{R})\times B(\mathbb{R})$ is defined as the sigma-algebra generated by A, as the comments in the link suggest, then $\displaystyle B(\mathbb{R})\times B(\mathbb{R})= B(\mathbb{R}\times\mathbb{R})$.