Monotone class theorem

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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[edit]

A monotone class in a set is a collection of subsets of which contains and is closed under countable monotone unions and intersections, i.e. if and then , and similarly for intersections of decreasing sequences of sets.

Monotone class theorem for sets[edit]

Statement[edit]

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[edit]

Statement[edit]

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[edit]

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 π − λ theorem, . (2) implies contains all simple functions, and then (3) implies that contains all bounded functions measurable with respect to .

Results and Applications[edit]

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[edit]

  1. ^ Durrett, Rick (2010). Probability: Theory and Examples (4th ed.). Cambridge University Press. p. 100. ISBN 978-0521765398. 

See also[edit]

This article was advanced during a Wikipedia course held at Duke University, which can be found here: Wikipedia and Its Ancestors