In probability, events are considered to be closed under countable union and complement, so mathematically they are modeled by $\sigma$-algebra. I was wondering why events are considered to be closed under countably union and complement?
In Nate Eldredge's post, he has done an excellent job on explaining this, by using whether questions are answered or not as an analogy to whether events occur or not, if I understand his post correctly. However, if someone could explain plainly without analogy, it could be clearer to me.
I was particularly curious why events are not considered to be closed under infinite (possibly uncountably) union, but instead just under countably union? So possibly to model events using the power set? I think this is not addressed in Nate Eldredge's post.
My guess would be that the reason might be related to the requirement on the likelihood of any event to occur to be "measurable" in some sense. But how exactly to understand this requirement is unclear to me.
PS: This post is related to my previous one Interpretation of sigma algebra, but the questions asked in these two are not the same.
Thanks and regards!