Question

proof that intersection of sigma algebra is the smallest sigma algebra

Answer 1

Fix a set Ω. A σ-algebra on Ω is a non-empty collection of subsets of Ω closed under taking complements and countable unions.

I'd like to prove that (1) for finite Ω, 2Ω is a σ-algebra and that (2) the intersection of a family of σ-algebras is a σ-algebra.