Quick question about $\sigma$-algebras. I have a quick question concerning $\sigma$-algebras. If A is a collection of subsets of a set X and Y is the $\sigma$-algebra generated by A, then can I conclude that every element of Y is either (1) an element of A, (2) a countable union of elements of A, (3) a countable intersection of elements of A, or (4) some countable union or intersection of the complements of elements of A? I feel like this can state more simply. Any input would be great! Thank You.
 A: No, you can't conclude anything about the general form of an element in the $\sigma$ algebra in such an easy manner. This is a common point of confusion for students familiar with algebraic objects generated by sets, where the generated object can usually be reconstructed by finite sums of finite products etc. The temptation to feel like everything should be reconstructable with countably many atoms is understandable.
To prove things about the $\sigma$ algebra generated by a set, you will majorly be appealing to the fact that it is contained in any $\sigma$ algebra containing the generators. General elements of the algebra can potentially be very weird: one could be a countable intersection of countable unions of sets in the algebra, some of which are complements and some of which aren't, unionized with ... and so on and so forth.
I suspect you have a concrete question you are struggling with regarding the $\sigma$ algebra generated by a set, and you are trying to gain a foothold in what the elements look like. If that's the case, post it as a new question and include a pointer to this question, and explain your starting difficulties. Good luck!
