Monotone class theorem

In measure theory and probability, the monotone class theorem connects monotone classes and sigma-algebras. The theorem says that the smallest monotone class containing an algebra of sets G is precisely the smallest σ-algebra containing G. It is used as a type of transfinite induction to prove many other theorems, such as Fubini's theorem.

Definition of a monotone class

A monotone class is a class M of sets that is closed under countable monotone unions and intersections, i.e. if and then , and similarly in the other direction.

Monotone class theorem for 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).

Monotone class theorem for functions

Statement

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

(1) If , then

(2) If , then and for any real number

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

Then contains all bounded functions that are measurable with respect to , the sigma-algebra generated by

Proof

The following argument originates in Rick Durrett's Probability: Theory and Examples. [1]

The assumption , (2) and (3) imply that is a λ-system. By (1) and the πλ theorem, . (2) implies contains all simple functions, and then (3) implies that contains all bounded functions measurable with respect to .

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.

References

  1. Durrett, Rick (2010). Probability: Theory and Examples (4th ed.). Cambridge University Press. p. 276. ISBN 978-0521765398.
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