# Thread: Product sigma algebra of a non-separable space

1. ## Product sigma algebra of a non-separable space

Hi, following is the question I have been thinking about for a while -

Is $\displaystyle \mathcal{B}(l_\infty \times l_\infty) = \sigma(\mathcal{B}(l_\infty) \times \mathcal{B}(l_\infty))?$

Can anyone help me on this? Thanks a lot.

3. ## Re: Product sigma algebra of a non-separable space

Thanks girdav. I have taken a look at this link but it doesn't specifically answer the question about $\displaystyle l_\infty$

4. ## Re: Product sigma algebra of a non-separable space

Michael Greinecker's answer can be used since $\displaystyle \ell^{\infty}$ has the same cardinality as $\displaystyle \mathbb R$.

5. ## Re: Product sigma algebra of a non-separable space

Right. Thanks a lot. I will read the bits in detail.

6. ## Re: Product sigma algebra of a non-separable space

Originally Posted by girdav
Michael Greinecker's answer can be used since $\displaystyle \ell^{\infty}$ has the same cardinality as $\displaystyle \mathbb R$.
Hi girdav, I read the link again. The doubt I have is the theorems says that for spaces with cardinality greater than 'c' we have the non-equality. But from what I understood, he doesn't prove the equality for all spaces with cardinality equal to 'c'. Am I missing something?