Thread: Sigma-Algebras

I am studying probability theory at the moment and I got a problem:
I don't clearly understand what a Sigma-Algebra is.
Can somebody explain this to me and give a comprehensible example, and tell me when an algebra is not a sigma-algebra, and why?

Thanks.

I’m not that good on probability theory, but a σ algebra is defined as follows.

Consider a nonempty set S and a collection Σ of subsets os S. (In other words, Σ is a subset of the power set of S.) Then Σ is a σ algebra over S iff the following properties hold:

(i) Σ is nonempty.
(ii) If X is a subset of S such that X is in Σ, then S\X (the complement of X in S) is in Σ.
(iii) The union of countably many subsets of S which are in Σ is in Σ.

Yeah this is what I have learned insofar too about Sigma-algebras.
However I cannot visualize how something like this looks like or what purpose there is to define a set as such.
Not to speak of an example.
Thanks though.

I don't think it's possible, or worthwhile, imagining what a sigma algebra "looks like". Instead, just think of a sigma algebra as being the set of possible events in your probability space. So the complement axiom is just saying "if P is an event, then 'not P', the complement of P, is also an event". The countable union axiom is saying "if P1, P2, ... are events, then 'P1 or P2 or P3...', the union of the Pi's, is an event".

So, for example, suppose I choose an integer. Then the sigma algebra of events should contain each event "serpicojr has chosen the integer n" for each integer n. But then the event "serpicojr has not chosen the integer n" is also an event--this is the complement. I can also take any countable union of "serpicojr has chosen the integer n" events--for example, take the union over n of "serpicojr has chosen the integer 2n", and now we have the event "serpicojr has chosen an even integer".

It's difficult (impossible?) to come up with a nontrivial, explicit example of a sigma algebra. The above sigma algebra is actually the power set of the integers--it's every subset. This isn't very interesting. The power set of any set is trivially a sigma algebra.

The simplest sigma algebra we usually consider on the real number line is the sigma algebra "generated" by the open intervals, which we call the Borel sigma algebra. So this contains all open sets by the countable union property, all closed sets by the complement property... and then a hierarchy of weirder and weirder sets. Stuff gets so weird in classifying these sets that we leave the realm of probability and enter the world of set theory and logic.

@serpicojr:
Thanks for your help.
I think I understand it a little better now.
And yes, that's exactly the problem I have, not having studied set theory and all that stuff already.
But that just would take too much time to deal with that now.
Maybe if I just go on reading the book further some things will be clarified by themselves.

You don't need the set theory--all you need to understand are the basic set operations and how they interact (e.g., De Morgan's laws).

What book are you working through? And, of course, if you have more questions, please come back here and ask--Jane and I, among others, are always willing to help!

And a suggestion: make sure you work through the exercises in the book, and, again, feel free to check your answers here.

Set theory is not that difficult once you get the hang of it. What you need to do, essentially, is to stop trying to visualize everything, and start learning to think in abstract symbols instead.

Let me just jump in and say there are two uses of the term "set theory" being used here. The first is the basic set theory that every working mathematician should know. This includes set operations, basic things about cardinals, and, I suppose, understanding the distinction between sets and proper classes. The second is the subdiscipline of set theory, which goes into the details of axiomatizing sets and describing various hierarchies of sets depending on these axioms. I've been using set theory to mean the latter, and it looks like algebraic topology (the user, not the subject) and mastermind have been using it to mean the former.

I do mean set theory of your second kind. Why did you think I meant something else?

Probability theory requiring sigma algebras is a very advanced branch of mathematics. It's not something that's likely to be offered in an elementary course on probability. In order to reach that advanced level, you would really need to be proficient with set theory of the first kind anyway. So why would I be offering advice on something I trust Mastermind should already have a working knowledge of?

Sigh. Let me rephrase my whole post again.

Set theory which goes into the details of axiomatizing sets and describing various hierarchies of sets depending on these axioms is not that difficult once you get the hang of it. What you need to do, essentially, is to stop trying to visualize everything, and start learning to think in abstract symbols instead.

There. I hope I've now made it so clear that it's impossible even for dyslexics to misunderstand me.

Whoa there, dude, my bad. I didn't mean to insult in any way.

I don't know, Martin's axiom still confuses me on occasion...