# Product sigma algebra of a non-separable space

• May 19th 2012, 04:52 AM
avkuvalekar
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.
• May 19th 2012, 07:30 AM
girdav
Re: Product sigma algebra of a non-separable space
• May 19th 2012, 08:02 AM
avkuvalekar
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 $\displaystyle l_\infty$
• May 19th 2012, 08:05 AM
girdav
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 $\displaystyle \mathbb R$.
• May 19th 2012, 09:45 AM
avkuvalekar
Re: Product sigma algebra of a non-separable space
Right. Thanks a lot. I will read the bits in detail. :)
• May 23rd 2012, 01:45 PM
avkuvalekar
Re: Product sigma algebra of a non-separable space
Quote:

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?