What is the significance of $\sigma$-fields in probability theory?

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 are the class of sets called events. So all the events that you will be dealing with in probability theory are sets which are in a sigma field. – Samrat Mukhopadhyay Feb 20 '14 at 19:59

$\sigma$-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:

1. If we can assign a probability to two events, then we can assign a probability that both events happen.
2. If we can assign a probability to two events, then we can assign a probability that at least one of them happens.
3. 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.

• Awesome. That' helped me fill in the gaps. The lecturer said that infinite sample spaces are beyond our course scope. Why are they difficult to understand? – M.K. Feb 20 '14 at 20:17
• Because everything infinite is harder to grasp. Also because for infinite spaces you start having to keep track of continuity, topology (or rather Borel structure) and so on. It's not much harder if you studied calculus; but if you haven't or if the course wasn't designed for that, then it will require a lot more work. Finite sample spaces is really just combinatorics. – Asaf Karagila Feb 20 '14 at 20:22
• Also when you say above "closed under intersection...", is this similar to the idea in group theory when we say something is closed under a certain operator? But in this case the operators are $\cap, \cup,$ and complement and the operands are sets? – M.K. Feb 20 '14 at 20:27
• Yes. Closure under an operation $*$ means if $x,y$ are in the collection, then so is $x*y$ (and similarly for a single variable operation). – Asaf Karagila Feb 20 '14 at 20:29
• Because groups are probably the least workable structure in some sense. You have just one "nicely behaving operation". So it turns out as a lot of the structure in a lot of the places. It's like asking whey sets appear everywhere. Mathematics, generally, is very intertwined, you just don't see it very well for the better part of the undergrad studies. – Asaf Karagila Feb 20 '14 at 20:40