B - Some results from analysis

Summary

The monotone class theorem

The monotone class theorem is a result from measure theory used in the proof of the Fubini theorem.

Definition B.1 ℳ is a monotone class if ℳ is a collection of subsets of X such that

1. (1) if A₁ ⊂ A₂ ⊂ …, A = ∪_iA_i, and each A_i ∈ ℳ, then A ∊ ℳ;

2. (2) if A₁ ⊃ A₂ ⊃ …, A = ∊ ℳ∩A_i, and each A_i ∈ ℳ, then A ∈ ℳ.

Recall that an algebra of sets is a collection A of sets such that if A₁,…, A_n ∈ A, then A₁ ∪ · ∪ A_n and A₁ ∩ · ∩ A_n are also in A, and if A ∈ A, then A_c ∈ A.

The intersection ofmonotone classes is a monotone class, and the intersection of all monotone classes containing a given collection of sets is the smallest monotone class containing that collection.

Theorem B.2Suppose A₀is an algebra of sets, A is the smallest σ-field containing A₀, and ℳ is the smallest monotone class containing A₀. Then ℳ = A.

Proof A σ-algebra is clearly a monotone class, so ℳ ⊂ A. We must show A ⊂ ℳ.

Let N₁ ={A ∈ ℳ : A_c ℳ}. Note N₁ is contained in ℳ, contains A₀, and is a monotone class. Since ℳ is the smallest monotone class containing A₀, then N = A, and therefore ℳ is closed under the operation of taking complements.