Hello,

You just have to play with the axioms of a sigma algebra.

By combining the property of complements and the property of countable unions, we get, by de Morgan's law, that a sigma algebra is closed under countable intersections.

The complement in of is and belongs to the sigma algebra.

Consider

(1) So for z>x, (x,z] belongs to the sigma algebra.

Okay, what about this... ?

Let's consider, for y>x,

A sigma algebra is closed under countable union.

So belongs to the sigma algebra.

Thus belongs to the sigma algebra.

Intersect with U to get (2)

Let's take a<b. So by (1) and (2), (a,b] and (a,b) belong to the sigma algebra.

Intersect (a,b] with the complement of (a,b) and you will get

So for any x in , belongs to the sigma algebra.

I hope all of this is clear... The letters are quite confusing. I suggest you do it on your own once you've catched the tricks...

Edit : and sorry for the mistakes.