I am struggling with just the very last stage of the solution and any suggestions would be much appriciated!! Thanks!
I am trying to show that given a set and a collection of subsets of , and given that
(ii) implies that the complement
( is closed under complements)
(iii) is closed under finite (or, case 2: countable) intersections, (ie, )
I am trying to show that is an algebra (or, case 2: a sigma algebra)
Handling case 1, closed under finite intersections, I think I have come up with a proof using some set theory but I was hoping someone might be able to verify.
So, firstly, I think all we need to show is that is closed under finite unions to show that it is an algebra.
To do this, take . Then since is closed under complements and also under finite intersections, .
Then by Demorgans Law, . Since the left-hand side is in , the right-hand side will be as well. Also, Since, we are closed under complements, .
Here is where I am stuck. Can I show that ? Because then I would be done (we would have shown it is closed under finite unions). And I believe a similar argument would work for countable unions and sigma algebras?
Again, any help appriciated!! Thank you!