I just wanted to confirm I have understood to a basic level why we need a sigma-algebra.
When the sample space for our experiment is uncountable, and we are thinking about events, we cannot say that "all possible subsets of the sample space are events," because there are certain peculiar events for that we don't know how to assign a probability to. ("non-measurable subsets of the sample space")
As a result, we take only measurable subsets of the original sample space (= events). Since measurable subsets are closed under complementation and countable unions, we can generate a "universe" which contains only measurable events - this is what I've understood a sigma-algebra to be.
Now all events (elements of my sigma-algebra) will have well-defined probabilities and we can continue?
Addition: E.g. If I have $\mathbb R$ as my sample space, then I can use the borel sigma-algebra which is measurable and it practically contains all "nice/normal" subsets of $\mathbb R$ that I would consider calculating probabilities for?
Is this the gist of it?
One thing I don't understand is - I believe the power set is a sigma algebra, however it would contain non-measurable sets if the sample space was e.g. the real line, right? But I think on Wikipedia it says a $\sigma$-algebra is a non-empty collection of measurable sets closed under complementation and countable unions