What's the Difference Between Algebras and Sigma-Algebras in Probability?

[MathNerd]

I'm currently taking a college level probability course and I am stuck on a couple questions involving algebras and sigma-algebras.

Let S be a fixed set.

1. What is the difference between an algebra on S and a sigma-algebra on S?

2. Why do we require an event space to be a sigma algebra instead of an algebra?

3. Find a set S and an algebra A on S such that A is not a sigma-algebra on S.


Also, I have a proof that I could use some hints on how to start and the general form in which I should go about it.

Prove that every sigma-algebra on S is an algebra on S.

Thanks.

 

[Elucidus]

I suspect it has something to do with sigma-algebras being limited to only finite intersections.

Continuous random variables cause problems of you assume countable additivity of probability for point events.

--Elucidus

 

[Preno]

I'm currently taking a college level probability course and I am stuck on a couple questions involving algebras and sigma-algebras.

Let S be a fixed set.

1. What is the difference between an algebra on S and a sigma-algebra on S?

A sigma algebra is an algebra closed wrt countable unions.

2. Why do we require an event space to be a sigma algebra instead of an algebra?

Because we want probability/measure to be sigma-additive.

3. Find a set S and an algebra A on S such that A is not a sigma-algebra on S.

The system of finite and co-finite subsets of any infinite set forms an algebra which is not a sigma algebra.

Prove that every sigma-algebra on S is an algebra on S.

What definitions of algebra and sigma-algebra are you using? Under most definitions, this would be trivial (perhaps requiring the use of de Morgan's laws).

 

[g_edgar]

Why do we require an event space to be a sigma algebra instead of an algebra?

Because we want probability/measure to be sigma-additive.

Perhaps not entirely responsive... the next question would be "why do we want that?"

So ... How about:

In order to do probability calculations involving limits of sequences. Either sequences of events or more generally sequences of random variables.