For a counterexample, take , , and .
The set is not a -algebra because is not in .
I think that you will have better luck with intersections.
Suppose that I have a set over which I define a collection of -algebra: ; that is, each is a -algebra of . Is the countable union of those also a -algebra? That is, is a -algebra over ? How about countable intersection?
1 - It seems pretty clear that and are contained in .
2 - It is also clear that if , then
So the first two conditions of a -algebra have been fulfilled: contains the full sample space / empty set and it is closed under complements.
3 - But if , is it true that ? I am unable to prove this fact. Yet I cannot find a counterexample.
A counterexample or hint of proof would be very appreciated - thanks.