Smallest Sigma-algebra (and algebra)

[tharindra]

1. what is the difference between algebra and sigma algebra?
2. How to construct the smallest algebra generated by a given collection of sets?
3. How to check that a given collection of sets is an algebra?
4. How to check that a given collection is the smallest algebra containing a collection of sets?
5. How to construct the smallest sigma-algebra generated by a given collection of sets, especially a collection of singletons?
6. How to check that a given collection of sets is a sigma-algebra?
7. How to prove that the sigma-algebra generated by two different collections of sets is the same. sigma-algebra?
8. How to check that a given collection is the smallest sigma-algebra containing a collection of sets?

Could you please give me some hints, for solve these kind of problems?
Thanks

 

[CompuChip]

The questions are rather straight-forward, once you have the definition of an algebra and of a sigma-algebra right. Can you look them up and perhaps write them here?

Then notice the differences (from the name "sigma algebra" you already suspect that a sigma algebra is just an algebra, but a special one - i.e. it satisfies all the requirements of any algebra but with some more axioms).

 

[Focus]

If you want the smallest sigma-algebra on a collection of sets, then make a set of all the sigma algebras containing this collection (this is non empty as the powerset will contain them), then just intersect everything you got.

Most of the other questions are of no concern really. You never actually know what the sigma algebra is, it is enough to know it is just a sigma algebra. The smallest sigma algebra comes into play in Borel sets which are the smallest sigma-algebra on a topology, containing all the open sets.

If you want to show that two generated sigma-algebras are the same then you try to obtain one from the other. As an example, you can generate the Borel sigma-algebra on R with sets of the form (a,b) or (a,b]. If you want to see they are the same then from (a,b) you obtain (a,b] by intersecting (a,b+1/n), and you obtain (a,b) from (a,b] by union of (a,b-1/n].

Hope this helps.

 

[tharindra]

I just need to know what is the difference between getting finite unions and countable unions. According to definition of sigma-algebra we can't produce a sigma-algebra A s.t.;

• {{2},{3},{4},...} is a subset of A
• {1} is not an element of A

But we can generate an algebra which satisfies above requirements because we just considering finite unions.

Am I correct?

 

[Focus]

I'm guessing your set is the naturals. If you have all but {1} in the sigma algebra, then by countable union you have {2,3,4,...} in the sigma algebra. Use the complement property to deduce that {1} is in the sigma algebra too.

Algebra can satisfy that as you do not have the set {2,3,4,...} in the algebra.