, are nonempty sets. Let and be -algebras.
Let
And
Prove that is an algebra.
It was not hard to prove that it is colsed with respect to finite unions. However I failed to prove it is closed with respect to complement!
Frankly speaking I was working on some probability theory theorem when this problem occured. I'm 99% sure is algebra but the full proof seems bit too long.... If somebody knows a short proof, please post it.
PS I think I can prove that is closed under finite intersections. Than I take complement of set that belongs to and I get finite intersection - the only problem is to find a simple proof that the "intersected" sets belong to .