1. ## Sigma - fields

Hi all,

need a bit of help on this question.

Let

omega = {1, . . . , 6} and A = {{1, 3, 5}, {1, 2, 3}}.
a) Describe F = sigma(A), the sigma-field generated by A.
Hint: For a finite set omega
, the number of elements of a sigma-field on omega
is always a power of 2.

I have grasped the concept that F denotes a collection of subsets on omega, and also A is a collection of events that generates a sigma-field. However full appreciation of this concept hasn't sunk in yet.

Is there a smart soul out there who could please enlighten me?

2. Hello,

The hint is here to let you know if you've forgotten any element.
We know that the empty set and omega belong to the sigma-field. That makes 2.

{1,3,5} and {1,2,3} obviously belong to the sigma-field, since they generate it.

We know a sigma-field is closed under union. So {1,3,5} U {1,2,3}={1,2,3,5} belong to the sigma-field.

Then we know it's closed under the complement : find the complement of {1,3,5}, {1,2,3} and {1,2,3,5} in Omega : {2,4,6}, {4,5,6}, {4,6}. That makes 3 more.

But now you can also do {1,3,5} U {4,5,6}={1,3,4,5,6} and its complement {2}
{1,2,3} U {2,4,6}={1,2,3,4,6} and its complement {5}
{2,4,6} U {4,5,6}={2,4,5,6} and its complement {1,3}

And the remaining union : {2} U {5}={2,5} and its complement {1,3,4,6}

That makes 16=2^4

You have to try finding all the possibilities...

3. Thanks for such a clear answer. Wow.......Deeply appreciated!