MathOverflow will be down for maintenance for approximately 3 hours, starting Monday evening (06/24/2013) at approximately 9:00 PM Eastern time (UTC-4).

## Simple: An algebra on a set of reals that is not a sigma-algebra [closed]

Hello All,

I am aware that this question is probably very simple (as I have scrolled around looking for similar questions) and therefore you can probably tell I am new to set theory and real analysis. So up front I am looking to understand a homework problem I did not get - to find:

1.) an infinite collection of subsets of $\mathbb{R}$ that contains $\mathbb{R}$ 2.) is closed under formation of countable unions 3.) is closed under formation of countable intersections 4.) is NOT a $\sigma$-algebra

So seeing this I immediately thought of a set that did not support closure under complementation, as this is the only remaining requirement in the list of a $\sigma$-algebra. I am quite lacking in knowledge about countability and the standard number systems, and am aware of the "standard" example utilizing the positive integers where the subsets are finite or their complements are finite - I understand that. The problem is that this is on the reals, which is uncountable (not countably infinite) and the only poor example I can think of is $\mathbb{R} - \varnothing$.

Any help would be greatly appreciated.

Brett

-
This question seems like a good question for math.stackexchange.com a similar site; here due to the rather narrow scope of the site it seems a bit out of scope. Vote to close. [Note: the tag is a bit surpsing, the keywords you mention and perhas measure theory seem better; also on math.SE, not here, there is a homework tag.] – quid Sep 6 2011 at 21:09
Why not use the algebra of the finite subsets of R, augmented by R? Gerhard "Ask Me About System Design" Paseman, 2011.09.06 – Gerhard Paseman Sep 6 2011 at 21:10
Gerhard, 'closed under formation of countable unions' – quid Sep 6 2011 at 21:26
Thank you. Perhaps I should specify countable next time. Gerhard "Shouldn't Do Math Before Coffee" Paseman, 2011.09.06 – Gerhard Paseman Sep 7 2011 at 1:43
I'm afraid homework problems are not in the scope of this site, and answers to homework problems are discouraged. – S. Carnahan Sep 7 2011 at 4:20