Monotone class theorem

Definition of a monotone class

A monotone class in a set R is a collection M of subsets of R which contains R and is closed under countable monotone unions and intersections, i.e. if A i ∈ M and A 1 ⊂ A 2 ⊂ … then ∪ i = 1 ∞ A i ∈ M , and similarly for intersections of decreasing sequences of sets.

Statement

Let G be an algebra of sets and define M(G) to be the smallest monotone class containing G. Then M(G) is precisely the σ-algebra generated by G, i.e. σ(G) = M(G)

Statement

Let A be a π-system that contains Ω and let H be a collection of functions from Ω to R with the following properties:

(1) If A ∈ A , then 1 A ∈ H

(2) If f , g ∈ H , then f + g and c f ∈ H for any real number c

(3) If f n ∈ H is a sequence of non-negative functions that increase to a bounded function f , then f ∈ H

Then H contains all bounded functions that are measurable with respect to σ ( A ) , the sigma-algebra generated by A

Proof

The following argument originates in Rick Durrett's Probability: Theory and Examples.

The assumption Ω ∈ A , (2) and (3) imply that G = { A : 1 A ∈ H } is a λ-system. By (1) and the π − λ theorem, σ ( A ) ⊂ G . (2) implies H contains all simple functions, and then (3) implies that H contains all bounded functions measurable with respect to σ ( A ) .

Results and Applications

As a corollary, if G is a ring of sets, then the smallest monotone class containing it coincides with the sigma-ring of G.

By invoking this theorem, one can use monotone classes to help verify that a certain collection of subsets is a sigma-algebra.

The monotone class theorem for functions can be a powerful tool that allows statements about particularly simple classes of functions to be generalized to arbitrary bounded and measurable functions.