Sigma-Algebra

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

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?

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 :