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 = ∪iAi, and each Ai ∈ ℳ, then A ∊ ℳ;

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

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