I'm an undergraduate level maths student and I've just done a number of exercises on elementary set theory and sigma fields. We've just started the courses and are discussing probability events as sets. I can't see the link between sigma fields and probability theory. Can anyone explain this intuitively at an undergrad level?
$\sigma$-fields, also known as $\sigma$-algebras, are used to model "events we can assign a probability to". We want some properties from these sort of events:
- If we can assign a probability to two events, then we can assign a probability that both events happen.
- If we can assign a probability to two events, then we can assign a probability that at least one of them happens.
- If we can assign a probability to an event, then we can assign a probability that it doesn't happen.
These three effectively describe intersection, union and complements. So if we want to think of events as "sets of possible outcomes" this means that a collection of sets that models events should be closed under intersection, union and complements.
But why countable unions (or equally, countable intersections, or countable disjoint unions)? Well. Modern probability doesn't limit itself just to something which has four, or five, or seven, or eighty million possible outcomes. Even if you are interested in a finite object, you are not usually interested in limiting its size right from the start.
So even if you are not interested in the infinite case, asking what happens when your probability space grows larger and larger, and whether or not there is some "uniform behavior", that it eventually has a predictable behavior, is a good question that is worth asking and worth answering. And for that purpose, infinite sets become so very useful.
Closure under countable unions/disjoint unions/intersections ensures that if we have a sequence of events, then its union, or intersection, is still an event itself. So we can talk about the limit of events (as a limit of sets) as an event of its own rights, and therefore assign probability to that event.