Well being closed under complements is a property of an algebra which is, thus the first part is a moot point.
As to countable unions, consider which is a countable increasing sequence of sets.
Prove that an algebra is a -algebra iff is closed under countable increasing unions.
Proof so far:
Suppose that is a -algebra, then if , then I have
Now, if , then since . This direction is simple.
On the other hand, if A is closed under countable increasing unions, that means that for:
, then I have And I need to show that is a -algebra.
Suppose that , I need to show that:
1.
2.
How should I proceed from here? Thanks.