1. ## Sigma Algebras

The problem is:

Let U={a,b,c}.

Write the power set and an explicitly list of all sigma algebras of subsets of U.

I got the power set to be {{empty set},{a},{b}{c},{a,b},{ac},{abc}}

I know the power set is a sigma algebra. I'm not sure what others are sigma algebras.

Could another one be {{a},{b},{c}}?

Also, if you have a collection of all open intervals on the real line together with all their complememts, would this be a sigma algebra?

Any help on this would be appreciated. Thanks

2. Originally Posted by taypez
The problem is:

Let U={a,b,c}.

Write the power set and an explicitly list of all sigma algebras of subsets of U.

I got the power set to be {{empty set},{a},{b},{c},{a,b},{ac},{abc}}

I know the power set is a sigma algebra. I'm not sure what others are sigma algebras.

Could another one be {{a},{b},{c}}?

Also, if you have a collection of all open intervals on the real line together with all their complememts, would this be a sigma algebra?

Any help on this would be appreciated. Thanks
i dunno what sigma algebras are, just wanted to point out that your power set is incomplete. you are missing the subset {b,c}

3. Originally Posted by taypez
The problem is:

Let U={a,b,c}.

Write the power set and an explicitly list of all sigma algebras of subsets of U.

I got the power set to be {{empty set},{a},{b}{c},{a,b},{ac},{abc}}

I know the power set is a sigma algebra. I'm not sure what others are sigma algebras.

Could another one be {{a},{b},{c}}?
No because the complement of one of your elements $\displaystyle \{ a \} ^C = \{ b, c \}$ is not in your set. Also the countable union $\displaystyle \{ a \} \cup \{ b \} = \{ a, b \}$ is not in your set.

Originally Posted by taypez
Also, if you have a collection of all open intervals on the real line together with all their complememts, would this be a sigma algebra?
Should be. All the complements and finite unions of each element are in the set.

-Dan