Let be a partition of . Write out the smallest -algebra containing the sets and

Printable View

- August 7th 2009, 07:01 PMMrJack1990Sigma-Algebra
Let be a partition of . Write out the smallest -algebra containing the sets and

- August 7th 2009, 07:43 PMMrJack1990
I have no idea if this is correct, but this is my best estimation of what the question is asking.

A partition is such that

and i.e. is exhaustive

Then by definition of borel-algebra we have

(1)

(2) As

(3) As

Is this right? - August 7th 2009, 11:18 PMMoo
Hello,

(you're asked for a sigma algebra, not a Borel algebra :))

So yup, the empty set and belong to (first+second axiom)

A,B,C belong to

Their complement belong to .

But... you can have a more precise formula for these complements :

Do you agree ?

We don't need to include because it's ...

And thus the third axiom is automatically checked.

So the smallest -algebra containing A,B,C (also called the generated -algebra by A,B,C) is :