A question about $$\sigma$$ algebra

1. Jan 24, 2005

gimmytang

Hi guys,
I am new comer here. I am now stuyding mathematical analysis and have a bunch of questions. Is here the right place for seeking help from people?
Ok.. Here comes my real question in $$\sigma$$ algebra:
Let $$A_1,A_2$$ be two arbitrary subsets of $$\Omega$$, find the smallest $$\sigma$$ algebra containing $$A_1,A_2$$.

Thanks!
gimmy

2. Jan 24, 2005

matt grime

A sigma algebra is closed under what operations?

3. Jan 24, 2005

gimmytang

It is closed under complementation and finite union. Therefore the smallest $$\sigma$$ algebra $$\chi =\{\Phi,\Omega, A_{1}, A_{1}^{c},A_{2},A_{2}^{c}\}$$. Is it correct?

Last edited: Jan 24, 2005
4. Jan 24, 2005

matt grime

"closed under finite union" so where is (A_1)u(A_2)?

5. Jan 24, 2005

gimmytang

yeah, should be more sets inside like this:
$$\{\Phi,\Omega,A_1,A_{1}^{c},A_2,A_{2}^{c},A_{1}\cup A_{2},A_{1}\cup A_{2}^{c},A_{1}^{c}\cup A_{2},A_{1}^{c}\cup A_{2}^{c},A_{1}\cap A_{2},A_{1}\cap A_{2}^{c},A_{1}^{c}\cap A_{2},A_{1}^{c}\cap A_{2}^{c}\}$$

6. Jan 24, 2005

Hurkyl

Staff Emeritus
Aren't &sigma;-algebras closed under countable unions? (It doesn't matter for this example, though)

7. Jan 24, 2005

master_coda

Yes, they are.

8. Jan 25, 2005

gimmytang

But according to the answer in a book the number of members in the smallest $$\sigma$$-field containing $$A_1,...A_n$$ is $$2^{2^n}$$. For n=2, there will be 16 members, while I only get 14. What the other 2 memebers are supposed to be?

9. Jan 25, 2005