Econometrics I #1: Sigma-algebra

An interesting concept in this first econometrics lecture (besides the routine probability/statistics stuff) was the notion of the σ-algebra.

Definition: Let B denote a set of some subsets of the sample space Ω. Then B is a σ-algebra, if

1. Φ is an element of B;
2. If x is in B, then the complement of x in Ω is in B;
3. A countable (finite/countably infinite) union of elements in B is an element in B.

Some direct results from this definition:

1. B is a subset of the power set of Ω, hence the power set of Ω is a σ-algebra.
2. Ω is in B, since Ω is the complement of Φ.
3. The trivial σ-algebra would be the set {Φ,Ω}.
4. A countable intersection of elements in B is another element in B, since this intersection is equivalent to the complement of the countable union of the complements of those elements. (De Morgan's Law)

For an arbitrary collection F of subsets of Ω, there exists a unique smallest σ-algebra B that contains F. B is called the σ-algebra generated by F, or σ(F).

Proof:

First the proof of the following lemma.

Lemma: For a fixed Ω, any nonempty intersection of its σ-algebras is a σ-algebra.

Clearly the intersection is a subset of the power set of Ω, containing Φ. (Property 1 checked) If x is an element in the intersection, x is an element of all the σ-algebras, and so is x-complement. Hence x-complement is in the intersection. (Property 2 checked) Similarly, take a countable union of elements in the intersection, since all these elements are in the intersection, they are in every σ-algebras, and so is their union. Hence this union is in the intersection. (Property 3 checked). QED

Main Proposition:

Let A be the set of all σ-algebras of Ω containing F. Clearly A is not empty because the power set of Ω is in A. Then consider the intersection of all elements in A, call it G. G is apparently nonempty. Since every element of A contains F, G contains F. Since every element is a σ-algebra, by the lemma, G is a σ-algebra. And lastly, every element in A contains G, hence G is the smallest element in A. Or put it in another way, G is the smallest σ-algebra containing F, a.k.a G=σ(F). QED

This is the first post concerning some technical stuff, and I find it quite interesting. I will post more on Borel sigma-algebra the next time. Enjoy.