Let X be a set and let {Fi | i ∈ I} be an arbitrary collection of σ- algebras. Showthat the collection F := {F | ∀i ∈ I, F ∈ Fi} is a σ-algebra.

To show that F is a σ-algebra, we need to verify three properties:

1. The empty set and X are in F.
2. If A is in F, then its complement is in F.
3. If {An} is a countable collection of sets in F, then the union of {An} is in F.

Let's verify each property:

1. Since each Fi is a σ-algebra, the empty set and X are in each Fi. Therefore, the empty set and X are in every set in the collection {Fi | i ∈ I}, so they are in F.

2. If A is in F, then A is in every Fi because F is the intersection of all Fi. Since each Fi is a σ-algebra, the complement of A is in each Fi. Therefore, the complement of A is in F.

3. If {An} is a countable collection of sets in F, then each An is in every Fi. Since each Fi is a σ-algebra, the union of {An} is in each Fi. Therefore, the union of {An} is in F.

Therefore, F is a σ-algebra.